case for $Q^{a}=0$. In the latter case, the system of PDEs resulting from the
condition $\frac{dI}{dt}=0$ is integrated directly without the need of
additional assumptions.

From the application of Theorem 11.4.1 in the rather simple –but widely
studied– case of autonomous conservative dynamical systems with two degrees of
freedom, we have achieved the following goals:
a. We have shown that one is possible to obtain the known integrable
potentials, which have been computed using other methods but the direct
method, in a simple, direct, and concrete geometrical approach.
b. We have found a new superintegrable potential whose integrability is
established only by means of autonomous and time-dependent CFIs.

##  Part IV Integrability of time-dependent dynamical systems

###  Chapter 12 Quadratic first integrals of time-dependent dynamical systems
of the form
$\ddot{q}^{a}=-\Gamma^{a}_{bc}(q)\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$

####  12.1 Introduction

As we have seen in the previous chapters, the standard way to determine the
FIs of a differential equation is the use of Lie/Noether symmetries, which
applies to the point as well as to the generalized Lie/Noether symmetries. The
relation of the Lie/Noether symmetries with the symmetries of the kinetic
metric has been considered, mostly, in the case of point symmetries for
autonomous conservative dynamical systems moving in a Riemannian space. In
particular, it has been shown (see e.g. [11, 39, 146, 170]) that the Lie point
symmetries are generated by the special projective algebra of the kinetic
metric, whereas the Noether point symmetries are generated by the homothetic
algebra of the kinetic metric; the latter being a subalgebra of the projective
algebra111A recent clear statement of these results is discussed in [171]..

In addition to the autonomous conservative systems, this method has been
applied to the time-dependent potentials $W(t,q)=\omega(t)V(q)$, that is, for
dynamical equations of the form
$\ddot{q}^{a}=-\Gamma_{bc}^{a}(q)\dot{q}^{b}\dot{q}^{c}-\omega(t)V^{,a}(q)$
(see e.g. [40, 172, 173, 174, 175, 176, 177]). In this case, it has been shown
that the Lie point symmetries, the Noether point symmetries and the associated
FIs are computed in terms of the collineations of the kinetic metric, plus a
set of constraints involving the time-dependent potential and the collineation
vectors. These time-dependent potentials are important because (among others)
they contain the time-dependent oscillator (see e.g. [162, 169, 173, 175,
178]) and the time-dependent Kepler potential (see e.g. [12, 76, 177, 179]). A
further development in the same line is the extension of this method to time-
dependent potentials $W(t,q)$ with linear damping terms [177]. It has been
shown that under a suitable time transformation the damping term can be
removed and the problem reduces to a time-dependent potential of the form
$W(t,q)=\bar{\omega}(t)V(q)$, where $\bar{\omega}(t)$ is a ‘frequency’
different from the original. Finally, the method of the Lie/Noether symmetries
has been applied to the study of PDEs (see e.g. [11, 128, 180, 181]).

In this chapter, we shall use again the direct method (see section 3.5), which
has been employed in the literature (see e.g. [38, 62, 76, 182]) both for
autonomous (see Part III) and time-dependent dynamical systems, in order to
compute the QFIs of time-dependent dynamical systems of the form
$\ddot{q}^{a}=-\Gamma_{bc}^{a}(q)\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$.
Because many well-known dynamical systems fall in this category, we intend to
recover in a direct single approach all the known results, which are scattered
in a large number of papers and have been derived mainly from the method of
the Lie/Noether symmetries.

The application of the direct method implies that the symmetric tensor
$K_{ab}$ of the considered QFI (7.3) is a KT of the kinetic metric. In
general, the computation of the KTs of a metric is a major task. However, for
spaces of constant curvature this problem has been solved (see sections 2.6
and 2.7). Therefore, in this chapter, we restrict our discussion to Euclidean
spaces only. Since the KT $K_{ab}$ is a function of $t$ and $q^{a}$, we
suggest two procedures of work: a. The polynomial method, and b. The basis
method.

In the polynomial method, one assumes a general polynomial form in the
variable $t$ both for the KT $K_{ab}$ and the vector $K_{a}$, and replaces
these expressions in the system of PDEs resulting from the condition
$\frac{dI}{dt}=0$. On the other hand, in the basis method, one computes first
a basis of the KTs of order two of the kinetic metric and, then, expresses in
this basis the KT $K_{ab}$ with the coefficients to be functions of $t$. The
vector $K_{a}$ and the FIs follow from the solution of the system of PDEs,
which is the same for both methods. We note that _both methods are suitable
for autonomous dynamical systems, but for time-dependent systems it appears
that the basis method is preferable._

Concerning the quantities $\omega(t)$ and $Q^{a}(q)$, again, there are two
ways to proceed:
a) Consider a general form for the function $\omega(t)$ and let the quantities
$Q^{a}$ unspecified. In this case, the quantities $Q^{a}$ act as constraints.
b) Specify the quantities $Q^{a}$ and determine for which functions
$\omega(t)$ the resulting dynamical system admits QFIs.

In the following, we shall consider both the polynomial method and the basis
method, starting from the former. As a first application, we assume the KT
$K_{ab}=N(t)\gamma_{ab}$, where $N(t)$ is an arbitrary function, and show that
we recover all the point Noether FIs found in [177]. As a second application,
we assume that $\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell}$ with
$b_{\ell}\neq 0$ and $\ell\geq 1$, while the quantities $Q^{a}$ are
unspecified. We find that, in this case, the system admits two families of
independent QFIs as stated in Theorem 12.5.1.

Subsequently, we consider the basis method. This is carried out in two steps:
In the first step, we assume that we know a basis $\\{C_{(N)ab}(q)\\}$ of the
vector space of KTs of the kinetic metric, and require that $K_{ab}$ has the
form $K_{ab}(t,q)=\sum_{N=1}^{m}\alpha_{N}(t)C_{(N)ab}(q)$. In the second
step, we specify the generalized forces to be conservative with the time-
dependent Newtonian generalized Kepler potential
$V=-\frac{\omega(t)}{r^{\nu}}$, where $\nu$ is a non-zero real constant and
$r=\sqrt{x^{2}+y^{2}+z^{2}}$. This potential for $\nu=-2,1$ includes,
respectively, the 3d time-dependent oscillator and the time-dependent Kepler
potential. For other values of $\nu$, it reduces to other important dynamical
systems; for example, for $\nu=2$, one obtains the Newton-Cotes potential
[183]. We determine the QFIs of the time-dependent generalized Kepler
potential and recover in a systematic way the known results concerning the
QFIs of the 3d time-dependent oscillator, the time-dependent Kepler potential
and the Newton-Cotes potential.

Using the well-known result that by a reparameterization the linear damping
term $\phi(t)\dot{q}^{a}$ of a dynamical system is absorbed to a time-
dependent force of the form $\omega(t)Q^{a}(q)$, we also study the
integrability of the non-linear differential equation
$\ddot{x}=-\omega(t)x^{\mu}+\phi(t)\dot{x}$ $(\mu\neq-1)$. Specifically, we
compute the relation between the coefficients $\omega(t)$ and $\phi(t)$ for
which QFIs are admitted. It is found that a family of ‘frequencies’
$\bar{\omega}(s)$ is admitted, which for $\mu=0,1,2$ is parameterized with
functions, whereas for $\mu\neq-1,0,1,2$ is parameterized with constants. As a
further application, we study the integrability of the well-known generalized
Lane-Emden equation.

####  12.2 The system of PDEs

We consider the time-dependent dynamical system

$\ddot{q}^{a}=-\Gamma^{a}_{bc}(q)\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$
(12.1)

where $\Gamma^{a}_{bc}$ are the Riemannian connection coefficients determined
by the kinetic metric $\gamma_{ab}$ of the system and $-\omega(t)Q^{a}(q)$ are
the time-dependent generalized forces.

Next, we consider a function $I(t,q^{a},\dot{q}^{a})$ of the form

$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ (12.2)

where $K_{ab}$ is a symmetric tensor, $K_{a}$ is a vector and $K$ is an
invariant.

We demand $(\ref{FI.5})$ to be a QFI of (12.1) by imposing the condition
$\frac{dI}{dt}=0$ along trajectories of the system. Using the dynamical
equations (12.1) to replace $\ddot{q}^{a}$ whenever it appears, we find the
following system of PDEs:

$\displaystyle K_{(ab;c)}$ $\displaystyle=$ $\displaystyle 0$ (12.3)
$\displaystyle K_{ab,t}+K_{(a;b)}$ $\displaystyle=$ $\displaystyle 0$ (12.4)
$\displaystyle-2\omega K_{ab}Q^{b}+K_{a,t}+K_{,a}$ $\displaystyle=$
$\displaystyle 0$ (12.5) $\displaystyle K_{,t}-\omega K_{a}Q^{a}$
$\displaystyle=$ $\displaystyle 0$ (12.6) $\displaystyle
K_{a,tt}+\omega\left(K_{b}Q^{b}\right)_{,a}-2\omega_{,t}K_{ab}Q^{b}-2\omega
K_{ab,t}Q^{b}$ $\displaystyle=$ $\displaystyle 0$ (12.7) $\displaystyle
K_{[a;b],t}-2\omega\left(K_{[a|c|}Q^{c}\right)_{;b]}$ $\displaystyle=$
$\displaystyle 0$ (12.8)

where equations (12.7) and (12.8) express the integrability conditions
$K_{,[at]}=0$ and $K_{,[ab]}=0$, respectively, for the scalar $K$.

Equation (12.3) implies that $K_{ab}$ is a KT of order two (possibly zero) of
the kinetic metric $\gamma_{ab}$.

The solution of the system (12.3) - (12.8) requires the function $\omega(t)$
and the quantities $Q^{a}(q)$; both being quantities which are characteristic
of the given dynamical system. There are two ways to proceed:
a) Consider a general form for the function $\omega(t)$ and let the quantities
$Q^{a}(q)$ unspecified. In this case, the quantities $Q^{a}(q)$ act as
constraints.
b) Specify the quantities $Q^{a}(q)$ and determine for which functions
$\omega(t)$ the resulting dynamical system admits QFIs.

However, before continuing with this kind of considerations, we first proceed
with the simple geometric choice $K_{ab}=N(t)\gamma_{ab}$, where $N(t)$ is an
arbitrary smooth function. By specifying the KT $K_{ab}$ like this, both the
function $\omega(t)$ and the quantities $Q^{a}(q)$ stay unspecified and can
act as constraints.

####  12.3 The point Noether FIs of the time-dependent dynamical system (12.1)

We consider the simplest choice

$K_{ab}=N(t)\gamma_{ab}$ (12.9)

where $N(t)$ is an arbitrary smooth function. This choice is purely geometric;
therefore, the function $\omega(t)$ and the quantities $Q^{a}(q)$ are
unspecified and act as constraints, whereas the vector $K_{a}$ is identified
with one collineation of the kinetic metric. With this $K_{ab}$, the system of
PDEs (12.3) - (12.8) becomes (eq. (12.3) vanishes trivially):

$\displaystyle N_{,t}\gamma_{ab}+K_{(a;b)}$ $\displaystyle=$ $\displaystyle 0$
(12.10) $\displaystyle-2\omega NQ_{a}+K_{a,t}+K_{,a}$ $\displaystyle=$
$\displaystyle 0$ (12.11) $\displaystyle K_{,t}-\omega K_{a}Q^{a}$
$\displaystyle=$ $\displaystyle 0$ (12.12) $\displaystyle
K_{a,tt}+\omega\left(K_{b}Q^{b}\right)_{,a}-2\omega_{,t}NQ_{a}-2\omega
N_{,t}Q_{a}$ $\displaystyle=$ $\displaystyle 0$ (12.13) $\displaystyle
K_{[a;b],t}-2\omega NQ_{[a;b]}$ $\displaystyle=$ $\displaystyle 0.$ (12.14)

We consider the following cases.

#####  12.3.1 Case $K_{a}=K_{a}(q)$ is the HV of $\gamma_{ab}$ with homothetic
factor $\psi$

In this case, $K_{a,t}=0$ and $K_{(a;b)}=\psi\gamma_{ab}$, where $\psi$ is an
arbitrary constant.

Equation (12.10) gives $N_{,t}=-\psi\implies N=-\psi t+c$, where $c$ is an
arbitrary constant.

Equation (12.14) implies that (take $\omega\neq 0$) $Q_{[a;b]}=0\implies
Q_{a}=V_{,a}$, where $V=V(q)$ is an arbitrary potential.

Replacing in (12.11), we find that

$K_{,a}=2\omega(-\psi t+c)V_{,a}\implies K=2\omega(-\psi t+c)V+M(t)$

where $M(t)$ is an arbitrary function.

Substituting the function $K(t,q)$ in (12.12), we get

$\omega K_{a}V^{,a}-2\omega_{,t}(-\psi t+c)V+2\omega\psi V-M_{,t}=0.$ (12.15)

The remaining condition (12.13) is just the partial derivative of (12.15) and,
hence, is satisfied trivially.

Moreover, since $\omega\neq 0$, equation (12.15) can be written in the form

$K_{a}V^{,a}-2(\ln\omega)_{,t}(-\psi t+c)V+2\psi V-\frac{M_{,t}}{\omega}=0$
(12.16)

which implies the conditions:

$\displaystyle 2(\ln\omega)_{,t}(-\psi t+c)$ $\displaystyle=$ $\displaystyle
c_{1}$ (12.17) $\displaystyle M_{,t}$ $\displaystyle=$ $\displaystyle
c_{2}\omega$ (12.18)

where $c_{1}$ and $c_{2}$ are arbitrary constants.

Therefore, equation (12.16) becomes

$K_{a}V^{,a}+(2\psi-c_{1})V-c_{2}=0.$ (12.19)

The QFI is

$I_{1}=(-\psi
t+c)\gamma_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}(q)\dot{q}^{a}+2\omega(-\psi
t+c)V+M(t)$ (12.20)

where $Q_{a}=V_{,a}$ and the quantities $\omega(t),M(t),V(q),K_{a}(q)$ satisfy
the conditions (12.17) - (12.19).

#####  12.3.2 Case $K_{a}=-M(t)S_{,a}(q)$ where $S_{,a}$ is the gradient HV of
$\gamma_{ab}$

In this case, $S_{;ab}=\psi\gamma_{ab}$ and $M(t)\neq 0$ is an arbitrary
function.

Equation (12.10) implies $N_{,t}=\psi M$.

From equation (12.14), we find that there exists a potential function $V(q)$
such that $Q_{a}=V_{,a}$.

Replacing the above results in (12.11), we obtain

$K_{,a}=2\omega NV_{,a}+M_{,t}S_{,a}\implies K=2\omega NV+M_{,t}S+C(t)$

where $C(t)$ is an arbitrary function.

Substituting in (12.12), we get (take $\omega M\neq 0$)

$\omega MS_{,a}V^{,a}+2\omega_{,t}NV+2\omega\psi MV+M_{,tt}S+C_{,t}=0\implies$
$S_{,a}V^{,a}+2\psi V+\frac{2(\ln\omega)_{,t}N}{M}V+\frac{M_{,tt}}{\omega
M}S+\frac{C_{,t}}{\omega M}=0$

which implies that:

$\displaystyle\frac{2(\ln\omega)_{,t}N}{M}$ $\displaystyle=$ $\displaystyle
d_{1}$ (12.21) $\displaystyle\frac{M_{,tt}}{\omega M}$ $\displaystyle=$
$\displaystyle m$ (12.22) $\displaystyle\frac{C_{,t}}{\omega M}$
$\displaystyle=$ $\displaystyle k$ (12.23) $\displaystyle
S_{,a}V^{,a}+(2\psi+d_{1})V+mS+k$ $\displaystyle=$ $\displaystyle 0$ (12.24)

where $d_{1},m,k$ are arbitrary constants. The remaining condition (12.13) is
satisfied identically.

The QFI is

$I_{2}=N\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-MS_{,a}\dot{q}^{a}+2\omega
NV+M_{,t}S+C(t)$ (12.25)

where $Q_{a}=V_{,a}$, $N_{,t}=\psi M$ and the conditions (12.21) - (12.24)
must be satisfied.

#####  12.3.3 Case $Q_{a}=V_{,a}$ and $K_{a}=-M(t)V_{,a}(q)$ where $V_{,a}$ is
the gradient HV of $\gamma_{ab}$

Equation (12.10) implies $N_{,t}=\psi M$, where $\psi$ is the homothetic
factor of $V_{,a}$.

From equation (12.11), we obtain

$K_{,a}=2\omega NV_{,a}+M_{,t}V_{,a}\implies K=2\omega NV+M_{,t}V+C(t)$

where $C(t)$ is an arbitrary function.

Substituting in (12.12), we get (take $\omega M\neq 0$)

$\omega MV_{,a}V^{,a}+2\omega_{,t}NV+2\omega\psi MV+M_{,tt}V+C_{,t}=0\implies$
$V_{,a}V^{,a}+2\psi V+\frac{2(\ln\omega)_{,t}N}{M}V+\frac{M_{,tt}}{\omega
M}V+\frac{C_{,t}}{\omega M}=0$

which implies that:

$\displaystyle\frac{M_{,tt}}{\omega M}+\frac{2(\ln\omega)_{,t}N}{M}$
$\displaystyle=$ $\displaystyle d_{2}$ (12.26)
$\displaystyle\frac{C_{,t}}{\omega M}$ $\displaystyle=$ $\displaystyle k$
(12.27) $\displaystyle V_{,a}V^{,a}+(2\psi+d_{2})V+k$ $\displaystyle=$
$\displaystyle 0$ (12.28)

where $d_{2}$ and $k$ are arbitrary constants. The remaining conditions are
satisfied identically.

The QFI is

$I_{3}=N\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-MV_{,a}\dot{q}^{a}+\left(2\omega
N+M_{,t}\right)V+C$ (12.29)

where $Q_{a}=V_{,a}$, $N_{,t}=\psi M$ and the conditions (12.26) - (12.28)
must be satisfied.

The above results reproduce Theorem 2 of [177] which states that _the point
Noether symmetries of the time-dependent potentials of the form
$\omega(t)V(q)$ are generated by the homothetic algebra of the kinetic metric
(provided the Lagrangian is regular)_.

It is interesting to observe that _the QFIs ( 12.20), (12.25) and (12.29)
produced by point Noether symmetries can be also produced by generalized
(gauged) Noether symmetries using the Inverse Noether Theorem 3.4.1 (see last
comments in section 7.2.2). This proves that a Noether FI is not associated
with a unique Noether symmetry._

####  12.4 The polynomial method for computing the QFIs

In the polynomial method, one assumes a polynomial form in $t$ for the KT
$K_{ab}(t,q)$ and the vector $K_{a}(t,q)$ and, then, solves the resulting
system for given $\omega(t)$ and $Q^{a}(q)$. For example, this method is
applied in chapter 9, where a general theorem is given (see Theorem 9.4.1)
which allows the finding of the QFIs of an autonomous holonomic dynamical
system with a linear damping term. In the present chapter, we follow the
assumptions made in section 9.3, and assume that the KT $K_{ab}(t,q)$ and the
vector $K_{a}(t,q)$ are given by equations (9.18) and (9.19), respectively.

