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

Antonios Mitsopoulos1,a) and Michael Tsamparlis1,b)

1Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics,
University of Athens, Panepistemiopolis, Athens 157 83, Greece
a)Author to whom correspondence should be addressed<EMAIL_ADDRESS>
b)Email<EMAIL_ADDRESS>
###### Abstract

We consider the time-dependent dynamical system
$\ddot{q}^{a}=-\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$ where
$\omega(t)$ is a non-zero arbitrary function and the connection coefficients
$\Gamma^{a}_{bc}$ are computed from the kinetic metric (kinetic energy) of the
system. In order to determine the quadratic first integrals (QFIs) $I$ we
assume that $I=K_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}\dot{q}^{a}+K$ where the
unknown coefficients $K_{ab},K_{a},K$ are tensors depending on $t,q^{a}$ and
impose the condition $\frac{dI}{dt}=0$. This condition leads to a system of
partial differential equations (PDEs) involving the quantities
$K_{ab},K_{a},K,$ $\omega(t)$ and $Q^{a}(q)$. From these PDEs, it follows that
$K_{ab}$ is a Killing tensor (KT) of the kinetic metric. We use the KT
$K_{ab}$ in two ways: a. We assume a general polynomial form in $t$ both for
$K_{ab}$ and $K_{a}$; b. We express $K_{ab}$ in a basis of the KTs of order 2
of the kinetic metric assuming the coefficients to be functions of $t$. In
both methods, this leads to a new system of PDEs whose solution requires that
we specify either $\omega(t)$ or $Q^{a}(q)$. We consider first that
$\omega(t)$ is a general polynomial in $t$ and find that in this case the
dynamical system admits two independent QFIs which we collect in a Theorem.
Next, we specify the quantities $Q^{a}(q)$ to be the generalized time-
dependent Kepler potential $V=-\frac{\omega(t)}{r^{\nu}}$ and determine the
functions $\omega(t)$ for which QFIs are admitted. We extend the discussion to
the non-linear differential equation
$\ddot{x}=-\omega(t)x^{\mu}+\phi(t)\dot{x}$ $(\mu\neq-1)$ and compute the
relation between the coefficients $\omega(t),\phi(t)$ so that QFIs are
admitted. We apply the results to determine the QFIs of the generalized Lane-
Emden equation.

##  1 Introduction

The equations of motion of a dynamical system define in the configuration
space a Riemannian structure with the metric of the kinetic energy (kinetic
metric). This metric is inherent in the structure of the dynamical system;
therefore, we expect that it will determine the first integrals (FIs) of the
system which are important in its evolution. On the other hand a metric is
fixed by its symmetries, that is, the linear collineations: Killing vectors
(KVs), homothetic vectors (HVs), conformal Killing vectors (CKVs), affine
collineations (ACs), projective collineations (PCs); and the quadratic
collineations: second order Killing tensors (KTs). The question then is how
the FIs of the dynamical system and the geometric symmetries of the kinetic
metric are related.

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 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. [1], [2], [3],
[4]) 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 algebra. A recent clear statement of these
results is discussed in [5].

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
equations of the form
$\ddot{q}^{a}=-\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)V^{,a}(q)$ (see
e.g. [6], [7], [8], [9], [10], [11], [12]). 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 constraint conditions 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. [8], [10],
[13], [14], [15]) and the time-dependent Kepler potential (see e.g. [12],
[16], [17], [18]). A further development in the same line is the extension of
this method to time-dependent potentials $W(t,q)$ with linear damping terms
[12]. 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 $W(t,q)=\bar{\omega}(t)V(q)$ but with different $\bar{\omega}(t)$.
Finally the Lie/Noether method has been applied to the study of partial
differential equations (PDEs) [4], [19], [20], [21].

Besides the aforementioned Lie/Noether method there is a different method
which computes the FIs in terms of the collineations of the kinetic metric
without using Lie symmetries. This method we shall apply in this paper. It has
as follows.

One assumes the generic quadratic first integral (QFI) to be of the form111The
linear FIs (LFIs) are also included for $K_{ab}=0$.

$I=K_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}\dot{q}^{a}+K$ (1)

where the coefficients $K_{ab},K_{a},K$ are tensors depending on the
coordinates $t,q^{a}$ and imposes the condition $\frac{dI}{dt}=0.$ Using again
the equations of motion to replace the quantities $\ddot{q}^{a}$ whenever they
appear, this condition leads to a system of PDEs involving the unknown
quantities $K_{ab},K_{a},K$ and the dynamical elements, i.e. the potential and
the generalized forces of the system. The solution of this system of PDEs
provides the QFIs (1). For future reference we shall call this method the
_direct method_.

The system of PDEs consists of two parts: a. The geometric part which is
independent of the dynamical quantities; and b. the dynamical part which
contains the scalar $K$ and the dynamical quantities. The main conclusion of
the geometric part is that the tensor $K_{ab}$ is a KT of the kinetic metric
whereas the vector $K_{a}$ is related to the linear collineations of that
metric. The dynamical part involves the scalar $K$ which is determined by a
set of constraint conditions which involve $K_{ab},K_{a},K$, the potential and
the generalized forces. Once $K$ is computed one gets the corresponding QFI
$I$.