Substituting (9.18) and (9.19) in the system of PDEs (12.3) - (12.8) (eq.
(12.3) is identically zero since $C_{(N)ab}$ are KTs), we obtain the system of
equations:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
C_{(1)ab}+C_{(2)ab}t+...+C_{(n)ab}t^{n-1}+L_{(0)(a;b)}+L_{(1)(a;b)}t+...+L_{(m)(a;b)}t^{m}$
(12.30) $\displaystyle 0$ $\displaystyle=$ $\displaystyle-2\omega
C_{(0)ab}Q^{b}-2\omega C_{(1)ab}Q^{b}t-...-2\omega
C_{(n)ab}Q^{b}\frac{t^{n}}{n}+L_{(1)a}+2L_{(2)a}t+...+$ (12.31)
$\displaystyle+mL_{(m)a}t^{m-1}+K_{,a}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle K_{,t}-\omega L_{(0)a}Q^{a}-\omega L_{(1)a}Q^{a}t-...-\omega
L_{(m)a}Q^{a}t^{m}$ (12.32) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(-2C_{(0)ab}Q^{b}-2C_{(1)ab}Q^{b}t-...-2C_{(n)ab}Q^{b}\frac{t^{n}}{n}\right)\omega_{,t}-2\omega
C_{(1)ab}Q^{b}-2\omega C_{(2)ab}Q^{b}t-...-$ (12.33) $\displaystyle-2\omega
C_{(n)ab}Q^{b}t^{n-1}+2L_{(2)a}+6L_{(3)a}t+...+m(m-1)L_{(m)a}t^{m-2}+\omega\left(L_{(0)b}Q^{b}\right)_{,a}+$
$\displaystyle+\omega\left(L_{(1)b}Q^{b}\right)_{,a}t+...+\omega\left(L_{(m)b}Q^{b}\right)_{,a}t^{m}$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\omega\left(C_{(0)[a\left|c\right|}Q^{c}\right)_{;b]}+2\omega\left(C_{(1)[a\left|c\right|}Q^{c}\right)_{;b]}t+...+2\omega\left(C_{(n)[a\left|c\right|}Q^{c}\right)_{;b]}\frac{t^{n}}{n}-L_{(1)\left[a;b\right]}-$
(12.34)
$\displaystyle-2L_{(2)\left[a;b\right]}t-...-mL_{(m)\left[a;b\right]}t^{m-1}.$

In this system of PDEs, the pairs $\omega(t),Q^{a}(q)$ are not specified. As
we explained in section 12.1, we shall fix a general form of $\omega$ and find
the admitted QFIs in terms of the (unspecified) $Q^{a}$. In the following
section, we choose $\omega(t)$ to be a general polynomial in $t$; however, any
other choice is possible.

####  12.5 The case $\mathbf{\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell}}$
with $\mathbf{b_{\ell}\neq 0}$ and $\mathbf{\ell\geq 1}$

We assume that

$\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell},\enskip b_{\ell}\neq
0,\enskip\ell\geq 1$ (12.35)

where $\ell$ is the degree of the polynomial. Substituting the function
(12.35) in the system of equations (12.30) - (12.34), we find222The proof of
Theorem 12.5.1 is given in appendix F. that there are two independent QFIs as
given in Theorem 12.5.1.

###### Theorem 12.5.1

The independent QFIs of the time-dependent dynamical system (12.1), where
$\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell}$ with $b_{\ell}\neq 0$ and
$\ell\geq 1$, are the following:

Integral 1.

$I_{n}=\left(C_{(0)ab}+\sum^{n}_{k=1}\frac{t^{k}}{k}C_{(k)ab}\right)\dot{q}^{a}\dot{q}^{b}+\sum^{n}_{k=0}t^{k}L_{(k)a}\dot{q}^{a}+\sum^{n}_{k=0}\sum^{\ell}_{r=0}\left(L_{(k)a}Q^{a}b_{r}\frac{t^{k+r+1}}{k+r+1}\right)+G(q)$

where $n=0,1,2,...$, $C_{(0)ab}$ is a KT, the KTs $C_{(N)ab}=-L_{(N-1)(a;b)}$
for $N=1,...,n$, $L_{(n)a}$ is a KV, $G(q)$ is an arbitrary function defined
by the condition

$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$ (12.36)

$c$ is an arbitrary constant defined by the condition

$L_{(n)a}Q^{a}=c$ (12.37)

and the following conditions are satisfied:

$\displaystyle 0$
$\displaystyle=\sum_{s=0}^{\ell-1}\left[-\frac{2(r+s)b_{(r+s\leq\ell)}}{n-s}C_{(n-s\geq
0)ab}Q^{b}-2b_{(r+s\leq\ell)}C_{(n-s>0)ab}Q^{b}+\right.$
$\displaystyle\quad\left.+b_{(r+s\leq\ell)}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right],\enskip r=1,2,...,\ell$ (12.38) $\displaystyle 0$
$\displaystyle=-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{n-s}C_{(n-s\geq
0)ab}Q^{b}\right]+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(n-s>0)ab}Q^{b}+b_{s}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right]$ (12.39) $\displaystyle 0$
$\displaystyle=k(k-1)L_{(k)a}-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{k-s-1}C_{(k-s-1\geq
0)ab}Q^{b}\right]+$
$\displaystyle\quad+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(k-s-1>0)ab}Q^{b}+b_{s}\left(L_{(k-s-2\geq
0)b}Q^{b}\right)_{,a}\right],\enskip k=2,3,...n.$ (12.40)

Integral 2.

$I_{e}=I_{e}(\ell=1)=-e^{\lambda t}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+\lambda
e^{\lambda
t}L_{a}\dot{q}^{a}+\left(b_{0}-\frac{b_{1}}{\lambda}\right)e^{\lambda
t}L_{a}Q^{a}+b_{1}te^{\lambda t}L_{a}Q^{a}$

where $L_{(a;b)}$ is a KT,
$\left(L_{b}Q^{b}\right)_{,a}=\frac{\lambda^{3}}{b_{1}}L_{a}$ and
$\lambda^{3}L_{a}=-2b_{1}L_{(a;b)}Q^{b}$.

We note that the FI $I_{e}$ exists only when $\omega(t)=b_{0}+b_{1}t$, that
is, for $\ell=1$.

####  12.6 Special cases of the QFI $I_{n}$

The parameter $n$ in the case Integral 1 of Theorem 12.5.1 runs over all
positive integers, i.e. $n=0,1,2,...$. This results in a sequence of QFIs
$I_{0},I_{1},I_{2},...$, that is, one QFI $I_{n}$ for each value $n$. A
significant characteristic of this sequence is that $I_{k}<I_{k+1}$, that is,
each QFI $I_{k}$, where $k=0,1,2,...$, can be derived from the next QFI
$I_{k+1}$ as a subcase.

In the following, we consider some special cases of the QFI $I_{n}$ for small
values of $n$.

#####  12.6.1 The QFI $I_{0}$

For $n=0$, the QFI is

$I_{0}=C_{(0)ab}\dot{q}^{a}\dot{q}^{b}+L_{(0)a}\dot{q}^{a}+b_{\ell}s\frac{t^{\ell+1}}{\ell+1}+...+b_{1}s\frac{t^{2}}{2}+b_{0}st$

where $C_{(0)ab}$ is a KT, $L_{(0)a}$ is a KV, $L_{(0)a}Q^{a}=s$ and
$C_{(0)ab}Q^{b}=0$.

This QFI consists of the independent FIs:

$I_{0a}=C_{(0)ab}\dot{q}^{a}\dot{q}^{b}\enskip\text{and}\enskip
I_{0b}=L_{(0)a}\dot{q}^{a}+b_{\ell}s\frac{t^{\ell+1}}{\ell+1}+...+b_{1}s\frac{t^{2}}{2}+b_{0}st.$

#####  12.6.2 The QFI $I_{1}$

For $n=1$, the conditions (12.37) - (12.40) become:

$\displaystyle L_{(1)a}Q^{a}$ $\displaystyle=$ $\displaystyle s$ (12.41)
$\displaystyle\left(L_{(0)b}Q^{b}\right)_{,a}$ $\displaystyle=$
$\displaystyle-2(\ell+1)L_{(0)(a;b)}Q^{b}$ (12.42) $\displaystyle
kb_{k}C_{(0)ab}Q^{b}$ $\displaystyle=$
$\displaystyle-(\ell-k+1)b_{k-1}L_{(0)(a;b)}Q^{b},\enskip k=1,...,\ell.$
(12.43)

Since $b_{\ell}\neq 0$, the last condition for $k=\ell$ gives
$C_{(0)ab}Q^{b}=$ $-\frac{b_{\ell-1}}{\ell b_{\ell}}L_{(0)(a;b)}Q^{b}$ and the
remaining equations become

$\left[(\ell-k+1)b_{k-1}-\frac{kb_{k}b_{\ell-1}}{\ell
b_{\ell}}\right]L_{(0)(a;b)}Q^{b}=0,\enskip k=1,...,\ell-1.$

The last set of equations exist only for $\ell\geq 2$. By using mathematical
induction and after successive substitutions, we find

$\left(b_{0}-\frac{b^{\ell}_{\ell-1}}{\ell^{\ell}b^{\ell-1}_{\ell}}\right)L_{(0)(a;b)}Q^{b}=0.$

The QFI is ($I_{0}<I_{1}$)

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left(-tL_{(0)(a;b)}+C_{(0)ab}\right)\dot{q}^{a}\dot{q}^{b}+tL_{(1)a}\dot{q}^{a}+L_{(0)a}\dot{q}^{a}+sb_{\ell}\frac{t^{\ell+2}}{\ell+2}+\left(sb_{\ell-1}+b_{\ell}L_{(0)a}Q^{a}\right)\frac{t^{\ell+1}}{\ell+1}+...+$
$\displaystyle+\left(sb_{0}+b_{1}L_{(0)a}Q^{a}\right)\frac{t^{2}}{2}+b_{0}L_{(0)a}Q^{a}t+G(q)$

where $C_{(0)ab}$ and $L_{(0)(a;b)}$ are KTs, $L_{(1)a}$ is a KV,
$L_{(1)a}Q^{a}=s$,
$\left(L_{(0)b}Q^{b}\right)_{,a}=-2(\ell+1)L_{(0)(a;b)}Q^{b}$,
$C_{(0)ab}Q^{b}=$ $-\frac{b_{\ell-1}}{\ell b_{\ell}}L_{(0)(a;b)}Q^{b}$,
$\left[(\ell-k+1)b_{k-1}-\frac{kb_{k}b_{\ell-1}}{\ell
b_{\ell}}\right]L_{(0)(a;b)}Q^{b}=0$ with $k=1,...,\ell-1$, and
$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$.

For some values of the degree $\ell$ of the polynomial $\omega(t)$, we have
the following:

1) For $\ell=1$.

We have $\omega=b_{0}+b_{1}t$ and the QFI is

$I_{1}=\left(-tL_{(0)(a;b)}+C_{(0)ab}\right)\dot{q}^{a}\dot{q}^{b}+tL_{(1)a}\dot{q}^{a}+L_{(0)a}\dot{q}^{a}+sb_{1}\frac{t^{3}}{3}+\left(sb_{0}+b_{1}L_{(0)a}Q^{a}\right)\frac{t^{2}}{2}+b_{0}L_{(0)a}Q^{a}t+G(q)$

where $C_{(0)ab}$ and $L_{(0)(a;b)}$ are KTs, $L_{(1)a}$ is a KV,
$L_{(1)a}Q^{a}=s$, $\left(L_{(0)b}Q^{b}\right)_{,a}=-4L_{(0)(a;b)}Q^{b}$,
$C_{(0)ab}Q^{b}=-\frac{b_{0}}{b_{1}}L_{(0)(a;b)}Q^{b}$, and
$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$.

2) For $\ell=2$.

We have $\omega=b_{0}+b_{1}t+b_{2}t^{2}$ and the QFI is

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left(-tL_{(0)(a;b)}+C_{(0)ab}\right)\dot{q}^{a}\dot{q}^{b}+tL_{(1)a}\dot{q}^{a}+L_{(0)a}\dot{q}^{a}+sb_{2}\frac{t^{4}}{4}+\left(sb_{1}+b_{2}L_{(0)a}Q^{a}\right)\frac{t^{3}}{3}+$
$\displaystyle+\left(sb_{0}+b_{1}L_{(0)a}Q^{a}\right)\frac{t^{2}}{2}+b_{0}L_{(0)a}Q^{a}t+G(q)$

where $C_{(0)ab}$ and $L_{(0)(a;b)}$ are KTs, $L_{(1)a}$ is a KV,
$L_{(1)a}Q^{a}=s$, $\left(L_{(0)b}Q^{b}\right)_{,a}=-6L_{(0)(a;b)}Q^{b}$,
$C_{(0)ab}Q^{b}=-\frac{b_{1}}{2b_{2}}L_{(0)(a;b)}Q^{b}$,
$\left(b_{0}-\frac{b_{1}^{2}}{4b_{2}}\right)L_{(0)(a;b)}Q^{b}=0$, and
$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$.

3) For $\ell=3$.

We have $\omega=b_{0}+b_{1}t+b_{2}t^{2}+b_{3}t^{3}$ and the QFI is

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left(-tL_{(0)(a;b)}+C_{(0)ab}\right)\dot{q}^{a}\dot{q}^{b}+tL_{(1)a}\dot{q}^{a}+L_{(0)a}\dot{q}^{a}+sb_{3}\frac{t^{5}}{5}+\left(sb_{2}+b_{3}L_{(0)a}Q^{a}\right)\frac{t^{4}}{4}+$
$\displaystyle+\left(sb_{1}+b_{2}L_{(0)a}Q^{a}\right)\frac{t^{3}}{3}+\left(sb_{0}+b_{1}L_{(0)a}Q^{a}\right)\frac{t^{2}}{2}+b_{0}L_{(0)a}Q^{a}t+G(q)$

where $C_{(0)ab}$ and $L_{(0)(a;b)}$ are KTs, $L_{(1)a}$ is a KV,
$L_{(1)a}Q^{a}=s$, $\left(L_{(0)b}Q^{b}\right)_{,a}=-8L_{(0)(a;b)}Q^{b}$,
$C_{(0)ab}Q^{b}=$ $-\frac{b_{2}}{3b_{3}}L_{(0)(a;b)}Q^{b}$,
$\left(b_{0}-\frac{b_{1}b_{2}}{9b_{3}}\right)L_{(0)(a;b)}Q^{b}=0$,
$\left(b_{1}-\frac{b_{2}^{2}}{3b_{3}}\right)L_{(0)(a;b)}Q^{b}=0$, and
$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$.

####  12.7 The basis method for computing QFIs

As it has been explained in section 12.1, in the basis method, instead of
considering the KT $K_{ab}$ to be given as a polynomial in $t$ with
coefficients arbitrary KTs (see eq. (9.18) ), one defines the KT $K_{ab}(t,q)$
by the requirement

$K_{ab}(t,q)=\sum_{N=1}^{m}\alpha_{N}(t)C_{(N)ab}(q)$ (12.44)

where $\alpha_{N}(t)$ are arbitrary smooth functions and the $m$ linearly
independent KTs $C_{(N)ab}(q)$ constitute a basis of the vector space of KTs
of the kinetic metric $\gamma_{ab}(q)$. In this case, one does not assume a
form for the vector $K_{a}(t,q)$, which is determined from the resulting
system of equations (12.3) - (12.8).

The basis method has been used previously by Katzin and Levine in [76], in
order to determine the QFIs for the time-dependent Kepler potential. As we
shall apply the basis method to 3d Newtonian systems, we need a basis of KTs
(and other collineations) of the Euclidean space $E^{3}$. This basis and, in
general, the geometric quantities of $E^{3}$ have been already computed in
section 2.9.

####  12.8 The time-dependent Newtonian generalized Kepler potential

The time-dependent Newtonian generalized Kepler potential is
$V=-\frac{\omega(t)}{r^{\nu}}$, where $\nu$ is a non-zero real constant and
$r=(x^{2}+y^{2}+z^{2})^{\frac{1}{2}}$. This potential contains (among others)
the 3d time-dependent oscillator [162, 169, 173, 175, 178] for $\nu=-2$, the
time-dependent Kepler potential [12, 177, 179, 76] for $\nu=1$, and the
Newton-Cotes potential for $\nu=2$ [183]. The integrability of these systems
has been studied in numerous works over the years using various methods;
mainly the Noether symmetries. Our purpose is to recover the results of these
works –and also new ones– using the basis method.

The Lagrangian of the system is

$L=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})+\frac{\omega(t)}{r^{\nu}}$
(12.45)

and the corresponding E-L equations are:

$\ddot{x}=-\frac{\nu\omega(t)}{r^{\nu+2}}x,\enskip\ddot{y}=-\frac{\nu\omega(t)}{r^{\nu+2}}y,\enskip\ddot{z}=-\frac{\nu\omega(t)}{r^{\nu+2}}z.$
(12.46)

For this dynamical system, the $Q^{a}=\frac{\nu q^{a}}{r^{\nu+2}}$ where
$q^{a}=(x,y,z)$, whereas the $\omega(t)$ is unspecified. We shall determine
the ‘frequencies’ $\omega(t)$ for which the resulting LFIs/QFIs are not
combinations of the angular momentum.

The LFIs/QFIs of the autonomous generalized Kepler potential, that is,
$\omega(t)=k=const$, have been determined in chapter 7 using the direct method
and are listed in Table LABEL:T1 (see also Table LABEL:Table.QFIs.Kepler.gen).

In Table LABEL:T1, $H_{\nu}$ is the Hamiltonian of the system, $L_{i}$ are the
components of the angular momentum, $R_{i}$ are the components of the Runge-
Lenz vector, and $B_{ij}$ are the components of the Jauch-Hill-Fradkin tensor.

Table 12.1:  The LFIs/QFIs of the autonomous generalized Kepler potential for $\omega(t)=k=const$. $V=-\frac{k}{r^{\nu}}$ | LFIs and QFIs
---|---
$\forall$ $\nu$ |  $L_{1}=y\dot{z}-z\dot{y}$, $L_{2}=z\dot{x}-x\dot{z}$, $L_{3}=x\dot{y}-y\dot{x}$, $H_{\nu}=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-\frac{k}{r^{\nu}}$
$\nu=-2$ | $B_{ij}=\dot{q}_{i}\dot{q}_{j}-2kq_{i}q_{j}$
$\nu=-2$, $k>0$ | $I_{3a\pm}=e^{\pm\sqrt{2k}t}(\dot{q}_{a}\mp\sqrt{2k}q_{a})$
$\nu=-2$, $k<0$ | $I_{3a\pm}=e^{\pm i\sqrt{-2k}t}(\dot{q}_{a}\mp i\sqrt{-2k}q_{a})$
$\nu=1$ | $R_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r}q_{i}$
$\nu=2$ |  $I_{1}=-H_{2}t^{2}+t(\dot{q}^{i}q_{i})-\frac{r^{2}}{2}$, $I_{2}=-H_{2}t+\frac{1}{2}(\dot{q}^{i}q_{i})$

Replacing $Q^{a}=\frac{\nu q^{a}}{r^{\nu+2}}$ in the conditions (12.3) -
(12.8), we obtain the following system of PDEs [76]:

$\displaystyle K_{(ab;c)}$ $\displaystyle=$ $\displaystyle 0$ (12.47)
$\displaystyle K_{(a;b)}+K_{ab,t}$ $\displaystyle=$ $\displaystyle 0$ (12.48)
$\displaystyle K_{,a}-\frac{2\nu\omega}{r^{\nu+2}}K_{ab}q^{b}+K_{a,t}$
$\displaystyle=$ $\displaystyle 0$ (12.49) $\displaystyle
K_{,t}-\frac{\nu\omega}{r^{\nu+2}}K_{a}q^{a}$ $\displaystyle=$ $\displaystyle
0$ (12.50) $\displaystyle
K_{a,tt}+\nu\omega\left(\frac{K_{b}q^{b}}{r^{\nu+2}}\right)_{,a}-\frac{2\nu\omega_{,t}}{r^{\nu+2}}K_{ab}q^{b}-\frac{2\nu\omega}{r^{\nu+2}}K_{ab,t}q^{b}$
$\displaystyle=$ $\displaystyle 0$ (12.51) $\displaystyle
K_{[a;b],t}-2\nu\omega\left(\frac{K_{[a|c|}q^{c}}{r^{\nu+2}}\right)_{;b]}$
$\displaystyle=$ $\displaystyle 0.$ (12.52)

From the Lagrangian (12.45), we infer that the kinetic metric is
$\delta_{ij}=diag(1,1,1)$.

According to the basis approach, the KT $K_{ab}(t,q)$ of (12.47) is given by
(2.50) provided that the twenty arbitrary constants $a_{I}$ are assumed to be
time-dependent functions $a_{I}(t)$.