The direct method can always be related to the Noether symmetries. Indeed
assuming that the system has a regular Lagrangian (which is always the case
since we assume that there exists the kinetic energy) it can be shown by using
the inverse Noether theorem (see [22] and section II in [23]) that to each QFI
$I$ one determines an associated gauged generalized Noether symmetry with
generator $\eta_{a}=-2K_{ab}\dot{q}^{b}-K_{a}$ and Noether function
$f=-K_{ab}\dot{q}^{a}\dot{q}^{b}+K$ whose Noether integral is the considered
QFI. Therefore we conclude that all QFIs of the form (1) are Noetherian,
provided the Lagrangian is regular, that is, the dynamical equations can be
solved in terms of $\ddot{q}^{a}$.

Moreover, this method has been employed in the literature (see [17], [24],
[25], [26]) both for autonomous and time-dependent dynamical systems. A recent
account of this method in the case of autonomous conservative systems together
with relevant references can be found in [27]. This approach being geometric
is powerful and convenient because with minimal calculations it allows the
computation of the FIs by using known results from differential geometry.

The purpose of the present work is to apply the direct method to compute the
QFIs of time-dependent equations of the form
$\ddot{q}^{a}=-\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$.
Because many well-known dynamical systems fall in this category we intend to
recover in a direct single approach all the known results derived from the
Lie/Noether symmetry method, which are scattered in a large number of papers.

As explained above, the solution of the system requires that the tensor
$K_{ab}$ 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 [28], [29], [30]). Therefore, in this paper, we
restrict our discussion to Euclidean spaces only. Since the KT $K_{ab}$ is a
function of $t,q^{a}$ we suggest two procedures of work: a. The polynomial
method; 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 in
the equations of the relevant system. In the basis method, one computes first
a basis of the KTs of order 2 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. 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 integrals found in [12]. As a second
application, we assume that $\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell}$ with
$b_{\ell}\neq 0$ and $\ell\geq 1$ whereas the quantities $Q^{a}$ are
unspecified. We find that in this case the system admits two families of
independent QFIs as stated in Theorem 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
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 ste,p 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 three-dimensional (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 (see e.g. [31]). 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. For easier reference we collect all the results in Table
LABEL:T2 of section 14.

Using the well-known result that by a reparameterization the linear damping
term $\phi(t)\dot{q}^{a}$ of a dynamical equation is absorbed to a time-
dependent force of the form $\omega(t)Q^{a}(q)$, we also study the non-linear
differential equation $\ddot{x}=-\omega(t)x^{\mu}+\phi(t)\dot{x}$
$(\mu\neq-1)$ and compute the relation between the coefficients
$\omega(t),\phi(t)$ 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.

The structure of the paper is as follows. In section 2 we determine the system
of PDEs resulting form the condition $dI/dt=0$. In section 3, we assume that
the KT is proportional to the kinetic metric and derive the point Noether FIs
of the time-dependent dynamical system (2). In section 4, we consider the
polynomial method and define the general forms of the KT $K_{ab}$ and the
vector $K_{a}$ which lead to a new form of the system of PDEs. In section 5,
we assume that $\omega(t)$ is a general polynomial of $t$ and we find that the
resulting time-dependent system admits two independent QFIs as stated in
Theorem 1. In section 6, we discuss some special cases of the QFI $I_{n}$ of
Theorem 1. In section 7, we consider the basis method. In section 8, we find a
basis for the KTs in $E^{3}$ in order to apply the basis method to 3d
Newtonian systems. In sections 9 \- 13, we study the time-dependent
generalized Kepler potential and find for which functions $\omega(t)$ admits
QFIs. Particularly, in section 13, we study a special class of time-dependent
oscillators with frequency $\omega_{3O}(t)$ as given in equation (123). We
collect our results for the several values of $\nu$ in Table LABEL:T2 of
section 14. In section 15, we use the independent LFIs $I_{41i},I_{42i}$ given
in equations (125), (126) to integrate the equations of the time-dependent
oscillators defined in section 13; and the FIs $L_{i}$, $E_{2}$, $A_{i}$
determined in subsection 11.1 to integrate the time-dependent Kepler potential
with $\omega(t)=\frac{k}{b_{0}+b_{1}t}$ where $kb_{1}\neq 0$. In section 16,
we consider the second order non-linear time-dependent differential equation
(154) and show that it is integrable with an associated QFI given in equation
(175) iff the functions $\omega(t),\phi(t)$ are related as shown in equation
(174). For the special values $\mu=0,1,2$ we find also that there exist
additional relations between $\omega(t),\phi(t)$ for which the resulting
differential equation admits a QFI. For $\mu=1$ equation (154) admits the
general solution (166) provided that condition (165) is satisfied. We apply
these results in subsection 16.1 and we study the properties of the well-known
generalized Lane-Emden equation. Finally, in section 17, we draw our
conclusions and, in the appendix, we give the proof of Theorem 