Condition (12.48) gives $K_{a,b}+K_{b,a}=-2K_{ab,t}$ which implies:

$\displaystyle K_{1,1}$ $\displaystyle=$ $\displaystyle-K_{11,t}$ (12.53)
$\displaystyle K_{2,2}$ $\displaystyle=$ $\displaystyle-K_{22,t}$ (12.54)
$\displaystyle K_{3,3}$ $\displaystyle=$ $\displaystyle-K_{33,t}$ (12.55)
$\displaystyle K_{1,2}+K_{2,1}$ $\displaystyle=$ $\displaystyle-2K_{12,t}$
(12.56) $\displaystyle K_{1,3}+K_{3,1}$ $\displaystyle=$
$\displaystyle-2K_{13,t}$ (12.57) $\displaystyle K_{2,3}+K_{3,2}$
$\displaystyle=$ $\displaystyle-2K_{23,t}.$ (12.58)

From the first three conditions (12.53) - (12.55), we find:

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle-\frac{\dot{a}_{6}}{2}xy^{2}-\frac{\dot{a}_{1}}{2}xz^{2}-\dot{a}_{4}xyz-\dot{a}_{5}xy-\dot{a}_{2}xz-\dot{a}_{3}x+A(y,z,t)$
$\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle-\frac{\dot{a}_{6}}{2}yx^{2}-\frac{\dot{a}_{7}}{2}yz^{2}-\dot{a}_{14}xyz-\dot{a}_{15}xy-\dot{a}_{12}yz-\dot{a}_{13}y+B(x,z,t)$
$\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle-\frac{\dot{a}_{1}}{2}zx^{2}-\frac{\dot{a}_{7}}{2}zy^{2}-\dot{a}_{10}xyz-\dot{a}_{11}xz-\dot{a}_{8}yz-\dot{a}_{9}z+C(x,y,t)$

where $A,B,C$ are arbitrary functions.

Substituting these results in (12.56) - (12.58), we obtain:

$\displaystyle 0$
$\displaystyle=\dot{a}_{10}z^{2}-3\dot{a}_{6}xy-2\dot{a}_{4}xz-2\dot{a}_{14}yz-2\dot{a}_{5}x-2\dot{a}_{15}y+2\dot{a}_{16}z+2\dot{a}_{17}+A_{,2}+B_{,1}$
(12.59) $\displaystyle 0$
$\displaystyle=\dot{a}_{14}y^{2}-2\dot{a}_{4}xy-3\dot{a}_{1}xz-2\dot{a}_{10}yz-2\dot{a}_{2}x+2\dot{a}_{18}y-2\dot{a}_{11}z+2\dot{a}_{19}+A_{,3}+C_{,1}$
(12.60) $\displaystyle 0$
$\displaystyle=\dot{a}_{4}x^{2}-2\dot{a}_{14}xy-2\dot{a}_{10}xz-3\dot{a}_{7}yz-2(\dot{a}_{16}+\dot{a}_{18})x-2\dot{a}_{12}y-2\dot{a}_{8}z+2\dot{a}_{20}+$
$\displaystyle\quad+B_{,3}+C_{,2}.$ (12.61)

By taking the second partial derivatives of (12.59) wrt $x,y$, of (12.60) wrt
$x,z$, and of (12.61) wrt $y,z$, we find that: $a_{1}=c_{1}$, $a_{6}=c_{2}$
and $a_{7}=c_{3}$ are arbitrary constants.

Then, equations (12.59) - (12.61) become:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{10}z^{2}-2\dot{a}_{4}xz-2\dot{a}_{14}yz-2\dot{a}_{5}x-2\dot{a}_{15}y+2\dot{a}_{16}z+2\dot{a}_{17}+A_{,2}+B_{,1}$
(12.62) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{14}y^{2}-2\dot{a}_{4}xy-2\dot{a}_{10}yz-2\dot{a}_{2}x+2\dot{a}_{18}y-2\dot{a}_{11}z+2\dot{a}_{19}+A_{,3}+C_{,1}$
(12.63) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{4}x^{2}-2\dot{a}_{14}xy-2\dot{a}_{10}xz-2(\dot{a}_{16}+\dot{a}_{18})x-2\dot{a}_{12}y-2\dot{a}_{8}z+2\dot{a}_{20}+B_{,3}+C_{,2}.$
(12.64)

By suitable differentiations of the above equations, we obtain:
$A_{,22}=2\dot{a}_{14}z+2\dot{a}_{15}$,
$A_{,33}=2\dot{a}_{10}y+2\dot{a}_{11}$, $B_{,11}=2\dot{a}_{4}z+2\dot{a}_{5}$,
$B_{,33}=2\dot{a}_{10}x+2\dot{a}_{8}$, $C_{,11}=2\dot{a}_{4}y+2\dot{a}_{2}$
and $C_{,22}=2\dot{a}_{14}x+2\dot{a}_{12}$. Then,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\dot{a}_{14}zy^{2}+\dot{a}_{10}yz^{2}+\dot{a}_{15}y^{2}+\dot{a}_{11}z^{2}+\sigma_{1}(t)yz+\sigma_{2}(t)y+\sigma_{3}(t)z+\sigma_{4}(t)$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\dot{a}_{4}zx^{2}+\dot{a}_{10}xz^{2}+\dot{a}_{5}x^{2}+\dot{a}_{8}z^{2}+\tau_{1}(t)xz+\tau_{2}(t)x+\tau_{3}(t)z+\tau_{4}(t)$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\dot{a}_{4}yx^{2}+\dot{a}_{14}xy^{2}+\dot{a}_{2}x^{2}+\dot{a}_{12}y^{2}+\eta_{1}(t)xy+\eta_{2}(t)x+\eta_{3}(t)y+\eta_{4}(t)$

where $\sigma_{k}(t),\tau_{k}(t),\eta_{k}(t)$ for $k=1,2,3,4$ are arbitrary
functions.

Substituting the above results in (12.62) - (12.64), we find:

$\displaystyle(\ref{eq.TKNq9a})$ $\displaystyle\implies$ $\displaystyle
a_{10}=c_{4},\enskip\sigma_{1}=-\tau_{1}-2\dot{a}_{16},\enskip\sigma_{2}=-\tau_{2}-2\dot{a}_{17}$
$\displaystyle(\ref{eq.TKNq9b})$ $\displaystyle\implies$ $\displaystyle
a_{14}=c_{5},\enskip\eta_{1}=-\sigma_{1}-2\dot{a}_{18},\enskip\eta_{2}=-\sigma_{3}-2\dot{a}_{19}$
$\displaystyle(\ref{eq.TKNq9c})$ $\displaystyle\implies$ $\displaystyle
a_{4}=c_{6},\enskip\tau_{1}=-\eta_{1}+2(\dot{a}_{16}+\dot{a}_{18}),\enskip\tau_{3}=-\eta_{3}-2\dot{a}_{20}$

from which, we have finally: $a_{10}=c_{4}$, $a_{14}=c_{5}$, $a_{4}=c_{6}$,
$\tau_{1}=2\dot{a}_{18}$, $\eta_{1}=2\dot{a}_{16}$,
$\sigma_{1}=-2(\dot{a}_{16}+\dot{a}_{18})$,
$\tau_{2}=-\sigma_{2}-2\dot{a}_{17}$, $\eta_{2}=-\sigma_{3}-2\dot{a}_{19}$ and
$\eta_{3}=-\tau_{3}-2\dot{a}_{20}$, where $c_{4},c_{5},c_{6}$ are arbitrary
constants.

Therefore, the KT $K_{ab}$ is

$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{5}y+a_{2}z+a_{3}$
$\displaystyle K_{12}$ $\displaystyle=$
$\displaystyle\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17}$
$\displaystyle K_{13}$ $\displaystyle=$
$\displaystyle\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19}$
(12.65) $\displaystyle K_{22}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{15}x+a_{12}z+a_{13}$
$\displaystyle K_{23}$ $\displaystyle=$
$\displaystyle\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20}$
$\displaystyle K_{33}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{11}x+a_{8}y+a_{9}$

and the vector $K_{a}$ is

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\dot{a}_{15}y^{2}+\dot{a}_{11}z^{2}-\dot{a}_{5}xy-\dot{a}_{2}xz-2(\dot{a}_{16}+\dot{a}_{18})yz-\dot{a}_{3}x+\sigma_{2}y+\sigma_{3}z+\sigma_{4}$
$\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle\dot{a}_{5}x^{2}+\dot{a}_{8}z^{2}-\dot{a}_{15}xy+2\dot{a}_{18}xz-\dot{a}_{12}yz-(\sigma_{2}+2\dot{a}_{17})x-\dot{a}_{13}y+\tau_{3}z+\tau_{4}$
(12.66) $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle\dot{a}_{2}x^{2}+\dot{a}_{12}y^{2}+2\dot{a}_{16}xy-\dot{a}_{11}xz-\dot{a}_{8}yz-(\sigma_{3}+2\dot{a}_{19})x-(\tau_{3}+2\dot{a}_{20})y-\dot{a}_{9}z+\eta_{4}.$

Replacing the above quantities in the constraint (12.52), we find the
following set of conditions:

$a_{2}=a_{12},\enskip a_{5}=a_{8},\enskip a_{11}=a_{15},\enskip
a_{16}=a_{18}=0$ (12.67)
$(\nu-1)a_{2}=0,\enskip(\nu-1)a_{5}=0,\enskip(\nu-1)a_{11}=0$ (12.68)
$(\nu+2)a_{17}=0,\enskip(\nu+2)a_{19}=0,\enskip(\nu+2)a_{20}=0,\enskip(\nu+2)(a_{3}-a_{9})=0,\enskip(\nu+2)(a_{3}-a_{13})=0$
(12.69)
$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0,\enskip\dot{\sigma}_{2}=-\ddot{a}_{17},\enskip\dot{\sigma}_{3}=-\ddot{a}_{19},\enskip\dot{\tau}_{3}=-\ddot{a}_{20}.$
(12.70)

We consider three cases depending on the value of $\nu$:
\- $\forall\nu$. The general case.
\- $\nu=1$. Time-dependent Kepler potential.
\- $\nu=-2$. Time-dependent 3d oscillator.

The Newton-Cotes potential ($\nu=2$) is contained as a subcase of the general
case.

####  12.9 The general case

Because this case holds for any value of $\nu$, conditions (12.67) - (12.70)
give:
$a_{2}=a_{5}=a_{8}=a_{11}=a_{12}=a_{15}=a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=0$,
$a_{3}=a_{9}=a_{13}$, $\sigma_{2}=c_{7}$, $\sigma_{3}=c_{8}$ and
$\tau_{3}=c_{9}$, where $c_{7},c_{8},c_{9}$ are arbitrary constants.

Substituting these results in the constraint (12.51), we find that

$\dddot{a}_{3}=0,\enskip(\nu-2)\omega\dot{a}_{3}-2\dot{\omega}a_{3}=0$ (12.71)

and

$\ddot{\sigma}_{4}=\ddot{\tau}_{4}=\ddot{\eta}_{4}=0,\enskip\omega\sigma_{4}=\omega\tau_{4}=\omega\eta_{4}=0\implies\sigma_{4}=\tau_{4}=\eta_{4}=0.$

Therefore, the KT $K_{ab}$ becomes

$K_{ab}=\left(\begin{array}[]{ccc}\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{3}&\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz\\\
\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{3}&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz\\\
\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz&\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{3}\\\
\end{array}\right)$ (12.72)

and the vector

$K_{a}=\left(\begin{array}[]{c}-\dot{a}_{3}x+c_{7}y+c_{8}z\\\
-c_{7}x-\dot{a}_{3}y+c_{9}z\\\ -c_{8}x-c_{9}y-\dot{a}_{3}z\\\
\end{array}\right).$ (12.73)

Since the ten parameters $a_{3}(t)$ and $c_{A}$, where $A=1,2,...,9$, are
independent (i.e. they generate different FIs), we consider the following two
cases.

#####  12.9.1 $a_{3}(t)=0$

In this case, conditions (12.71) are satisfied identically leaving the
function $\omega(t)$ free to be any function.

The KT (12.72) becomes

$K_{ab}=\left(\begin{array}[]{ccc}\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz&\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz\\\
\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz\\\
\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz&\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy\\\
\end{array}\right)$

and the vector (12.73) becomes the general non-gradient KV
$K_{a}=\left(\begin{array}[]{c}c_{7}y+c_{8}z\\\ -c_{7}x+c_{9}z\\\
-c_{8}x-c_{9}y\\\ \end{array}\right)$.

Then, the constraint (12.50) implies that (since $K_{a}q^{a}=0$) $K=G(x,y,z)$,
which when replaced in (12.49) gives (since $K_{ab}q^{b}=0$) $G_{,a}=0$.
Hence, $K=const\equiv 0$.

The QFI $I=K_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}\dot{q}^{a}$ leads only to the
three components $L_{i}$ of the angular momentum. In particular, $I$ contains
nine independent parameters each of them defining a FI: a) $c_{7}$, $c_{8}$,
$c_{9}$ lead to the components $L_{1}=y\dot{z}-z\dot{y}$,
$L_{2}=z\dot{x}-x\dot{z}$, $L_{3}=x\dot{y}-y\dot{x}$ of the angular momentum
(LFIs); and b) $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, $c_{5}$, $c_{6}$ lead to
the products (QFIs depending on $L_{i}$) $L_{1}^{2}$, $L_{2}^{2}$,
$L_{3}^{2}$, $L_{1}L_{2}$, $L_{1}L_{3}$ and $L_{2}L_{3}$.

We have the following proposition.

###### Proposition 12.9.1

The time-dependent generalized Kepler potential
$V(t,q)=-\frac{\omega(t)}{r^{\nu}}$, for a general smooth function
$\omega(t)$, admits only the LFIs of the angular momentum $L_{i}$. Independent
QFIs in general do not exist; they are all quadratic combinations of $L_{i}$.

#####  12.9.2 $c_{A}=0$ where $A=1,2,...,9$

In this case, conditions (12.71) imply that $a_{3}(t)=b_{0}+b_{1}t+b_{2}t^{2}$
and

$\omega_{(\nu)}(t)=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$
(12.74)

where $k,b_{0},b_{1},b_{2}$ are arbitrary constants and the index $(\nu)$
denotes the dependence of $\omega(t)$ on the value of $\nu$.

Since $c_{A}=0$, the quantities (12.72) and (12.73) become, respectively,
$K_{ab}=a_{3}\delta_{ab}$ and $K_{a}=-\dot{a}_{3}q_{a}$. Substituting in the
remaining constraints (12.49) and (12.50), we find
$K=b_{2}r^{2}-\frac{2k(b_{0}+b_{1}t+b_{2}t^{2})^{\nu/2}}{r^{\nu}}$.

The QFI is

$J_{\nu}=(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k(b_{0}+b_{1}t+b_{2}t^{2})^{\frac{\nu-2}{2}}}{r^{\nu}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}.$
(12.75)

We note that the resulting time-dependent generalized Kepler potential

$V=-\frac{\omega_{\nu}(t)}{r^{\nu}},\enskip\omega_{\nu}=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$
(12.76)

is a subcase of the Case III potential of [12] if we set
$U\left(\frac{r}{\phi}\right)=$
$k_{1}\frac{r^{2}}{\phi^{2}}-\frac{k\phi^{\nu}}{r^{\nu}}$ with
$\phi=\sqrt{b_{0}+b_{1}t+b_{2}t^{2}}$ and
$k_{1}=\frac{b_{0}b_{2}}{2}-\frac{b_{1}^{2}}{8}$. Then, the associated QFI
(3.13) of [12] (for $K_{1}=K_{2}=0$) reduces to the QFI $J_{\nu}$.

For some values of $\nu$, we have the following results:

\- $\nu=1$ (time-dependent Kepler potential).

The $\omega_{(1)}(t)=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{-1/2}$ and the QFI
$J_{1}=E_{3}$ (see section 12.10.2 below).

\- $\nu=2$ (Newton-Cotes potential [183]).

The $\omega_{(2)}=k=const$ and the QFI is

$J_{2}=(b_{0}+b_{1}t+b_{2}t^{2})\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r^{2}}\right)-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}}{2}r^{2}=b_{0}H_{2}-b_{1}I_{2}-b_{2}I_{1}.$

This expression contains the independent QFIs:

$H_{2}=\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r^{2}},\enskip
I_{1}=-t^{2}H_{2}+tq^{i}\dot{q}_{i}-\frac{r^{2}}{2},\enskip
I_{2}=-tH_{2}+\frac{q^{i}\dot{q}_{i}}{2}$

where $H_{2}$ is the Hamiltonian of the system. These are the FIs found in
chapter 7 (see also Table LABEL:T1) in the case of the autonomous generalized
Kepler potential for $\nu=2$.

\- $\nu=-2$ (time-dependent oscillator).

The $\omega_{(-2)}=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{-2}$ and the QFI is

$J_{-2}=(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{2}}r^{2}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}.$

This is the trace of the QFIs (12.104) found below for
$a_{3}(t)=b_{0}+b_{1}t+b_{2}t^{2}$. Substituting this $a_{3}(t)$ in (12.103)
and (12.104), we find, respectively, that the $\omega=\omega_{(-2)}$ with
constant $k=-\frac{1}{8}(b_{1}^{2}-4b_{2}b_{0}+2c_{0})$ and the QFIs are

$I_{ij}=\Lambda_{ij}(a_{3}=b_{0}+b_{1}t+b_{2}t^{2})=(b_{0}+b_{1}t+b_{2}t^{2})\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-(b_{1}+2b_{2}t)q_{(i}\dot{q}_{j)}+b_{2}q_{i}q_{j}.$ (12.77)

Therefore, the trace $Tr[I_{ij}]=I_{11}+I_{22}+I_{33}=2J_{-2}$. We note that
$r^{2}=q^{i}q_{i}$.

We infer the following new general result, which includes the time-dependent
Kepler potential and the time-dependent oscillator as subcases.

######  Proposition 12.9.2 (3d time-dependent generalized Kepler potentials
which admit FIs)

For all
functions $\omega(t)$ the time-dependent generalized Kepler potential
$V(t,q)=-\frac{\omega(t)}{r^{\nu}}$ admits the LFIs of the angular momentum
and QFIs, which are products of the components of the angular momentum.
However, for the function
$\omega(t)=\omega_{(\nu)}(t)=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$
the resulting time-dependent generalized Kepler potential admits the
additional QFI $J_{\nu}$ given by (12.75).

####  12.10 The time-dependent Kepler potential

In this case, $\nu=1$ and conditions (12.67) - (12.70) give:
$a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=0$, $a_{5}=a_{8}$, $a_{2}=a_{12}$,
$a_{3}=a_{9}=a_{13}$,$a_{11}=a_{15}$,
$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0$, $\sigma_{2}=c_{7}$,
$\sigma_{3}=c_{8}$ and $\tau_{3}=c_{9}$.

Then, the constraint (12.51) gives $\dddot{a}_{3}=0$,
$\sigma_{4}=\tau_{4}=\eta_{4}=0$, $a_{3}\omega^{2}=c_{10}$,
$a_{2}\omega=c_{11}$, $a_{5}\omega=c_{12}$ and $a_{11}\omega=c_{13}$, where
$c_{10},c_{11},c_{12},c_{13}$ are arbitrary constants. Finally, we have:

$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{5}y+a_{2}z+a_{3}$
$\displaystyle K_{12}$ $\displaystyle=$
$\displaystyle\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz-\frac{a_{5}}{2}x-\frac{a_{11}}{2}y$
$\displaystyle K_{13}$ $\displaystyle=$
$\displaystyle\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz-\frac{a_{2}}{2}x-\frac{a_{11}}{2}z$
$\displaystyle K_{22}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{11}x+a_{2}z+a_{3}$
$\displaystyle K_{23}$ $\displaystyle=$
$\displaystyle\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz-\frac{a_{2}}{2}y-\frac{a_{5}}{2}z$
$\displaystyle K_{33}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{11}x+a_{5}y+a_{3}$

and

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\dot{a}_{11}y^{2}+\dot{a}_{11}z^{2}-\dot{a}_{5}xy-\dot{a}_{2}xz-\dot{a}_{3}x+c_{7}y+c_{8}z$
$\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle\dot{a}_{5}x^{2}+\dot{a}_{5}z^{2}-\dot{a}_{11}xy-\dot{a}_{2}yz-
c_{7}x-\dot{a}_{3}y+c_{9}z$ $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle\dot{a}_{2}x^{2}+\dot{a}_{2}y^{2}-\dot{a}_{11}xz-\dot{a}_{5}yz-
c_{8}x-c_{9}y-\dot{a}_{3}z$

where

$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0,\enskip\dddot{a}_{3}=0,\enskip
a_{3}\omega^{2}=c_{10},\enskip a_{2}\omega=c_{11},\enskip
a_{5}\omega=c_{12},\enskip a_{11}\omega=c_{13}.$ (12.78)

From the last conditions, it follows that in order QFIs to be allowed the
function $\omega(t)$ can have only three possible forms:
\- $\omega(t)$ a general function;
\- $\omega(t)=\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$, where
$c_{11}b_{1}\neq 0$; and
\- $\omega(t)=\omega_{3K}(t)=\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$, where
$k\neq 0$ and $b_{1}^{2}-4b_{2}b_{0}\neq 0$.

This result confirms the results found previously in [12, 76, 177]. We note
that the time-dependent Kepler potential $V=-\frac{\omega_{2K}(t)}{r}$ is a
subcase of the Case II potential of [12] for $\mu_{0}=c_{11}$ and
$\phi=b_{0}+b_{1}t$, whereas the potential $V=-\frac{\omega_{3K}(t)}{r}$ is a
subcase of the Case III potential of [12] (see section 12.9.2).

In the following, we discuss the cases for the special functions
$\omega_{2K}(t)$ and $\omega_{3K}(t)$ because the case for a general function
$\omega(t)$ reproduces the results of the subsection 12.9.1.

#####  12.10.1 $\omega(t)=\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$,
$c_{11}b_{1}\neq 0$

In this case, conditions (12.78) give: $a_{2}=b_{0}+b_{1}t$,
$a_{3}=\frac{c_{10}}{c_{11}^{2}}(b_{0}+b_{1}t)^{2}$,
$a_{5}=\frac{c_{12}}{c_{11}}(b_{0}+b_{1}t)$ and
$a_{11}=\frac{c_{13}}{c_{11}}(b_{0}+b_{1}t)$.

Substituting the resulting vector $K_{a}$ and the KT $K_{ab}$ in (12.50), we
find the scalar $K=-\frac{2c_{10}b_{1}t}{c_{11}r}+G(q)$. Replacing this
function in the remaining constraint (12.49), we find

$G(x,y,z)=-\frac{2c_{10}b_{0}}{c_{11}r}-\frac{c_{13}x+c_{12}y+c_{11}z}{r}+\frac{c_{10}b_{1}^{2}}{c_{11}^{2}}r^{2}.$

Therefore,

$K(x,y,z,t)=\frac{c_{10}b_{1}^{2}r^{2}}{c_{11}^{2}}-\frac{2c_{10}(b_{0}+b_{1}t)}{c_{11}r}-\frac{c_{13}x+c_{12}y+c_{11}z}{r}.$

The QFI is

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{c_{3}}{2}L_{1}^{2}+\frac{c_{1}}{2}L_{2}^{2}+\frac{c_{2}}{2}L_{3}^{2}-c_{4}L_{1}L_{2}-c_{5}L_{1}L_{3}-c_{6}L_{2}L_{3}-c_{9}L_{1}+c_{8}L_{2}-c_{7}L_{3}+$
$\displaystyle+\frac{2c_{10}}{c_{11}^{2}}E_{2}+\frac{c_{13}}{c_{11}}A_{1}+\frac{c_{12}}{c_{11}}A_{2}+A_{3}$

where $\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$ and

$\displaystyle L_{i}$ $\displaystyle\equiv$ $\displaystyle
q_{i+1}\dot{q}_{i+2}-q_{i+2}\dot{q}_{i+1}$ (12.79) $\displaystyle E_{2}$
$\displaystyle\equiv$
$\displaystyle(b_{0}+b_{1}t)^{2}\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{c_{11}}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)q^{i}\dot{q}_{i}+\frac{b_{1}^{2}r^{2}}{2}$
(12.80) $\displaystyle\tilde{R}_{i}$ $\displaystyle\equiv$
$\displaystyle(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{c_{11}}{r(b_{0}+b_{1}t)}q_{i}$
(12.81) $\displaystyle A_{i}$ $\displaystyle\equiv$
$\displaystyle(b_{0}+b_{1}t)\tilde{R}_{i}+b_{1}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right).$
(12.82)

We note that $i=1,2,3$, $q_{i}=(x,y,z)$ and $q_{i}\equiv q_{i+3k}$ for all
$k\in\mathbb{N}$, that is, $x=q_{1}=q_{4}=$ $q_{7}=...,$ $y=q_{2}=$
$q_{5}=q_{8}=...,$ and $z=q_{3}=q_{6}$ $=q_{9}=...~{}$.

The QFI $I$ contains the already found LFIs $L_{i}$ of the angular momentum;
the QFI $E_{2}$, which for $b_{1}=0$ reduces to the Hamiltonian of the Kepler
potential $V=-\frac{c_{11}}{b_{0}r}$; and the QFIs $A_{i}$, which may be
considered as a generalization of the Runge-Lenz vector
$R_{i}\left(k=\frac{c_{11}}{b_{0}}\right)$ for time-dependence
$\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$. Indeed, we have
$A_{i}(b_{1}=0)=b_{0}R_{i}\left(k=\frac{c_{11}}{b_{0}}\right)$.

The expressions (12.80) - (12.82) are written compactly as follows:

$\displaystyle E_{2}$ $\displaystyle\equiv$ $\displaystyle
c_{11}^{2}\left[\frac{1}{\omega_{2K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{2K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{2K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{2K}}\right)^{2}\frac{r^{2}}{4}\right]$
(12.83) $\displaystyle\tilde{R}_{i}$ $\displaystyle\equiv$
$\displaystyle(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{\omega_{2K}}{r}q_{i}$
(12.84) $\displaystyle A_{i}$ $\displaystyle\equiv$ $\displaystyle
c_{11}\left[\frac{1}{\omega_{2K}}\tilde{R}_{i}-\frac{(\ln\omega_{2K})^{\cdot}}{\omega_{2K}}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right)\right]$
(12.85)

where $\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$.

We remark that only five of the seven FIs $E_{2}$, $L_{i}$, $A_{i}$ are
(functionally) independent because they are related as follows:

$\mathbf{A}\cdot\mathbf{L}=0,\enskip
2E_{2}\mathbf{L}^{2}+c_{11}^{2}=\mathbf{A}^{2}.$ (12.86)

For $b_{1}=0$ and $b_{0}\neq 0$, we have
$\omega_{2K}=\frac{c_{11}}{b_{0}}\equiv k=const$, $E_{2}=b_{0}^{2}H$,
$\tilde{R}_{i}=R_{i}$ and $A_{i}=b_{0}R_{i}$, where $H$ is the Hamiltonian and
$R_{i}$ the Runge-Lenz vector for the Kepler potential $V=-\frac{k}{r}$. Then,
as expected, equation (12.86) reduces to the well-known relation

$2H\mathbf{L}^{2}+k^{2}=\mathbf{R}^{2}.$ (12.87)

#####  12.10.2
$\omega(t)=\omega_{3K}(t)=\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$, $k\neq
0$, $b_{1}^{2}-4b_{2}b_{0}\neq 0$

In this case, conditions (12.78) give333Observe that if
$b_{1}^{2}-4b_{2}b_{0}=0$, this case reduces to the case of the section
12.10.1 because equation $b_{0}+b_{1}t+b_{2}t^{2}=0$ has a double root $t_{0}$
and can be factored in the form $b_{2}(t-t_{0})^{2}$.: $a_{2}=a_{5}=a_{11}=0$,
$c_{11}=c_{12}=c_{13}=0$ and
$a_{3}=\frac{c_{10}}{k^{2}}(b_{0}+b_{1}t+b_{2}t^{2})$.

Substituting the resulting quantities $K_{a}$ and $K_{ab}$ in (12.50), we find
the scalar $K=-\frac{2c_{10}}{r\omega_{3K}}+G(q)$. When this scalar is
replaced in the remaining constraint (12.49), it gives
$G(x,y,z)=\frac{b_{2}c_{10}}{k^{2}}r^{2}$. Therefore,

$K(x,y,z,t)=\frac{b_{2}c_{10}}{k^{2}}r^{2}-\frac{2c_{10}}{r\omega_{3K}}.$

The QFI is

$I=\frac{c_{3}}{2}L_{1}^{2}+\frac{c_{1}}{2}L_{2}^{2}+\frac{c_{2}}{2}L_{3}^{2}-c_{4}L_{1}L_{2}-c_{5}L_{1}L_{3}-c_{6}L_{2}L_{3}-c_{9}L_{1}+c_{8}L_{2}-c_{7}L_{3}+\frac{2c_{10}}{k^{2}}E_{3}$

where

$E_{3}\equiv(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}$
(12.88)

is the only new independent QFI. This QFI is written equivalently as

$E_{3}=k^{2}\left[\frac{1}{\omega_{3K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{3K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{3K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{3K}}\right)^{2}\frac{r^{2}}{4}\right].$
(12.89)

For $b_{1}=b_{2}=0$, $E_{2}$ reduces to the well-known Hamiltonian of the
time-independent Kepler potential.

We note also that the QFIs (12.80) and (12.88) can be written compactly as
(see eq. (2.86) in [76])

$E_{\mu}=k^{2}\left[\frac{1}{\omega_{\mu
K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{\mu
K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{\mu
K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{\mu
K}}\right)^{2}\frac{r^{2}}{4}\right]$ (12.90)

where $\mu=2,3$, $\omega_{2K}(t)=\frac{k}{b_{0}+b_{1}t}$ and
$\omega_{3K}(t)=\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$.

######  Proposition 12.10.1 (Time-dependent Kepler potentials which admit
additional FIs [76])

The
time-dependent Kepler potential $V(t,q)=-\frac{\omega(t)}{r}$, for the
function $\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$, where $c_{11}b_{1}\neq
0$, and the function
$\omega_{3K}(t)=\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$, where $k\neq 0$
and $b_{1}^{2}-4b_{2}b_{0}\neq 0$, admits additional QFIs given by (12.80),
(12.82) and (12.88), respectively.

####  12.11 The 3d time-dependent oscillator

In this case, we have $\nu=-2$ and conditions (12.67) - (12.70) give:
$a_{2}=a_{5}=a_{8}=a_{11}=a_{12}=a_{15}=a_{16}=a_{18}=0$ and

$\enskip\dot{\sigma}_{2}=-\ddot{a}_{17},\enskip\dot{\sigma}_{3}=-\ddot{a}_{19},\enskip\dot{\tau}_{3}=-\ddot{a}_{20}.$
(12.91)

Then, the constraint (12.51) implies that:

$\ddot{\sigma}_{4}-2\omega\sigma_{4}=0,\enskip\ddot{\tau}_{4}-2\omega\tau_{4}=0,\enskip\ddot{\eta}_{4}-2\omega\eta_{4}=0,$
(12.92)
$\dddot{a}_{3}-8\omega\dot{a}_{3}-4\dot{\omega}a_{3}=0,\enskip\dddot{a}_{9}-8\omega\dot{a}_{9}-4\dot{\omega}a_{9}=0,\enskip\dddot{a}_{13}-8\omega\dot{a}_{13}-4\dot{\omega}a_{13}=0,$
(12.93)
$\dddot{a}_{17}-8\omega\dot{a}_{17}-4\dot{\omega}a_{17}=0,\enskip\dddot{a}_{19}-8\omega\dot{a}_{19}-4\dot{\omega}a_{19}=0,\enskip\dddot{a}_{20}-8\omega\dot{a}_{20}-4\dot{\omega}a_{20}=0.$
(12.94)

Therefore,

$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{3}$
$\displaystyle K_{12}$ $\displaystyle=$
$\displaystyle\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz+a_{17}$
$\displaystyle K_{13}$ $\displaystyle=$
$\displaystyle\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz+a_{19}$
$\displaystyle K_{22}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{13}$
(12.95) $\displaystyle K_{23}$ $\displaystyle=$
$\displaystyle\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz+a_{20}$
$\displaystyle K_{33}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{9}$

and

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle-\dot{a}_{3}x+\sigma_{2}y+\sigma_{3}z+\sigma_{4}$ $\displaystyle
K_{2}$ $\displaystyle=$
$\displaystyle-(\sigma_{2}+2\dot{a}_{17})x-\dot{a}_{13}y+\tau_{3}z+\tau_{4}$
(12.96) $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle-(\sigma_{3}+2\dot{a}_{19})x-(\tau_{3}+2\dot{a}_{20})y-\dot{a}_{9}z+\eta_{4}.$

Before we proceed with considering various subcases, it is important that we
discuss the ODEs (12.93) and (12.94).

#####  12.11.1 The Lewis invariant

Equations of the form

$\dddot{a}-8\omega\dot{a}-4\dot{\omega}a=0$ (12.97)

where $a=a(t)$, can be written as follows:

$a\ddot{a}-\frac{1}{2}\dot{a}^{2}-4\omega a^{2}=c_{0}=const.$ (12.98)

By putting $a=-\rho^{2}$, where $\rho=\rho(t)$, equation (12.98) becomes

$\ddot{\rho}-2\omega\rho-\frac{c_{0}}{2\rho^{3}}=0.$ (12.99)

For $2\omega(t)=-\psi^{2}(t)$, equation (12.99) is written as

$\ddot{\rho}+\psi^{2}\rho-\frac{c_{0}}{2\rho^{3}}=0.$ (12.100)

Equation (12.100) is the auxiliary equation (see [10, 164, 173]) that should
be introduced in order to derive the Lewis invariant for the 1d time-dependent
oscillator

$\ddot{x}+\psi^{2}x=0.$ (12.101)

By eliminating the $\psi^{2}$, using (12.101), and by multiplying with the
factor $x\dot{\rho}-\rho\dot{x}$, equation (12.100) gives

$\ddot{\rho}-\frac{\rho}{x}\ddot{x}-\frac{c_{0}}{2\rho^{3}}=0\implies\left[\frac{1}{2}\left(x\dot{\rho}-\rho\dot{x}\right)^{2}+\frac{c_{0}}{4}\left(\frac{x}{\rho}\right)^{2}\right]^{\cdot}=0\implies$
$I\equiv\frac{1}{2}\left(x\dot{\rho}-\rho\dot{x}\right)^{2}+\frac{c_{0}}{4}\left(\frac{x}{\rho}\right)^{2}=const$
(12.102)

which is the well-known Lewis invariant for the 1d time-dependent harmonic
oscillator or, equivalently, a FI for the 2d time-dependent system with
equations of motion (12.100) and (12.101).

#####  12.11.2 The system of equations (12.91) - (12.94)

Conditions (12.92) are not involved into conditions (12.91), (12.93) and
(12.94). This means that the parameters $\sigma_{4},\tau_{4},\eta_{4}$ give
different independent FIs from the remaining parameters
$a_{3},a_{9},a_{13},a_{17},a_{19},a_{20}$. Therefore, without loss of
generality, they can be treated separately. This leads to the following two
cases.

###### Case: $a_{3}\neq 0$ and $\sigma_{4}=\tau_{4}=\eta_{4}=0$

Because the ODEs (12.93) and (12.94) are independent (i.e. each one leads to a
different FI) and of the same form, without loss of generality, we assume:
$a_{9}=k_{1}a_{3}$, $a_{13}=k_{2}a_{3}$, $a_{17}=k_{3}a_{3}$,
$a_{19}=k_{4}a_{3}$ and $a_{20}=k_{5}a_{3}$, where
$k_{1},k_{2},k_{3},k_{4},k_{5}$ are arbitrary constants.

From the discussion of section 12.11.1 and the assumption $a_{3}\neq 0$,
condition (12.93) concerning $a_{3}(t)$ becomes (see eq. (9.2) in [173])

$\dddot{a}_{3}-8\omega\dot{a}_{3}-4\dot{\omega}a_{3}=0\implies
a_{3}\ddot{a}_{3}-\frac{1}{2}\dot{a}_{3}^{2}-4\omega
a_{3}^{2}=c_{0}\implies\omega(t)=\frac{\ddot{a}_{3}}{4a_{3}}-\frac{1}{8}\left(\frac{\dot{a}_{3}}{a_{3}}\right)^{2}-\frac{c_{0}}{4a_{3}^{2}}$
(12.103)

where $c_{0}$ is an arbitrary constant and $a_{3}(t)$ is an arbitrary non-zero
function.

Moreover, conditions (12.91) become $\sigma_{2}=-\dot{a}_{17}$,
$\sigma_{3}=-\dot{a}_{19}$ and $\tau_{3}=-\dot{a}_{20}$ because any additional
constant (in general $\sigma_{2}=-\dot{a}_{17}+m_{1}$, where $m_{1}$ is a
constant) leads to the usual LFIs of the angular momentum.

Then, the KT (12.95) and the vector (12.96) become444We set
$c_{1}=...=c_{6}=0$ because they generate the already found FIs of the angular
momentum.:

$K_{ab}=a_{3}\left(\begin{array}[]{ccc}1&k_{3}&k_{4}\\\ k_{3}&k_{2}&k_{5}\\\
k_{4}&k_{5}&k_{1}\end{array}\right),\enskip
K_{a}=-\dot{a}_{3}\left(\begin{array}[]{c}x+k_{3}y+k_{4}z\\\
k_{3}x+k_{2}y+k_{5}z\\\ k_{4}x+k_{5}y+k_{1}z\end{array}\right).$

Substituting in the constraints (12.49) and (12.50), we find

$K=\frac{\dot{a}_{3}^{2}+2c_{0}}{4a_{3}}\left(x^{2}+k_{2}y^{2}+k_{1}z^{2}+2k_{3}xy+2k_{4}xz+2k_{5}yz\right).$

Using equation (12.103), we can write
$\frac{\dot{a}_{3}^{2}+2c_{0}}{4a_{3}}=\frac{\ddot{a}_{3}}{2}-2\omega a_{3}$.

The QFI is

$\displaystyle I$ $\displaystyle=$ $\displaystyle
a_{3}\left(\dot{x}^{2}+k_{2}\dot{y}^{2}+k_{1}\dot{z}^{2}+2k_{3}\dot{x}\dot{y}+2k_{4}\dot{x}\dot{z}+2k_{5}\dot{y}\dot{z}\right)-\dot{a}_{3}(x+k_{3}y+k_{4}z)\dot{x}-$
$\displaystyle-\dot{a}_{3}(k_{3}x+k_{2}y+k_{5}z)\dot{y}-\dot{a}_{3}(k_{4}x+k_{5}y+k_{1}z)\dot{z}+$
$\displaystyle+\left(\frac{\ddot{a}_{3}}{2}-2\omega
a_{3}\right)\left(x^{2}+k_{2}y^{2}+k_{1}z^{2}+2k_{3}xy+2k_{4}xz+2k_{5}yz\right).$

This expression contains six QFIs, which are the components of the symmetric
tensor (see eqs. (1.4) and (6.24) in [173])

$\Lambda_{ij}=a_{3}\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{a}_{3}q_{(i}\dot{q}_{j)}+\frac{\ddot{a}_{3}}{2}q_{i}q_{j}.$
(12.104)

This tensor for $a_{3}=const\neq 0$ reduces to the Jauch-Hill-Fradkin tensor
$B_{ij}$ with $\omega=-\frac{c_{0}}{4a_{3}^{2}}=const$.

If we make the transformation (see section 12.11.1) $a_{3}(t)=-\rho^{2}(t)$
and $2\omega(t)=-\psi^{2}(t)$, equation (12.46) becomes

$\ddot{q}^{a}-2\omega q^{a}=0\implies\ddot{q}^{a}+\psi^{2}q^{a}=0$ (12.105)

and the QFIs (12.104) give

$\Lambda_{ij}=-\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\left(\rho\dot{q}_{j}-\dot{\rho}q_{j}\right)-\frac{c_{0}}{2}\rho^{-2}q_{i}q_{j}$
(12.106)

where condition (12.103) takes the form (12.100).

The symmetric tensor (12.106) may be thought of as a 3d generalization of the
1d Lewis invariant (12.102). Moreover, equation (12.106) coincides with eq.
(8) in [178] and eq. (1.4) in [173] when $c_{0}=2$.

###### Case: $a_{3}=a_{9}=a_{13}=a_{17}=a_{19}=a_{20}=0$ and $\sigma_{4}\neq
0$

In this case, conditions (12.93) and (12.94) vanish identically, and
conditions (12.91) imply that $\sigma_{2}=c_{7}$, $\sigma_{3}=c_{8}$ and
$\tau_{3}=c_{9}$.

Since the remaining ODEs (12.92) are all independent (i.e. each one generates
an independent FI) and of the same form, without loss of generality, we assume
$\tau_{4}=k_{1}\sigma_{4}$ and $\eta_{4}=k_{2}\sigma_{4}$, where $k_{1}$ and
$k_{2}$ are arbitrary constants.

From (12.92) for $\sigma_{4}\neq 0$, we get

$\omega(t)=\frac{\ddot{\sigma}_{4}}{2\sigma_{4}}.$ (12.107)

The parameters $c_{A}$, where $A=1,2,...,9$, produce the FIs of the angular
momentum and we fix them to zero. Therefore, $K_{ab}=0$ and
$K_{a}=\sigma_{4}\left(1,k_{1},k_{2}\right)$. Substituting in the remaining
constraints (12.49) and (12.50), we find
$K=-\dot{\sigma}_{4}\left(x+k_{1}y+k_{2}z\right)$.

The QFI is
$I=\sigma_{4}\dot{x}-\dot{\sigma}_{4}x+k_{1}\left(\sigma_{4}\dot{y}-\dot{\sigma}_{4}y\right)+k_{2}\left(\sigma_{4}\dot{z}-\dot{\sigma}_{4}z\right)$,
which contains the irreducible LFIs (see eq. (6.25) in [173])

$I_{4i}=f\dot{q}_{i}-\dot{f}q_{i}$ (12.108)

where $f(t)$ is an arbitrary non-zero function satisfying (12.107) for
$\sigma_{4}=f$. We note that the LFIs (12.108) can be derived directly from
the equations of motion for $\omega(t)=\frac{\ddot{f}}{2f}$.

From the above two cases, we arrive at the following conclusion.

######  Proposition 12.11.1 (3d time-dependent oscillators which admit
additional FIs)

For the function
$\omega(t)=\frac{\ddot{a}_{3}}{4a_{3}}-\frac{1}{8}\left(\frac{\dot{a}_{3}}{a_{3}}\right)^{2}-\frac{c_{0}}{4a^{2}_{3}}$,
where $a_{3}(t)\neq 0$ and $c_{0}$ is an arbitrary constant, and the function
$\omega(t)=\frac{\ddot{f}}{2f}$, where $f(t)\neq 0$, the resulting 3d time-
dependent oscillator $V(t,q)=-\omega(t)r^{2}$ admits the QFIs (12.104) and the
LFIs (12.108), respectively.

####  12.12 A special class of time-dependent oscillators

In Proposition 12.11.1, it has been shown that the time-dependent oscillator
($\nu=-2$) for the frequency

$\omega_{1O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}$
(12.109)

where $f(t)$ is an arbitrary non-zero function, admits the six QFIs

$\Lambda_{ij}=f(t)\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{f}q_{(i}\dot{q}_{j)}+\frac{\ddot{f}}{2}q_{i}q_{j}$
(12.110)

and for the frequency

$\omega_{2O}(t)=\frac{\ddot{g}}{2g(t)}$ (12.111)

where $g(t)$ is an arbitrary non-zero function, admits the three LFIs

$I_{4i}=g(t)\dot{q}_{i}-\dot{g}q_{i}.$ (12.112)

We consider the class of the 3d time-dependent oscillators for which
$\omega_{1O}(t)=\omega_{2O}(t)$. These oscillators admit both the six QFIs
$\Lambda_{ij}$ and the three LFIs $I_{4i}$.

The condition $\omega_{1O}(t)=\omega_{2O}(t)$ relates the functions $f(t)$ and
$g(t)$ as follows:

$\omega_{3O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}=\frac{\ddot{g}}{2g(t)}.$
(12.113)

It can be easily proved that both the choices

$g=f^{1/2}\cos\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{f(t)}$
(12.114)

and

$g=f^{1/2}\sin\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{f(t)}$
(12.115)

satisfy the requirement (12.113) for any non-zero function $f(t)$. In other
words, all the time-dependent oscillators with frequency

$\omega_{3O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}$
(12.116)

admit the six QFIs

$\Lambda_{ij}=f(t)\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{f}q_{(i}\dot{q}_{j)}+\frac{\ddot{f}}{2}q_{i}q_{j}$
(12.117)

and the six LFIs

$\displaystyle I_{41i}$ $\displaystyle=$
$\displaystyle\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\sin\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\cos\theta$
(12.118) $\displaystyle I_{42i}$ $\displaystyle=$
$\displaystyle-\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\cos\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\sin\theta.$
(12.119)

These are the LFIs $J^{k}_{3}$ and $J^{k}_{4}$ derived in eqs. (44) and (45)
in [175] using point Noether symmetries and Noether’s Theorem.

We note that $\frac{dI_{42i}}{d\theta}=I_{41i}$ and

$\Lambda_{ij}=I_{41i}I_{41j}+I_{42i}I_{42j}.$ (12.120)

Next, we consider the LFIs of the angular momentum
$L_{i}=q_{i+1}\dot{q}_{i+2}-q_{i+2}\dot{q}_{i+1}$. These LFIs can be expressed
equivalently as components of the totally antisymmetric tensor

$L_{ij}=q_{i}\dot{q}_{j}-q_{j}\dot{q}_{i}=\varepsilon_{ijk}L^{k}$ (12.121)

where $\varepsilon_{ijk}$ is the 3d Levi-Civita symbol and $L^{i}=L_{i}$ since
the kinetic metric $\gamma_{ij}=\delta_{ij}$. Then, (see eq. (51) in [175])

$L_{ij}=\left(\frac{2}{c_{0}}\right)^{1/2}\left(I_{41i}I_{42j}-I_{41j}I_{42i}\right).$
(12.122)

###### Proposition 12.12.1

For the class of 3d time-dependent oscillators with potential
$V(t,q)=-\omega(t)r^{2}$, where $\omega(t)$ is defined in terms of an
arbitrary non-zero (smooth) function $f(t)$ as in (12.116), the only
independent FIs are the LFIs $I_{41i}$ and $I_{42i}$.

In order to recover the results of [175], we assume a time-dependent
oscillator with $\omega_{3O}(t)$ given by (12.116), and we write the non-zero
function $f(t)$ in the form $f(t)=\rho^{2}(t)$. Then, equation (12.116)
becomes

$\omega_{3O}(t)=\frac{\ddot{\rho}}{2\rho}-\frac{c_{0}}{4\rho^{4}}.$ (12.123)

The relations (12.114) and (12.115) become:

$g=\rho\cos\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-2}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{\rho^{2}}$
(12.124)
$g=\rho\sin\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-2}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{\rho^{2}}$
(12.125)

and the LFIs (12.118) and (12.119) take the form:

$\displaystyle I_{41i}$ $\displaystyle=$
$\displaystyle\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-1}q_{i}\sin\theta+\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\cos\theta$
(12.126) $\displaystyle I_{42i}$ $\displaystyle=$
$\displaystyle-\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-1}q_{i}\cos\theta+\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\sin\theta.$
(12.127)

These latter expressions for $c_{0}=2$ coincide with the independent LFIs (44)
and (45) found in [175].

Finally, we note that if we consider in this special class of oscillators the
simple case $f=1$, we find $\omega_{3O}(t)=const=-\frac{c_{0}}{4}\equiv k$,
which is the 3d autonomous oscillator (for $k<0$). Then, it can be shown that
the exponential LFIs $I_{3i\pm}$ (see Table LABEL:T1) found in chapter 7 can
be written in terms of $I_{41i}$ and $I_{42j}$. Indeed, we have
$I_{3i\pm}(k>0)=I_{41i}\mp iI_{42i}$ and $I_{3i\pm}(k<0)=I_{41i}\pm iI_{42i}$.

####  12.13 Collection of results

We collect the results concerning the time-dependent generalized Kepler
potential for all values of $\nu$ in Table LABEL:T2. We note that for
$\nu=-2,1,2$ the dynamical system is the time-dependent 3d oscillator, the
time-dependent Kepler potential, and the Newton-Cotes potential, respectively.
Concerning notation, we have: $q^{i}=(x,y,z)$, $q_{i}\equiv q_{i+3k}$ for all
$k\in\mathbb{N}$ and
$\tilde{R}_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r(b_{0}+b_{1}t)}q_{i}$.

Table 12.2:  The LFIs/QFIs of the time-dependent generalized Kepler potential $V=-\frac{\omega(t)}{r^{\nu}}$. $\nu$ | $\omega(t)$ | LFIs and QFIs
---|---|---
$\forall$ $\nu$ |  $\forall$ $\omega$ |  $L_{i}=q_{i+1}\dot{q}_{i+2}-q_{i+2}\dot{q}_{i+1}$, $L_{ij}=q_{i}\dot{q}_{j}-q_{j}\dot{q}_{i}=\varepsilon_{ijk}L^{k}$
$k$ | $H_{\nu}=\frac{1}{2}\dot{q}^{i}\dot{q}_{i}-\frac{k}{r^{\nu}}$
$\omega_{\nu}=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$ | $J_{\nu}=(b_{0}+b_{1}t+b_{2}t^{2})\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{\nu}}{r^{\nu}}\right)-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}$
$-2$ | $k$ | $B_{ij}=\dot{q}_{i}\dot{q}_{j}-2kq_{i}q_{j}$
$k>0$ | $I_{3a\pm}=e^{\pm\sqrt{2k}t}(\dot{q}_{a}\mp\sqrt{2k}q_{a})$
$k<0$ | $I_{3a\pm}=e^{\pm i\sqrt{-2k}t}(\dot{q}_{a}\mp i\sqrt{-2k}q_{a})$
$\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{2}}$ | $I_{ij}=(b_{0}+b_{1}t+b_{2}t^{2})\left(\dot{q}_{i}\dot{q}_{j}-2\omega q_{i}q_{j}\right)-(b_{1}+2b_{2}t)q_{(i}\dot{q}_{j)}+b_{2}q_{i}q_{j}$
$\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}$ |  $L_{ij}=\left(\frac{2}{c_{0}}\right)^{1/2}\left(I_{41i}I_{42j}-I_{41j}I_{42i}\right)$, $\Lambda_{ij}=f(t)\left(\dot{q}_{i}\dot{q}_{j}-2\omega q_{i}q_{j}\right)-\dot{f}q_{(i}\dot{q}_{j)}+\frac{\ddot{f}}{2}q_{i}q_{j}=I_{41i}I_{41j}+I_{42i}I_{42j}$, $I_{41i}=\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\sin\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\cos\theta$, $I_{42i}=-\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\cos\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\sin\theta$ where $\theta=\left(\frac{c_{0}}{2}\right)^{1/2}\int f^{-1}dt$
$\frac{\ddot{g}}{2g(t)}$ | $I_{4i}=g(t)\dot{q}_{i}-\dot{g}q_{i}$
$1$ | $k$ | $R_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r}q_{i}$
$\frac{k}{b_{0}+b_{1}t}$ |  $E_{2}=(b_{0}+b_{1}t)^{2}\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)q^{i}\dot{q}_{i}+\frac{b_{1}^{2}r^{2}}{2}$, $A_{i}=(b_{0}+b_{1}t)\tilde{R}_{i}+b_{1}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right)$ where $\tilde{R}_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r(b_{0}+b_{1}t)}q_{i}$
$\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$ | $E_{3}=(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}$
$2$ | $k$ |  $I_{1}=-H_{2}t^{2}+t(\dot{q}^{i}q_{i})-\frac{r^{2}}{2}$, $I_{2}=-H_{2}t+\frac{1}{2}(\dot{q}^{i}q_{i})$

####  12.14 Integrating the dynamical equations

In this section, we use the independent LFIs $I_{41i}$ and $I_{42i}$ to
integrate the dynamical equations of the special class of 3d time-dependent
oscillators ($\nu=-2$) defined in section 12.12 with $\omega(t)$ given by
(12.116). We also use the FIs $L_{i}$, $E_{2}$, $A_{i}$ to integrate the time-
dependent Kepler potential ($\nu=1$) with $\omega(t)=\frac{k}{b_{0}+b_{1}t}$,
where $kb_{1}\neq 0$ (see section 12.10.1).

#####  12.14.1 The 3d time-dependent oscillator with $\omega(t)$ given by
(12.116)

Using the LFIs (12.118) and (12.119), we find

$q_{i}(t)=\left(\frac{2}{c_{0}}\right)^{1/2}f^{1/2}\Big{(}I_{41i}\sin\theta-
I_{42i}\cos\theta\Big{)}$ (12.128)

where $I_{41i}$ and $I_{42i}$, $i=1,2,3$, are arbitrary constants (real or
imaginary), and $\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int f^{-1}dt$.

The solution (12.128) coincides with the solution (52) of [175].

In the case of the 1d time-dependent oscillator, if we set
$2\omega(t)=-\psi^{2}(t)$, $c_{0}=2$ and $f(t)=\rho^{2}(t)$, equation (12.46)
and the defining relation (12.116) for $\omega(t)$ become:

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-\psi^{2}x$ (12.129)
$\displaystyle\ddot{\rho}$ $\displaystyle=$
$\displaystyle-\psi^{2}\rho+\rho^{-3}.$ (12.130)

The LFIs (12.126) and (12.127) become:

$\displaystyle I_{41}$ $\displaystyle=$
$\displaystyle\rho^{-1}x\sin\theta+\left(\rho\dot{x}-x\dot{\rho}\right)\cos\theta$
(12.131) $\displaystyle I_{42}$ $\displaystyle=$
$\displaystyle-\rho^{-1}x\cos\theta+\left(\rho\dot{x}-x\dot{\rho}\right)\sin\theta.$
(12.132)

The general solution (12.128) is

$x(t)=\rho(t)\Big{(}I_{41}\sin\theta-I_{42}\cos\theta\Big{)}$ (12.133)

where $\dot{\theta}=\rho^{-2}$ and $\rho(t)$ is a given non-zero function
which defines $\psi(t)$ through (12.130). This is the 1d solution (9) in
[175].

#####  12.14.2 The solution of the time-dependent Kepler potential with
$\omega_{2K}(t)=\frac{k}{b_{0}+b_{1}t}$, where $kb_{1}\neq 0$

In section 12.10.1, it is shown that this system admits the following FIs:

$L_{1}=y\dot{z}-z\dot{y},\enskip L_{2}=z\dot{x}-x\dot{z},\enskip
L_{3}=x\dot{y}-y\dot{x}$
$E_{2}=(b_{0}+b_{1}t)^{2}\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)q^{i}\dot{q}_{i}+\frac{b_{1}^{2}r^{2}}{2}$
$A_{i}=(b_{0}+b_{1}t)\tilde{R}_{i}+b_{1}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right)$

where
$\tilde{R}_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r(b_{0}+b_{1}t)}q_{i}$.
The components of the generalized Runge-Lenz vector are written as:

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}t)(\dot{y}L_{3}-\dot{z}L_{2})+b_{1}\left(zL_{2}-yL_{3}\right)-\frac{k}{r}x$
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}t)(\dot{z}L_{1}-\dot{x}L_{3})+b_{1}\left(xL_{3}-zL_{1}\right)-\frac{k}{r}y$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}t)(\dot{x}L_{2}-\dot{y}L_{1})+b_{1}\left(yL_{1}-xL_{2}\right)-\frac{k}{r}z.$

Since the angular momentum is a FI, the motion is on a plane. We choose,
without loss of generality, the plane $z=0$ and on that the polar coordinates
$x=r\cos\theta$ and $y=r\sin\theta$. Then,

$L_{1}=L_{2}=0,\enskip L_{3}=r^{2}\dot{\theta},\enskip
E_{2}=(b_{0}+b_{1}t)^{2}\left[\frac{\dot{r}^{2}+r^{2}\dot{\theta}^{2}}{2}-\frac{k}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)r\dot{r}+\frac{b_{1}^{2}r^{2}}{2}$
$A_{1}=L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\sin\theta+\Big{[}(b_{0}+b_{1}t)L_{3}r\dot{\theta}-k\Big{]}\cos\theta$
$A_{2}=-L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\cos\theta+\Big{[}(b_{0}+b_{1}t)L_{3}r\dot{\theta}-k\Big{]}\sin\theta,\enskip
A_{3}=0.$

Using the relation $\dot{\theta}=\frac{L_{3}}{r^{2}}$ to replace
$\dot{\theta}$, the above relations are written as:

$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}t)^{2}\left[\frac{\dot{r}^{2}}{2}+\frac{L_{3}^{2}}{2r^{2}}-\frac{k}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)r\dot{r}+\frac{b_{1}^{2}r^{2}}{2}$
(12.134) $\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\sin\theta+\Big{[}(b_{0}+b_{1}t)\frac{L_{3}^{2}}{r}-k\Big{]}\cos\theta$
(12.135) $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle-
L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\cos\theta+\Big{[}(b_{0}+b_{1}t)\frac{L_{3}^{2}}{r}-k\Big{]}\sin\theta.$
(12.136)

By multiplying equation (12.135) with $\cos\theta$ and (12.136) with
$\sin\theta$, we find that

$\frac{1}{r}=\frac{k}{L_{3}^{2}(b_{0}+b_{1}t)}\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)\implies
r=\frac{L_{3}^{2}(b_{0}+b_{1}t)}{k\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)}$
(12.137)

where $k_{1}\equiv\frac{A_{1}}{k}$ and $k_{2}\equiv\frac{A_{2}}{k}$.

Applying the transformation $k_{1}=\alpha\cos\beta$ and
$k_{2}=\alpha\sin\beta$, equation (12.137) is written (see also section 5 in
[76])

$\frac{1}{r}=\frac{\omega_{2K}}{L_{3}^{2}}\Big{[}1+\alpha\cos\left(\theta-\beta\right)\Big{]}\implies
r=\frac{L_{3}^{2}\omega_{2K}^{-1}}{1+\alpha\cos\left(\theta-\beta\right)}$
(12.138)

which for $\omega_{2K}(t)=const$ (standard Kepler problem) reduces to the
analytical equation of a conic section in polar coordinates. In that case,
$\alpha$ is the eccentricity.

It is also worthwhile to mention that the relation (12.86) becomes
$2E_{2}L_{3}^{2}=k^{2}(\alpha^{2}-1)$.

Moreover, equation (12.134) gives

$\left[\frac{d}{dt}\left(\frac{r}{b_{0}+b_{1}t}\right)\right]^{2}=-2(b_{0}+b_{1}t)^{-2}\left[\frac{L_{3}^{2}}{2r^{2}}-\frac{k}{r(b_{0}+b_{1}t)}-\frac{E_{2}}{(b_{0}+b_{1}t)^{2}}\right].$

Finally, in the polar plane, the equations of motion (12.46) for $\nu=1$
become:

$\displaystyle\ddot{r}-r\dot{\theta}^{2}+\frac{\omega_{2K}}{r^{2}}$
$\displaystyle=$ $\displaystyle 0$ (12.139) $\displaystyle
r\ddot{\theta}+2\dot{r}\dot{\theta}$ $\displaystyle=$ $\displaystyle 0.$
(12.140)

Equation (12.140) implies the FI of the angular momentum
$L_{3}=r^{2}\dot{\theta}$. It can be easily checked that the solution (12.137)
satisfies equation (12.139) by replacing $\ddot{\theta}$ from (12.140) and
$\dot{\theta}$ with $\frac{L_{3}}{r^{2}}$. The solution (12.137) into the LFI
$L_{3}$ gives

$\int\frac{k^{2}dt}{L_{3}^{3}(b_{0}+b_{1}t)^{2}}=\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}\implies\frac{k}{L_{3}^{2}(b_{0}+b_{1}t)}=-\frac{b_{1}L_{3}}{k}\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}.$
(12.141)

Substituting (12.141) in (12.137), we obtain the trajectory

$\frac{1}{r}=-\frac{b_{1}L_{3}}{k}\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}$
(12.142)

which coincides with eq. (5.17) in [76].

####  12.15 A class of 1d non-linear time-dependent equations

In this section, we use the well-known result [177] that the non-linear
dynamical system

$\ddot{q}^{a}=-\Gamma^{a}_{bc}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)+\phi(t)\dot{q}^{a}$
(12.143)

is equivalent to the dynamical system (without damping term)

$\frac{d^{2}q^{a}}{ds^{2}}=-\Gamma^{a}_{bc}\frac{dq^{b}}{ds}\frac{dq^{c}}{ds}-\bar{\omega}(s)Q^{a}(q)$
(12.144)

where $\phi(t)$ is an arbitrary function such that

$s(t)=\int
e^{\int\phi(t)dt}dt,\enskip\bar{\omega}(s)=\omega(t(s))\left(\frac{dt}{ds}\right)^{2}\iff\omega(t)=\bar{\omega}(s(t))e^{2\int\phi(t)dt}.$
(12.145)

We apply this result to the following problem:

_Consider the second order ODE_

$\ddot{x}=-\omega(t)x^{\mu}+\phi(t)\dot{x}$ (12.146)

_where the constant $\mu\neq-1$, and determine the relation between the
functions $\omega(t)$ and $\phi(t)$ for which the ODE (12.146) admits a QFI;
therefore, it is integrable._

This problem has been considered previously in [184, 185] (see eq. (28a) in
[184] and eq. (17) in [185]) and has been answered partially using different
methods. In [184], the author used the Hamiltonian formalism, where one looks
for a canonical transformation to bring the Hamiltonian in a time-separable
form; whereas in [185] the author used a direct method for constructing FIs by
multiplying the equation with an integrating factor. In [185], it is shown
that both methods are equivalent and that the results of [185] generalize
those of [184]. In the following, we shall generalize the results of [185]; in
addition, we discuss a number of applications.

Equation (12.146) is equivalent to equation

$\frac{d^{2}x}{ds^{2}}=-\bar{\omega}(s)x^{\mu},\enskip\mu\neq-1$ (12.147)

where the function $\bar{\omega}(s)$ is given by (12.145).

Replacing with $Q^{1}=x^{\mu}$ in the system of equations (12.3) - (12.8), we
find that555 In 1d Euclidean space, the KT condition (12.3) gives $K_{11,1}=0$
$\implies K_{11}=K_{11}(s)$, that is, the KT is an arbitrary function of $s$.
$K_{11}=K_{11}(s)$ and the following conditions:

$\displaystyle K_{1}(s,x)$ $\displaystyle=-\frac{dK_{11}}{ds}x+b_{1}(s)$
(12.148) $\displaystyle K(s,x)$
$\displaystyle=2\bar{\omega}K_{11}\frac{x^{\mu+1}}{\mu+1}+\frac{d^{2}K_{11}}{ds^{2}}\frac{x^{2}}{2}-\frac{db_{1}}{ds}x+b_{2}(s)$
(12.149) $\displaystyle 0$
$\displaystyle=\left(\frac{2\frac{d\bar{\omega}}{ds}K_{11}}{\mu+1}+\frac{2\bar{\omega}\frac{dK_{11}}{ds}}{\mu+1}+\bar{\omega}\frac{dK_{11}}{ds}\right)x^{\mu+1}-\bar{\omega}b_{1}x^{\mu}+\frac{d^{3}K_{11}}{ds^{3}}\frac{x^{2}}{2}-\frac{d^{2}b_{1}}{ds^{2}}x+\frac{db_{2}}{ds}$
(12.150)

where $b_{1}(s)$ and $b_{2}(s)$ are arbitrary functions. Then, the general QFI
(12.2) becomes

$I=K_{11}(s)\left(\frac{dx}{ds}\right)^{2}+K_{1}(s,x)\frac{dx}{ds}+K(s,x).$
(12.151)

We consider the solution of the system (12.148) - (12.150) for various values
of $\mu$.

As it will be shown, for $\mu\neq-1$, there results a family of ‘frequencies’
$\bar{\omega}(s)$ parameterized with constants; whereas, for the specific
values $\mu=0,1,2$, there results a family of ‘frequencies’ $\bar{\omega}(s)$
parameterized with functions.

1) Case $\mu=0$.

We find the QFI

$I=K_{11}\left(\frac{dx}{ds}\right)^{2}-\frac{dK_{11}}{ds}x\frac{dx}{ds}+b_{1}(s)\frac{dx}{ds}+c_{3}x^{2}+2\bar{\omega}(s)K_{11}x-\frac{db_{1}}{ds}x+\int
b_{1}(s)\bar{\omega}(s)ds$ (12.152)

where $K_{11}=c_{1}+c_{2}s+c_{3}s^{2}$, $c_{1},c_{2},c_{3}$ are arbitrary
constants, and the functions $b_{1}(s),\bar{\omega}(s)$ satisfy the condition

$\frac{d^{2}b_{1}}{ds^{2}}=2\frac{d\bar{\omega}}{ds}K_{11}+3\bar{\omega}\frac{dK_{11}}{ds}.$
(12.153)

Using the transformation (12.145), equations (12.152) and (12.153) become,
respectively, as follows:

$\displaystyle I$ $\displaystyle=$ $\displaystyle\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}\dot{x}^{2}-\left[c_{2}+2c_{3}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}x\dot{x}+$ (12.154)
$\displaystyle+b_{1}(s(t))e^{-\int\phi(t)dt}\dot{x}+c_{3}x^{2}+2\omega(t)\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}x-$
$\displaystyle-\dot{b}_{1}e^{-\int\phi(t)dt}x+\int
b_{1}(s(t))\omega(t)e^{-\int\phi(t)dt}dt$

and

$\displaystyle\ddot{b}_{1}-\phi\dot{b}_{1}$ $\displaystyle=$ $\displaystyle
2e^{-\int\phi(t)dt}\left(\dot{\omega}-2\phi\omega\right)\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int e^{\int\phi(t)dt}dt\right)^{2}\right]+$
(12.155) $\displaystyle+3\omega\left[c_{2}+2c_{3}\int
e^{\int\phi(t)dt}dt\right].$

2) Case $\mu=1$.

We derive again the results of the time-dependent oscillator (see Table
LABEL:T2 for $\nu=-2$) in one dimension. Using the transformation (12.145), we
deduce that the original equation

$\ddot{x}=-\omega(t)x+\phi(t)\dot{x}$ (12.156)

for the frequency

$\omega(t)=-\rho^{-1}\ddot{\rho}+\phi(\ln\rho)^{\cdot}+\rho^{-4}e^{2\int\phi(t)dt}$
(12.157)

admits the general solution

$x(t)=\rho(t)\left(A\sin\theta+B\cos\theta\right)$ (12.158)

where $\rho(t)\equiv\rho(s(t))$ and
$\theta(s(t))=\int\rho^{-2}(t)e^{\int\phi(t)dt}dt$.

3) Case $\mu=2$.

We find the function $\bar{\omega}=K_{11}^{-5/2}$ and the QFI

$I=K_{11}(s)\left(\frac{dx}{ds}\right)^{2}-\frac{dK_{11}}{ds}x\frac{dx}{ds}+(c_{4}+c_{5}s)\frac{dx}{ds}+\frac{2}{3}K_{11}^{-3/2}x^{3}+\frac{d^{2}K_{11}}{ds^{2}}\frac{x^{2}}{2}-c_{5}x$
(12.159)

where $c_{4},c_{5}$ are arbitrary constants and the function $K_{11}(s)$ is
given by

$\frac{d^{3}K_{11}}{ds^{3}}=2(c_{4}+c_{5}s)K_{11}^{-5/2}.$ (12.160)

Using the transformation (12.145), the above results become:

$\omega(t)=K_{11}^{-5/2}e^{2\int\phi(t)dt}$ (12.161) $\displaystyle I$
$\displaystyle=$ $\displaystyle
K_{11}e^{-2\int\phi(t)dt}\dot{x}^{2}-\dot{K}_{11}e^{-2\int\phi(t)dt}x\dot{x}+\left[c_{4}+c_{5}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}\dot{x}+\frac{2}{3}K_{11}^{-3/2}x^{3}+$
(12.162)
$\displaystyle+\left(\ddot{K}_{11}-\phi\dot{K}_{11}\right)e^{-2\int\phi(t)dt}\frac{x^{2}}{2}-c_{5}x$

and

$\dddot{K}_{11}-3\phi\ddot{K}_{11}-\dot{\phi}\dot{K}_{11}+2\phi^{2}\dot{K}_{11}=2\left[c_{4}+c_{5}\int
e^{\int\phi(t)dt}dt\right]e^{3\int\phi(t)dt}K_{11}^{-5/2}$ (12.163)

where the function $K_{11}=K_{11}(s(t))$.

We note that for $\mu=2$ equation (12.146), or to be more specific its
equivalent (12.147), arises in the solution of Einstein field equations when
the gravitational field is spherically symmetric and the matter source is a
shear-free perfect fluid (see e.g. [135, 186, 187, 188, 189, 190]).

4) Case $\mu\neq-1$.

In this case $b_{1}=b_{2}=0$, $K_{11}=c_{1}+c_{2}s+c_{3}s^{2}$ and
$\bar{\omega}(s)=(c_{1}+c_{2}s+c_{3}s^{2})^{-\frac{\mu+3}{2}}$, where
$c_{1},c_{2},c_{3}$ are arbitrary constants.

The QFI (12.151) becomes

$I=(c_{1}+c_{2}s+c_{3}s^{2})\left(\frac{dx}{ds}\right)^{2}-(c_{2}+2c_{3}s)x\frac{dx}{ds}+\frac{2}{\mu+1}(c_{1}+c_{2}s+c_{3}s^{2})^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}$
(12.164)

and the function

$\bar{\omega}(s)=(c_{1}+c_{2}s+c_{3}s^{2})^{-\frac{\mu+3}{2}}.$ (12.165)

It can be checked that (12.164) and (12.165) for $\mu=0,1,2$ give results
compatible with the ones we found for these values of $\mu$.

Using the transformation (12.145), we deduce that the original system (12.146)
is integrable iff the functions $\omega(t),\phi(t)$ are related as follows:

$\omega(t)=\left[c_{1}+c_{2}\int e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]^{-\frac{\mu+3}{2}}e^{2\int\phi(t)dt}.$
(12.166)

In this case, the associated QFI (12.164) is

$\displaystyle I$ $\displaystyle=$ $\displaystyle\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}\dot{x}^{2}-\left[c_{2}+2c_{3}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}x\dot{x}+$ (12.167)
$\displaystyle+\frac{2}{\mu+1}\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}.$

These expressions generalize the ones given in [185]. Indeed, if we introduce
the notation $\omega(t)\equiv\alpha(t)$ and $\phi(t)\equiv-\beta(t)$, then
equations (12.166) and (12.167) for $c_{3}=0$ become eqs. (25) and (26) of
[185].

#####  12.15.1 The generalized Lane-Emden equation

Consider the 1d generalized Lane-Emden equation (see eq. (6) in [191])

$\ddot{x}=-\omega(t)x^{\mu}-\frac{k}{t}\dot{x}$ (12.168)

where $k$ is an arbitrary constant. This equation is well-known in the
literature because of its _many applications in astrophysical problems_ (see
citations in [191]). In general, to find explicit analytic solutions of
equation (12.168) is a major task. For example, such solutions have been found
only for the special values $\mu=0,1,5$, in the case that the function
$\omega(t)=1$ and the constant $k=2$. New exact solutions, or at least the
Liouville integrability, of equation (12.168) are guaranteed, if we find a way
to determine its FIs. We see that equation (12.168) is a subcase of the
original equation (12.146) for $\phi(t)=-\frac{k}{t}$; therefore, we can apply
the results found earlier in section 12.15.

In what follows, we discuss only the fourth case where $\mu\neq-1$ in order to
compare our results with those found in666In [191], the authors used point
Noether symmetries and Noether’s Theorem. Table 1 of [191]. In particular, for
$\phi(t)=-\frac{k}{t}$ the function (12.166) and the associated QFI (12.167)
become:

$\omega(t)=t^{-2k}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)^{-\frac{\mu+3}{2}}$
(12.169)

and

$I=t^{2k}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)\dot{x}^{2}-t^{k}\left(c_{2}+2c_{3}M\right)x\dot{x}+\frac{2}{\mu+1}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}$
(12.170)

where the function $M(t)=\int t^{-k}dt$.

Concerning the form of the function $M(t)$, there are two cases to be
considered: a) $k=1$, and b) $k\neq 1$.

a) Case $k=1$.

We have $M=\ln t$. Then, equations (12.169) and (12.170) become:

$\omega(t)=t^{-2}\left[c_{1}+c_{2}\ln t+c_{3}(\ln
t)^{2}\right]^{-\frac{\mu+3}{2}}$ (12.171)

and

$I=t^{2}\left[c_{1}+c_{2}\ln t+c_{3}(\ln
t)^{2}\right]\dot{x}^{2}-t\left(c_{2}+2c_{3}\ln
t\right)x\dot{x}+\frac{2}{\mu+1}\left[c_{1}+c_{2}\ln t+c_{3}(\ln
t)^{2}\right]^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}.$ (12.172)

We consider the following subcases:

-$c_{2}=c_{3}=0$ and $c_{1}\neq 0$.

Equations (12.171) and (12.172) give, respectively, the function
$\omega(t)=At^{-2}$ and the QFI (divide $I$ with $2c_{1}$)
$I=\frac{t^{2}}{2}\dot{x}^{2}+\frac{A}{\mu+1}x^{\mu+1}$ where the constant
$A=c_{1}^{-\frac{\mu+3}{2}}$. This is the Case 5 in Table 1 of [191].

\- $c_{1}=c_{3}=0$ and $c_{2}\neq 0$.

Equations (12.171) and (12.172) give, respectively, the function
$\omega(t)=At^{-2}(\ln t)^{-\frac{\mu+3}{2}}$ and the QFI (divide $I$ with
$2c_{2}$) $I=\frac{1}{2}t^{2}(\ln
t)\dot{x}^{2}-\frac{t}{2}x\dot{x}+\frac{A}{\mu+1}(\ln
t)^{-\frac{\mu+1}{2}}x^{\mu+1}$ where the constant
$A=c_{2}^{-\frac{\mu+3}{2}}$. This is the Case 6 in Table 1 of [191].

\- $c_{1}=c_{2}=0$ and $c_{3}\neq 0$.

Equations (12.171) and (12.172) give, respectively, the function
$\omega(t)=At^{-2}(\ln t)^{-\mu-3}$ and the QFI (divide $I$ with $2c_{3}$)
$I=\frac{1}{2}(t\ln t)^{2}\dot{x}^{2}-t(\ln t)x\dot{x}+\frac{A}{\mu+1}(\ln
t)^{-\mu-1}x^{\mu+1}+\frac{x^{2}}{2}$ where the constant
$A=c_{3}^{-\frac{\mu+3}{2}}$. This is the Case 7 in Table 1 of [191].

b) Case $k\neq 1$.

We have $M=\frac{t^{1-k}}{1-k}$. Then, equations (12.169) and (12.170) become,
respectively, as follows:

$\omega(t)=t^{-2k}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]^{-\frac{\mu+3}{2}}$
(12.173)

and

$\displaystyle I$ $\displaystyle=$ $\displaystyle
t^{2k}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]\dot{x}^{2}-t^{k}\left(c_{2}+\frac{2c_{3}}{1-k}t^{1-k}\right)x\dot{x}+$
(12.174)
$\displaystyle+\frac{2}{\mu+1}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}.$

We consider the following subcases:

\- $c_{2}=c_{3}=0$ and $c_{1}\neq 0$.

Equations (12.173) and (12.174) give, respectively, the function
$\omega(t)=At^{-2k}$ and the QFI (divide $I$ with $2c_{1}$)
$I=\frac{t^{2k}}{2}\dot{x}^{2}+\frac{A}{\mu+1}x^{\mu+1}$ where the constant
$A=c_{1}^{-\frac{\mu+3}{2}}$. This is the Case 2 in Table 1 of [191].

\- $c_{1}=c_{3}=0$ and $c_{2}\neq 0$.

Equations (12.173) and (12.174) give, respectively, the function
$\omega(t)=At^{\frac{1}{2}(k\mu-k-\mu-3)}$ and the QFI (multiply $I$ with
$\frac{1-k}{c_{2}}$)
$I=t^{k+1}\dot{x}^{2}+(k-1)t^{k}x\dot{x}+\frac{2A}{\mu+1}t^{\frac{1}{2}(\mu+1)(k-1)}x^{\mu+1}$
where the constant $A=\left(\frac{c_{2}}{1-k}\right)^{-\frac{\mu+3}{2}}$. This
is the Case 3 in Table 1 of [191].

We note also that for $k=\frac{\mu+3}{\mu-1}$ where $\mu\neq 1$ the function
$\omega(t)=A=const$. This reproduces the first subcase of Case 1 in Table 1 of
[191], which is the Case 5.1 of [192].

\- $c_{1}=c_{2}=0$ and $c_{3}\neq 0$.

Equations (12.173) and (12.174) give, respectively, the function
$\omega(t)=At^{k\mu+k-\mu-3}$ and the QFI (multiply $I$ with
$\frac{(1-k)^{2}}{2c_{3}}$)
$I=\frac{t^{2}}{2}\dot{x}^{2}+(k-1)tx\dot{x}+\frac{A}{\mu+1}t^{(\mu+1)(k-1)}x^{\mu+1}+\frac{1}{2}(k-1)^{2}x^{2}$
where the constant $A=\left(\frac{1-k}{\sqrt{c_{3}}}\right)^{\mu+3}$. This is
the Case 4 in Table 1 of [191].

We note also that for $k=\frac{\mu+3}{\mu+1}$ the function
$\omega(t)=A=const$. This recovers the second subcase of Case 1 in Table 1 of
[191], which is the Case 5.2 of [192].

We conclude that the seven cases 1-7 found in Table 1 of [191] are just
subcases of the above two general cases a) and b). To compare with these
results, one may adopt the notation: $\omega=f$, $k=n$ and $\mu=p$.

###  Chapter 13 New conservation laws and exact cosmological solutions in
Brans-Dicke cosmology with an extra scalar field

####  13.1 Introduction

The detailed analysis of the recent cosmological observations indicates that
the universe has been through two accelerating phases [193, 194, 195, 196].
The current acceleration era is assumed to be driven by an unknown source
known as dark energy whose main characteristic is the negative pressure, which
provides an anti-gravity effect [197]. On the other hand, the early-universe
acceleration era, known as inflation, is described by a scalar field, the
inflaton, which is used to explain the homogeneity and isotropy of the present
universe. In particular, this scalar field dominates the dynamics and explains
the expansion era [198, 199]. Nevertheless, the scalar field inflationary
models are mainly defined on homogeneous spacetimes, or on background spaces
with small inhomogeneities [200, 201]. In [202], it was found that the
presence of a positive cosmological constant in Bianchi cosmologies leads to
expanding Bianchi spacetimes, evolving towards the de Sitter universe. That
was the first result to support the cosmic ‘no-hair’ conjecture [203, 204].
This latter conjecture states that all expanding universes with a positive
cosmological constant admit as asymptotic solution the de Sitter universe. The
necessity of the de Sitter expansion is that it provides a rapid expansion for
the size of the universe such that the latter, effectively, loses its memory
on the initial conditions, which implies that the de Sitter expansion solves
the ‘flatness’, ‘horizon’ and monopole problem [205, 206].

In the literature, scalar fields have been introduced in the gravitational
theory in various ways. The simplest scalar field model is the quintessence
model, which consists of a scalar field minimally coupled to gravity [207,
208]. Another family of scalar fields are those which belong to the scalar-
tensor theory. In this theory, the scalar field is non-minimally coupled to
gravity, which makes it essential for the physical state of the theory.
Another important characteristic of the scalar-tensor theories is that they
are in consistence with Mach’s principle. The most common scalar-tensor theory
is the Brans-Dicke theory [209] which is considered in this study. For other
scalar-tensor theories and generalizations, we refer the reader to [210, 211,
212, 213, 214, 215, 216, 217] and references therein.

The Einstein field equations of general relativity are a set of ten nonlinear
second-order PDEs with independent variables the spacetime coordinates and
dependent variables the components of the metric tensor. However, by assuming
specific forms for the metric tensor and the existence of collineations, the
field equations are simplified by reducing the number of the independent
variables (see e.g. [93, 218, 219, 220, 221] and references therein).
According to the cosmological principle, in large scale the universe is
assumed to be homogeneous, isotropic, and spatially flat. This implies that
the background space is described by the Friedmann - Lemaître - Robertson -
Walker (FLRW) spacetime. This spacetime is characterized by the scale factor,
which defines the radius of the 3d Euclidean space. Since General Relativity
is a second order theory, the field equations involve second order derivatives
of the scale factor. For simple cosmological fluids, such are the ideal gas or
the cosmological constant, the field equations can be solved explicitly [222].
However, when additional degrees of freedom are introduced, such as a scalar
field, the field equations cannot be solved with the use of closed-form
functions and techniques of analytic mechanics, and one looks for FIs which
establish their (Liouville) integrability [223, 224, 225, 226]. In the
following, we shall determine FIs (i.e. conservation laws) of the field
equations, by applying the direct method of chapter 12.

In this chapter, we consider a cosmological model in which the gravitational
action integral is that of Brans-Dicke theory with an additional scalar field
minimally coupled to gravity [227, 228]. This two-scalar field model belongs
to the family of multi-scalar field models, which have been used as unified
dark energy models [229, 230, 231] or as alternative models for the
description of the acceleration phases of the universe [232, 233, 234, 235].
Indeed, multifield inflationary models provide an alternative mechanism for
the description of the early acceleration phase of the universe. The mechanism
for the end of the inflation is much more simple. Specifically, the scalar
fields at the beginning, and at the end, of the inflation are not necessary
the same. Thus, this can lead to different number of e-folds and affect the
curvature perturbations [236, 237]. The latter, in the non-adiabatic
perturbations, can provide detectable non-Gaussianities in the power spectrum
[238]. As far as the late-time acceleration phase, multifield cosmological
models have been introduced to describe dark energy models with varying
equation of state parameter, which can cross the phantom divide line without
the appearance of ghosts [238]. Such models can solve the Hubble-tension
problem [233]. Furthermore, multi-scalar field models can attribute the
additional degrees of freedom provided by the alternative theories of gravity
[239, 240, 241].

####  13.2 Cosmological model

For the gravitational action integral, we consider that of Brans-Dicke scalar
field theory with an additional matter source. We have the following
expression [209, 210]:

$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}\phi
R-\frac{1}{2}\frac{\omega_{BD}}{\phi}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}+L_{\psi}\left(\psi,\psi_{;\mu}\right)\right]+S_{m}$
(13.1)

where $x^{k}$ are the spacetime coordinates, $g_{\mu\nu}$ is the metric,
$\phi\left(x^{\kappa}\right)$ denotes the Brans-Dicke scalar field and
$\omega_{BD}$ is the Brans-Dicke parameter. The action $S_{m}$ is assumed to
describe an ideal gas with constant equation of state parameter and the
Lagrangian function $L_{\psi}\left(\psi,\psi_{;\mu}\right)$ corresponds to the
second scalar field $\psi\left(x^{\kappa}\right)$, which is assumed to be that
of quintessence and minimally coupled to the Brans-Dicke scalar field. With
these assumptions the action integral (13.1) takes the following form:

$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}\phi
R-\frac{1}{2}\frac{\omega_{BD}}{\phi}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}-\frac{1}{2}g^{\mu\nu}\psi_{;\mu}\psi_{;\nu}-V\left(\psi\right)\right]+S_{m}.$
(13.2)

The gravitational field equations follow from the variation (see
$\delta_{0}$-variation in chapter 1) of the action integral (13.2) wrt the
metric tensor $g_{\mu\nu}$. They are

$G_{\mu\nu}=\frac{\omega_{BD}}{\phi^{2}}\left(\phi_{;\mu}\phi_{;\nu}-\frac{1}{2}g_{\mu\nu}g^{\kappa\lambda}\phi_{;\kappa}\phi_{;\lambda}\right)+\frac{1}{\phi}\left(\phi_{;\mu\nu}-g_{\mu\nu}g^{\kappa\lambda}\phi_{;\kappa\lambda}\right)+\frac{1}{\phi}T_{\mu\nu}$
(13.3)

where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor.
The total energy momentum tensor
$T_{\mu\nu}={}^{\psi}T_{\mu\nu}+{}^{m}T_{\mu\nu}$, where ${}^{m}T_{\mu\nu}$
corresponds to the ideal gas and ${}^{\psi}T_{\mu\nu}$ provides the
contribution of the field $\psi\left(x^{k}\right)$ in the field equations.

Concerning the equations of motion for the matter source and the two scalar
fields, we have ${}^{m}T_{\mu\nu;\sigma}g^{\mu\sigma}=0,$ while variation wrt
the fields $\phi\left(x^{\kappa}\right)$ and $\psi\left(x^{\kappa}\right)$
provides the second order PDEs:

$g^{\mu\nu}\phi_{;\mu\nu}-\frac{1}{2\phi}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}+\frac{\phi}{2\omega_{BD}}R=0$
(13.4) $g^{\mu\nu}\psi_{;\mu\nu}-\frac{dV}{d\psi}=0.$ (13.5)

We assume the background space to be the FLRW spacetime with line element

$ds^{2}=-dt^{2}+a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right)$ (13.6)

where $a(t)$ is the scale factor of the universe and
$H\left(t\right)=\frac{\dot{a}}{a}$ is the Hubble function. We note that a dot
indicates derivative wrt the cosmic time $t$.

From the line element (13.6), it follows that the Ricci scalar is
$R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right]$.
Replacing in the gravitational field equations (13.3), we obtain:

$3\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\omega_{BD}}{2}\left(\frac{\dot{\phi}}{\phi}\right)^{2}-3\frac{\dot{a}}{a}\frac{\dot{\phi}}{\phi}+\frac{\rho_{m}+\rho_{\psi}}{\phi}$
(13.7)
$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{\omega_{BD}}{2}\left(\frac{\dot{\phi}}{\phi}\right)^{2}-2\frac{\dot{a}}{a}\frac{\dot{\phi}}{\phi}-\frac{\ddot{\phi}}{\phi}-\frac{p_{m}+p_{\psi}}{\phi}$
(13.8)

where $\rho_{m}$ and $p_{m}$ are the mass density and the isotropic pressure,
respectively, of the ideal gas. For the quintessence field, we have:

$\rho_{\psi}=\frac{1}{2}\dot{\psi}^{2}+V\left(\psi\right),\enskip
p_{\psi}=\frac{1}{2}\dot{\psi}^{2}-V\left(\psi\right).$ (13.9)

For the equations of motion of the scalar fields, we find:

$\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}=\frac{\left(\rho_{m}-3p_{m}\right)+\left(\rho_{\psi}-3p_{\psi}\right)}{2\omega_{BD}+3}$
(13.10)

and

$\ddot{\psi}+3H\dot{\psi}+\frac{dV}{d\psi}=0.$ (13.11)

Finally, for the matter source, the continuity equation
${}^{m}T_{\mu\nu;\sigma}g^{\mu\sigma}=0$ reads

$\dot{\rho}_{m}+3\frac{\dot{a}}{a}\left(\rho_{m}+p_{m}\right)=0.$ (13.12)

For an ideal gas, the equation of state is $p_{m}=w_{m}\rho_{m}$, where
$w_{m}$ is an arbitrary constant. Substituting in equation (13.12), we find
the solution

$\rho_{m}=\rho_{m0}a^{-3\left(1+w_{m}\right)}$ (13.13)

where $\rho_{m0}$ is an arbitrary constant.

The system of the ODEs that should be solved consists of the differential
equations (13.7), (13.8), (13.10) and (13.11).

####  13.3 Exact cosmological solutions

We can use the direct results111 To be consistent with the notation of this
chapter, we change the notation of section 12.15 as follows: the power $\mu$
changes into $n$, the new parameter $s$ into $\tau$, and the function
$\phi(t)$ into $\Phi(t)$.  of section 12.15 as an alternative to the Euler-
Duarte-Moreira method of integrability of the anharmonic oscillator [242] in
order to find exact solutions in the modified Brans-Dicke (BD) theory.

Specifically, we consider the equation of motion for the quintessence scalar
field $\psi\left(t\right)$ with potential function
$V(\psi)=\frac{\psi^{n+1}}{n+1}$, where $n\neq-1$. Then, equation (13.11)
becomes

$\ddot{\psi}=-\psi^{n}-3\frac{\dot{a}}{a}\dot{\psi}$ (13.14)

which is a subcase of (12.146) for $\omega(t)=1$ and $\Phi(t)=-3(\ln
a)^{\cdot}$. Replacing in the transformation (12.145), we find that

$\tau(t)=\int a^{-3}(t)dt,\enskip\bar{\omega}(\tau(t))=a^{6}(t).$ (13.15)

In the new parameter $\tau$, equation (13.14) reads

$\psi^{\prime\prime}+a^{6}\psi^{n}=0$ (13.16)

where $\psi^{\prime}=\frac{d\psi\left(\tau\right)}{d\tau}$.

The line element of the background space becomes

$ds^{2}=-a^{6}\left(\tau\right)d\tau^{2}+a^{2}\left(\tau\right)\left(dx^{2}+dy^{2}+dz^{2}\right)$
(13.17)

which means that the rest of the field equations read:

$\displaystyle
6\phi\left(\frac{a^{\prime}}{a}\right)^{2}+6\frac{a^{\prime}}{a}\phi^{\prime}-\omega_{BD}\frac{\phi^{\prime
2}}{\phi}-(\psi^{\prime})^{2}-\frac{2}{n+1}a^{6}\psi^{n+1}$
$\displaystyle=2a^{6}\rho_{m}$ (13.18) $\displaystyle
4\phi\frac{a^{\prime\prime}}{a}-10\phi\left(\frac{a^{\prime}}{a}\right)^{2}-2\frac{a^{\prime}}{a}\phi^{\prime}+\omega_{BD}\frac{\left(\phi^{\prime}\right)^{2}}{\phi}+2\phi^{\prime\prime}+(\psi^{\prime})^{2}-\frac{2}{n+1}a^{6}\psi^{n+1}$
$\displaystyle=-2a^{6}p_{m}$ (13.19) $\displaystyle
6\phi\frac{a^{\prime\prime}}{a}-\omega_{BD}\left[2\phi^{\prime\prime}-\frac{(\phi^{\prime})^{2}}{\phi}\right]-12\phi\left(\frac{a^{\prime}}{a}\right)^{2}$
$\displaystyle=0.$ (13.20)

Next, we apply the results of section 12.15 for equation (13.16) and we
determine for several cases of the parameter $n$ the corresponding QFIs. The
resulting QFIs are expressed in terms of the scale factor $a(\tau)$ and the
other arbitrary functions of $\tau$ (i.e. $K_{11}(\tau)$, $b_{1}(\tau)$,
$b_{2}(\tau)$), which satisfy additional conditions. Solving these conditions,
whenever it is possible, we find a scale factor $a(\tau)$ for which equation
(13.16) is integrable. Replacing this scale factor in the original equation of
motion (13.14) of the quintessence field, we end up with a new integrable
second-order ODE for $\psi$, which, most times, can be solved using standard
methods (e.g. Lie symmetries) from the symmetries of differential equations.
As a final step, for the computation of exact solutions, we replace the
solutions $a(t)$ and $\psi(t)$ in the remaining equations (13.7) - (13.10),
and we determine the BD scalar field $\phi(t)$ and the BD parameter
$\omega_{BD}$.

#####  13.3.1 Case $n=0$

For $n=0$, the associated QFI (12.152) becomes

$I=K_{11}\left(\psi^{\prime}\right)^{2}-K_{11}^{\prime}\psi\psi^{\prime}+b_{1}(\tau)\psi^{\prime}+c_{3}\psi^{2}+2a^{6}K_{11}\psi-
b_{1}^{\prime}\psi+\int b_{1}(\tau)a^{6}d\tau$ (13.21)

where $K_{11}=c_{1}+c_{2}\tau+c_{3}\tau^{2}$, the parameters
$c_{1},c_{2},c_{3}$ are arbitrary constants and the functions
$b_{1}(\tau),a(\tau)$ satisfy the condition

$b_{1}^{\prime\prime}=12a^{5}a^{\prime}K_{11}+3a^{6}K_{11}^{\prime}.$ (13.22)

We note that for $b_{1}=0$, we find the results of section 13.3.4 below when
$n=0$.

#####  13.3.2 Case $n=1$

Using transformation (13.15), equation $\psi^{\prime\prime}=-a^{6}\psi$ admits
the solution

$\psi(\tau)=\rho(\tau)\left(A\sin\theta+B\cos\theta\right)$ (13.23)

where $\theta=\int\rho^{-2}d\tau$ and the functions $\rho(t(\tau)),a(t(\tau))$
satisfy the condition

$\rho^{\prime\prime}+\rho a^{6}-\rho^{-3}=0.$ (13.24)

#####  13.3.3 Case $n=2$

For $n=2$, we have $K_{11}=a^{-12/5}$ and the associated QFI (12.159) becomes

$I=a^{-12/5}\left(\psi^{\prime}\right)^{2}+\frac{12}{5}a^{-17/5}a^{\prime}\psi\psi^{\prime}+(c_{4}+c_{5}\tau)\psi^{\prime}+\frac{2}{3}a^{18/5}\psi^{3}+\frac{6}{5}\left[\frac{17}{5}a^{-22/5}(a^{\prime})^{2}-a^{-17/5}a^{\prime\prime}\right]\psi^{2}-c_{5}\psi$
(13.25)

where $c_{4},c_{5}$ are arbitrary constants and the function $a(t(\tau))\equiv
a(\tau)$ is given by

$a^{\prime\prime\prime}-\frac{51}{5}\frac{a^{\prime}}{a}a^{\prime\prime}+\frac{374}{25}\left(\frac{a^{\prime}}{a}\right)^{2}a^{\prime}+\frac{5}{6}(c_{4}+c_{5}\tau)a^{47/5}=0.$
(13.26)

Substituting the given functions $\omega(t)$ and $\Phi(t)$ in equations
(12.161) - (12.163), we find equivalently that

$a(t)=K_{11}^{-\frac{5}{12}}$ (13.27)

and

$\displaystyle I$
$\displaystyle=K_{11}^{-3/2}\dot{\psi}^{2}-K_{11}^{-5/2}\dot{K}_{11}\psi\dot{\psi}+\left(c_{4}+c_{5}\int
K_{11}^{5/4}dt\right)K_{11}^{-5/4}\dot{\psi}+\frac{2}{3}K_{11}^{-3/2}\psi^{3}+$
$\displaystyle\quad+\left[\ddot{K}_{11}-\frac{5}{4}\left(\ln
K_{11}\right)^{\cdot}\dot{K}_{11}\right]K_{11}^{-5/2}\frac{\psi^{2}}{2}-c_{5}\psi.$
(13.28)

where the function $K_{11}=K_{11}(t)$ is given by the ODE

$\dddot{K}_{11}-\frac{15}{4}\left(\ln
K_{11}\right)^{\cdot}\ddot{K}_{11}-\frac{5}{4}\left(\ln
K_{11}\right)^{\cdot\cdot}\dot{K}_{11}+\frac{25}{8}\frac{\dot{K}_{11}^{3}}{K_{11}^{2}}=2\left[c_{4}+c_{5}\int
K_{11}^{5/4}dt\right]K_{11}^{5/4}.$ (13.29)

Equation (13.14) becomes $\ddot{\psi}=-\psi^{2}+\frac{5}{4}\left(\ln
K_{11}\right)^{\cdot}\dot{\psi}$. We note that for $c_{4}=c_{5}=0$ we retrieve
the results of section 13.3.4 below for $n=2$.

In the special case with $c_{5}=0$, we find for equation (13.29) the special
solution $K_{11}\left(t\right)=k_{0}t^{-12}$ with constraint
$c_{4}k_{0}^{1/4}=-192$, where $k_{0}$ is an arbitrary constant. Moreover,
from equation (13.27), the scale factor is

$a(t)=K_{11}^{-\frac{5}{12}}=k_{0}^{-5/12}t^{5}.$ (13.30)

Therefore, the Klein-Gordon equation (13.14) becomes

$\ddot{\psi}+\frac{15}{t}\dot{\psi}+\psi^{2}=0.$ (13.31)

The latter equation can be solved by quadratures. In particular, it admits the
Lie symmetries: $\Gamma^{1}=\psi\partial_{\psi}-\frac{t}{2}\partial_{t}$ and
$\Gamma^{2}=\left(3\psi
t^{2}-48\right)\partial_{\psi}-\frac{t^{3}}{2}\partial_{t}$. Using the vector
field $\Gamma^{1}$, we find the reduced equation
$\frac{1}{2}\frac{df^{2}}{d\lambda}+2\lambda\frac{df}{d\lambda}+12f+\lambda^{2}=0$,
where222It can be easily checked that if we replace
$\frac{df}{d\lambda}=\frac{df}{dt}\frac{dt}{d\lambda}$ in the reduced form, we
obtain again equation (13.31). $f\left(\lambda\right)=t^{3}\dot{\psi}$ and
$\lambda=t^{2}\psi$. The latter ODE is an Abel equation of second type.
Moreover, if we assume that $\lambda$ is a constant, $\lambda=\lambda_{0}$,
then we find $\psi=\lambda_{0}t^{-2}$. By replacing this $\psi(t)$ in (13.31),
it follows that $\lambda_{0}=24$. Therefore, we end up with the solution
$\psi=\frac{24}{t^{2}}$. Let us now find, for this particular exact solution,
the complete solution for the gravitational field equations.

Replacing these results in the rest of the field equations for dust fluid
source, that is, $p_{m}=0$ and $\rho_{m}=\rho_{0}a^{-3}$ where $\rho_{0}$ is a
constant, the evolution equation for the BD field becomes

$\ddot{\phi}+\frac{15}{t}\dot{\phi}=\frac{1}{2\omega_{BD}+3}\left(\rho_{0}a^{-3}-\dot{\psi}^{2}+\frac{4}{3}\psi^{3}\right)$

which admits the general solution

$\phi(t)=-\frac{1}{2\omega_{BD}+3}\left(\frac{2016}{5}t^{-4}+\frac{\rho_{0}k_{0}^{5/4}}{13}t^{-13}\right)+\frac{k_{1}}{14}t^{-14}$

where $k_{1}$ is an arbitrary constant. Finally, by replacing this solution in
the constraint (13.7), it follows that (eq. (13.8) is satisfied identically)
$\omega_{BD}=-\frac{45}{16}$ and $k_{1}=\rho_{0}=0$.

We conclude that the gravitational field equations for this model, with the
use of the QFI for equation (13.14), admit the following exact solution:

$\omega=-\frac{45}{16},\enskip
a(t)=k_{0}^{-5/12}t^{5},\enskip\psi(t)=24t^{-2},\enskip\phi(t)=\frac{768}{5}t^{-4}$
(13.32)

with physical quantities $\rho_{m}=p_{m}=0$, $\rho_{\psi}=5760t^{-6}$ and
$p_{\psi}=-3456t^{-6}$.

For solution (13.32), transformation (13.15) gives

$\tau=-\frac{k_{0}^{5/4}}{14}t^{-14}\implies
t=\left(-14k_{0}^{-5/4}\right)^{-1/14}\tau^{-1/14}.$ (13.33)

Then, the transformed field equations (13.16) and (13.18) - (13.20) admit the
solution:

$\omega=-\frac{45}{16},\enskip
a=k_{0}^{-5/12}(-14k_{0}^{-5/4})^{-5/14}\tau^{-5/14},\enskip\psi=24(-14k_{0}^{-5/4})^{1/7}\tau^{1/7},\enskip\phi=\frac{768}{5}(-14k_{0}^{-5/4})^{2/7}\tau^{2/7}.$
(13.34)

#####  13.3.4 Case $n\neq-1$

In this case, the associated QFI (12.164) becomes

$I=(c_{1}+c_{2}\tau+c_{3}\tau^{2})\left(\psi^{\prime}\right)^{2}-(c_{2}+2c_{3}\tau)\psi\psi^{\prime}+\frac{2}{n+1}(c_{1}+c_{2}\tau+c_{3}\tau^{2})^{-\frac{n+1}{2}}\psi^{n+1}+c_{3}\psi^{2}$
(13.35)

and the function

$a(\tau)=(c_{1}+c_{2}\tau+c_{3}\tau^{2})^{-\frac{n+3}{12}}.$ (13.36)

Substituting the given functions $\omega(t)$ and $\Phi(t)$ in the relation
(12.166), we find equivalently that

$a^{6}(t)=\left[c_{1}+c_{2}\int a^{-3}(t)dt+c_{3}\left(\int
a^{-3}(t)dt\right)^{2}\right]^{-\frac{n+3}{2}}$ (13.37)

and the associated QFI (12.167) becomes

$\displaystyle I$ $\displaystyle=\left[c_{1}+c_{2}\int
a^{-3}(t)dt+c_{3}\left(\int
a^{-3}(t)dt\right)^{2}\right]a^{6}(t)\dot{\psi}^{2}-\left[c_{2}+2c_{3}\int
a^{-3}(t)dt\right]a^{3}(t)\psi\dot{\psi}+$
$\displaystyle\quad+\frac{2}{n+1}\left[c_{1}+c_{2}\int
a^{-3}(t)dt+c_{3}\left(\int
a^{-3}(t)dt\right)^{2}\right]^{-\frac{n+1}{2}}\psi^{n+1}+c_{3}\psi^{2}.$
(13.38)

We consider the following special cases for which equation (13.14) admits a
closed-form solution for $n\neq-3,1$. In the case $n=-3$, the spacetime is
that of Minkowski. Hence, we omit the analysis.

###### Subcase $\left|\tau\right|<<1$

For small values of $\left|\tau\right|$ (i.e. $c_{1}=c_{3}=0$) the scale
factor (13.36) is approximated as
$a\left(\tau\right)\simeq\tau^{-\frac{n+3}{12}}$; therefore, it follows

$a(t)=B_{0}(t-t_{0})^{\frac{n+3}{3(n-1)}}$ (13.39)

where $B_{0}=\left[-\frac{c_{2}(n-1)}{4}\right]^{\frac{n+3}{3(n-1)}}$ and
$t_{0}$ is an arbitrary constant.

For this asymptotic solution, the equation of motion (13.14) for the second
field $\psi$ becomes

$\ddot{\psi}=-\psi^{n}-\frac{n+3}{n-1}\frac{1}{t-t_{0}}\dot{\psi}.$ (13.40)

For the latter equation, the QFI (13.38) is

$I=\left[-\frac{c_{2}(n-1)}{4}\right]^{\frac{2(n+1)}{n-1}}(t-t_{0})^{\frac{2(n+1)}{n-1}}\left(\dot{\psi}^{2}+\frac{2}{n+1}\psi^{n+1}\right)-c_{2}\left[-\frac{c_{2}(n-1)}{4}\right]^{\frac{n+3}{n-1}}(t-t_{0})^{\frac{n+3}{n-1}}\psi\dot{\psi}.$
(13.41)

This QFI, which corresponds to the scale factor (13.39), together with the
results of the cases $n=0,1,2$ produces new solutions $\psi(t)$ which have not
found before.

Furthermore, for the scale factor (13.39) the closed-form solution of (13.40)
is found to be

$\psi\left(t\right)=\psi_{0}(t-t_{0})^{-\frac{2}{n-1}}~{},~{}\psi_{0}=\left(\frac{2}{n-1}\right)^{\frac{2}{n-1}}$
(13.42)

whereas for the BD field $\phi\left(t\right)$ it follows that
$n=3,~{}\phi(t)=\frac{\phi_{0}}{(t-t_{0})^{2}}$ and
$\omega_{BD}=-\frac{3}{2}$. However, this value for the BD parameter
$\omega_{BD}$ is not physically accepted. Hence, we do not have any closed-
form solution. In all discussion above, we have considered $\rho_{m}=0$.

###### Subcase $\left|\tau\right|\gg 1$

For large values of $\tau$ (i.e. $c_{1}=c_{2}=0$), the scale factor (13.36) is
approximated as $a(\tau)\simeq\tau^{-\frac{n+3}{6}}$. Therefore, the original
equation (13.37) becomes

$a^{-\frac{6}{n+3}}=c_{3}^{\frac{1}{2}}\int a^{-3}dt$ (13.43)

which implies (see eq. (31) of [227])

$a(t)=A_{0}(t-t_{0})^{\frac{n+3}{3(n+1)}}$ (13.44)

where $A_{0}=\left[-\frac{\sqrt{c_{3}}(n+1)}{2}\right]^{\frac{n+3}{3(n+1)}}$
and $t_{0}$ is an arbitrary constant. The scale factor (13.44) describes a
scaling solution, where the effective cosmological fluid is that of an ideal
gas with effective parameter333 It must hold that
$a\propto(t-t_{0})^{\frac{2}{3\left(1+w_{eff}\right)}}$. Therefore, by
equating with the power of (13.44), we find that
$\frac{2}{3\left(1+w_{eff}\right)}=\frac{n+3}{3(n+1)}\implies
w_{eff}=\frac{n-1}{n+3}.$ for the equation of state $w_{eff}=\frac{n-1}{n+3}.$
Furthermore, for $-3<n<-1$ and $-1<n<0$, the scale factor describes an
accelerated universe. For $-1<n<0$, we have $-1<w_{eff}<-\frac{1}{3}$; while
for $-3<n<-1$, the effective parameter crosses the phantom divide line, that
is, $w_{eff}<-1$.

For this asymptotic solution, the equation of motion (13.14) for the scalar
field $\psi$ becomes

$\ddot{\psi}=-\psi^{n}-\frac{n+3}{n+1}\frac{1}{t-t_{0}}\dot{\psi}$ (13.45)

and the corresponding QFI (13.38) is written as

$I=c_{3}\left[\frac{(n+1)(t-t_{0})}{2}\dot{\psi}+\psi\right]^{2}+\frac{c_{3}(n+1)}{2}(t-t_{0})^{2}\psi^{n+1}$
(13.46)

where $t\neq t_{0}$.

However, the system admits the closed-form solution (see eq. (32) of [227])

$\psi\left(t\right)=\psi_{0}\left(t-t_{0}\right)^{-\frac{2}{n-1}}$ (13.47)

where $\psi_{0}$ is given by the expression
$\psi_{0}=(-2)^{\frac{3}{n-1}}\left[\left(n+1\right)\left(n-1\right)^{2}\right]^{\frac{1}{1-n}}$.
Replacing in the remaining equations (13.7) - (13.10) for the BD field, we
find

$\phi\left(t\right)=\phi_{0}(t-t_{0})^{-\frac{4}{n-1}}$ (13.48)

where

$\displaystyle\phi_{0}$
$\displaystyle=\frac{(n-1)^{\frac{4}{1-n}}}{2(n+3)(2\omega_{BD}+3)}\left[(-2)^{\frac{3(n+1)}{n-1}}(n+1)^{\frac{n+1}{1-n}}-(-2)^{\frac{6}{n-1}}(n+1)^{\frac{n-3}{n-1}}\right]$
(13.49) $\displaystyle\omega_{BD}$
$\displaystyle=\frac{b_{1}-3b_{2}}{1+2b_{2}}.$ (13.50)

We note that we have assumed that there is not any other matter source, i.e.
$\rho_{m}=0$. The constants $b_{1}$ and $b_{2}$ are given by the relations:

$\displaystyle b_{1}$
$\displaystyle=\frac{(n+3)(n-1)}{2(n+1)}\left[\frac{(n+3)(n-1)}{12(n+1)}-1\right]$
(13.51) $\displaystyle b_{2}$
$\displaystyle=\frac{n+3}{4}\cdot\frac{2(-2)^{\frac{6}{n-1}}(n+1)^{\frac{2}{1-n}}+(-2)^{\frac{3(n+1)}{n-1}}(n+1)^{\frac{2n}{1-n}}}{(-2)^{\frac{3(n+1)}{n-1}}(n+1)^{\frac{n+1}{1-n}}-(-2)^{\frac{6}{n-1}}(n+1)^{\frac{n-3}{n-1}}}.$
(13.52)

In the following section, we perform a detailed study on the stability (see
chapter 4) of the latter closed-form solutions.

####  13.4 Stability of scaling solutions

The analysis of the stability properties (see chapter 4) of the exact
solutions provides us with important information about the evolution of the
background space on the asymptotic solutions. In particular, we can infer if
an exact solution is stable, which can be seen as a future attractor for the
original dynamical system. On the other hand, in the cases of unstable
solutions, the behavior of the asymptotic solution and its dynamics give us
results for the curvature and the dynamics of the metric space.

According to the methods in [207, 243, 244], we consider the second order ODE

$F(\ddot{\psi},\dot{\psi},\psi)=0$ (13.53)

where $\psi(t)$ is a smooth function, which admits a singular power law
solution

$\psi_{c}(t)=\psi_{0}t^{\beta}$ (13.54)

where $\phi_{0}$ and $\beta$ are arbitrary constants. To examine the stability
of the solution $\psi_{c}$, we introduce the logarithmic time $T$ through
$t=e^{T}$. Using this transformation, we compute

$\dot{\psi}=e^{-T}\psi^{\prime},\enskip\ddot{\psi}=e^{-2T}(\psi^{\prime\prime}-\psi^{\prime}),\enskip\psi_{c}\left(t(T)\right)=\psi_{0}e^{\beta
T},\enskip\frac{\psi_{c}^{\prime}}{\psi_{c}}=\beta$ (13.55)

where in this discussion $\psi^{\prime}\equiv\frac{d\psi}{dT}$.

We introduce, also, the dimensionless function

$u(T)=\frac{\psi(T)}{\psi_{c}(T)}$ (13.56)

and the stability analysis is translated into the analysis of the stability of
the equilibrium point $u=1$ of a transformed dynamical system with
$\psi(T)=\psi_{c}(T)u(T)$.

In this section, we use a similar procedure for analyzing the stability of the
scaling solutions obtained in section 13.3.4.

#####  13.4.1 Case $\left|\tau\right|\gg 1$

For the analysis of the solution (13.47) of equation (13.45), we set $t_{0}=0$
by a time shift. Using (13.55), we have:

$\psi^{\prime\prime}=-\frac{2}{n+1}\psi^{\prime}-e^{2T}\psi^{n}$ (13.57)

and

$\psi_{c}\left(T\right)=\psi_{0}e^{-\frac{2T}{n-1}}$ (13.58)

where $\psi=\psi(t(T))$.

Introducing the parameter $p=-\frac{2}{n-1}$, we have:
$u^{\prime\prime}(T)=\frac{p^{2}e^{-pT}\psi(T)}{\psi_{0}}+\frac{e^{-pT}\psi^{\prime\prime}(T)}{\psi_{0}}-\frac{2pe^{-pT}\psi^{\prime}(T)}{\psi_{0}}$,
$u^{\prime}(T)=\frac{e^{-pT}\psi^{\prime}(T)}{\psi_{0}}-\frac{pe^{-pT}\psi(T)}{\psi_{0}}$
and $u(T)=\frac{e^{-pT}\psi(T)}{\psi_{0}}$. Inverting these relations, we
find:

$\psi^{\prime\prime}(T)=\psi_{0}e^{pT}\left[p^{2}u(T)+2pu^{\prime}(T)+u^{\prime\prime}(T)\right],\enskip\psi^{\prime}(T)=\psi_{0}e^{pT}\left[pu(T)+u^{\prime}(T)\right],\enskip\psi(T)=\psi_{0}e^{pT}u(T).$

Then, equation (13.57) becomes

$u^{\prime\prime}=-2\left(p+\frac{1}{n+1}\right)u^{\prime}-\psi_{0}^{n-1}e^{\left[(n-1)p+2\right]T}u^{n}-\frac{p(np+p+2)}{n+1}u.$
(13.59)

Substituting $p=-\frac{2}{n-1}$ and
$\psi_{0}=(-2)^{\frac{3}{n-1}}\left[\left(n+1\right)\left(n-1\right)^{2}\right]^{\frac{1}{1-n}}$
in (13.59), we obtain the second order ODE

$u^{\prime\prime}=\frac{2(n+3)}{n^{2}-1}u^{\prime}+\frac{8}{(n-1)^{2}(n+1)}\left(u^{n}-u\right).$
(13.60)

Introducing the variables $x=u(T)$ and $y=u^{\prime}(T)$, we obtain the
autonomous system:

$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle y$ (13.61)
$\displaystyle y^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2(n+3)}{n^{2}-1}y+\frac{8}{(n-1)^{2}(n+1)}\left(x^{n}-x\right).$
(13.62)

The scaling solution (13.47) is transformed into the equilibrium point
$P:=(x,y)=(1,0)$. The system (13.61) - (13.62) admits two additional
equilibrium points: a) The trivial solution $O:=(x,y)=(0,0)$, and b) when $n$
is odd, the symmetrical point of $P$, that is, $\bar{P}:=(x,y)=(-1,0)$. The
linearization matrix of the system (13.61) - (13.62) is
$J(x,y)=\left(\begin{array}[]{cc}0&1\\\
\frac{8\left(nx^{n-1}-1\right)}{(n-1)^{2}(n+1)}&\frac{2(n+3)}{n^{2}-1}\end{array}\right)$.

For $n>1$, $J(0,0)$ is real-valued with eigenvalues
$\left\\{\frac{4}{n^{2}-1},\frac{2}{n-1}\right\\}$. Then, the origin is an
unstable equilibrium point.

The eigenvalues of $J(1,0)$ are
$\left\\{-\frac{2}{n+1},\frac{4}{n-1}\right\\}$. Therefore, $P(1,0)$ is a sink
for $-1<n<1$ and a saddle for $n<-1$ or $n>1$.

If $n$ is an odd number, say $n=2k+1$ with $k\in\mathbb{Z}$, the eigenvalues
of $J(-1,0)$ are $\left\\{-\frac{1}{k+1},\frac{2}{k}\right\\}$ and, when they
exist, $\bar{P}$ is a saddle.

In Figures 13.1 and 13.2, we give the phase portraits of the system (13.61) -
(13.62) for some characteristic values of $n$.

Figure 13.1:  The phase portrait of the system (13.61) - (13.62) for $n=0$ and
$n=2$. For $n=0$ the only equilibrium point is the sink $P(1,0)$, whereas for
$n=2$ the point $O(0,0)$ is a source and the point $P(1,0)$ is a saddle.
Figure 13.2:  The phase portrait of the system (13.61) - (13.62) for $n=3$. In
this case, the origin $O(0,0)$ is a source and the points $\bar{P}(-1,0)$,
$P(1,0)$ are saddles.

####  13.5 Conclusions

We considered a cosmological model consisted by a Brans-Dicke scalar field and
a minimally coupled quintessence scalar field in a spatially flat FLRW
background space. For this cosmological model, the gravitational field
equations define a Hamiltonian system of six degrees of freedom, whose
dynamical variables are the scale factor and the two scalar fields.

For a power law potential function of the quintessence scalar field, we found
QFIs for the equation of motion of this field. Using these QFIs, we determined
exact solutions for the field equations. In particular, we found scaling
solutions for the scale factor which describe ideal gas solutions. Moreover,
we were able to recover previous published results in the literature and also
to find new QFIs.

Using well-known methods, we studied the stability of the scaling solution
(13.47). We showed that this solution is transformed to the equilibrium point
$P:=(x,y)=(1,0)$, which is a sink for $-1<n<1$ and a saddle for $n<-1$ or
$n>1$. Furthermore, we found that the associated dynamical system admits two
additional equilibrium points: a) The trivial solution $O:=(x,y)=(0,0)$ which
is a source for $n>1$, and b) the symmetrical point of $P$,
$\bar{P}:=(x,y)=(-1,0)$, which is a saddle when $n$ is odd with $n\neq-1,1$.

Until now, the majority of this kind of studies, have been done mainly with
the application of variational symmetries. Our approach is more general and
does not required the existence of a point-like Lagrangian, that is, of a
minisuperspace description. Therefore, that generic approach can be applied in
other gravitational models without minisuperspace; for example, such models
are the Class B Bianchi spacetimes.

###  Chapter 14 Integrable time-dependent central potentials

####  14.1 Introduction

As we have seen in previous chapters, there are various methods for the
determination of the FIs of the time-dependent conservative systems

$\ddot{q}^{a}=-U^{,a}(t,q)$ (14.1)

where $U(t,q)$ denotes the potential of the system and a general flat kinetic
metric $\gamma_{ab}(q)$ is used for raising/lowering the indices. These
methods include (see sections 3.5 and 12.1) e.g.: Noether’s theorem, the Lie
theory of extended groups, Ermakov’s method, theory of canonical
transformations, Inverse Noether Theorem, and the direct method (see e.g. [12,
37, 38, 55, 62, 67, 74, 76, 161, 173, 184, 245]).

The case of QFIs of the general form111 The case of LFIs also included for
$K_{ab}=0$.  (7.3) is the one that has been considered mostly in the
literature. There are two major methods which have been used in order to
determine the QFIs (7.3): a. The method of the Inverse Noether Theorem (see
Theorem 3.4.1), and b. The direct method. Both methods have been discussed in
section 3.5; however, it is necessary to recall some key points, essential for
the current discussion.

Concerning method a., one assumes that the dynamical equations (14.1) possess
a regular Lagrangian $L(t,q,\dot{q})$ and uses the Inverse Noether Theorem
3.4.1 to associate to each QFI of the form (7.3) the generalized gauged
Noether symmetry

$\xi=0,\enskip\eta_{a}=-2K_{ab}\dot{q}^{b}-K_{a},\enskip
f=-K_{ab}\dot{q}^{a}\dot{q}^{b}+K.$ (14.2)

From (14.2), it follows that the generator $\eta^{a}$ is a linear function of
the velocities, that is, $\eta^{a}=A_{b}^{a}(t,q)\dot{q}^{b}+B^{a}(t,q)$ where
$A^{a}_{b}(t,q)$ and $B^{a}(t,q)$ are tensor quantities. Subsequently, one
substitutes $\eta^{a}$ and $L$ in the generalized Killing equations (see
section 1.7) and finds a system of PDEs whose solution determines the time-
dependent potentials that admit QFIs.

In the present chapter, we address the problem of finding all time-dependent
central potentials of regular conservative Newtonian systems that admit QFIs
of the form (7.3); therefore, all such potentials that are (Liouville)
integrable. Previous work on this problem has been done in [12], where the
method of the Inverse Noether Theorem was used. It was concluded that there
are three classes of time-dependent central potentials, which were classified
as Cases I, II and III, given by the following expressions:

$V_{I}(t,r)=\frac{1}{2}\lambda(t)r^{2},\enskip
V_{II}(t,r)=-\frac{\ddot{\phi}}{2\phi}r^{2}-\frac{\mu_{0}}{\phi}r^{-1}\enskip\text{and}\enskip
V_{III}(t,r)=-\frac{\ddot{\phi}}{2\phi}r^{2}+\phi^{-2}F\left(\frac{r}{\phi}\right).$

Here $\phi(t)$ is an arbitrary smooth function, the frequency $\lambda(t)$ of
the time-dependent oscillator may be either the function
$\lambda_{1}(t)=\frac{K}{\phi^{4}}-\frac{\ddot{\phi}}{\phi}$ or the function
$\lambda_{2}(t)=-\frac{\ddot{\phi}}{\phi}$, $\mu_{0}$ and $K$ are arbitrary
constants, and $F$ is an arbitrary smooth function of its argument. For each
potential, the corresponding QFI was provided. We note that, in fact, there is
only one Case because Cases I and II are subcases of Case III for $F=0$,
$F=\frac{K}{2}\phi^{-2}r^{2}$, and $F=-\mu_{0}\frac{\phi}{r}$. As it will be
shown, this result is a partial answer to the problem. It is important also to
note that in [12] the dynamical system was considered to be 2d with variables
$r,\theta$.

In a different approach [246], the authors considered 1d Hamiltonian systems
of the form $H=\frac{1}{2}p^{2}+U(t,q)$ and, using the direct method for QFIs
of the form (7.3), determined all the time-dependent potentials $U(t,q)$ which
admit QFIs222Obviously, in this case, central motion makes no sense.. They
concluded that these potentials are

$U(t,r)=-\frac{\ddot{\rho}}{2\rho}q^{2}+\left(\alpha\frac{\ddot{\rho}}{\rho}-\ddot{\alpha}\right)q+\frac{1}{\rho^{2}}G\left(\frac{q-\alpha}{\rho}\right)$
(14.3)

where $\rho(t)$, $\alpha(t)$ and $G$ are arbitrary smooth functions of their
arguments. The associated QFI is

$I=\frac{1}{2}\left[\rho(\dot{\rho}-\dot{\alpha})-\dot{\rho}(q-\alpha)\right]^{2}+G\left(\frac{q-\alpha}{\rho}\right).$
(14.4)

Equivalently, the potential (14.3) may be written as

$U(t,r)=\frac{1}{2}\Omega^{2}(t)q^{2}-F_{1}(t)q+\frac{1}{\rho^{2}}\widetilde{G}\left(\frac{q-\alpha}{\rho}\right)$
(14.5)

where $\widetilde{G}$ is an arbitrary smooth function of its argument and the
functions $\Omega(t),F_{1}(t),\rho(t),\alpha(t)$ satisfy the conditions: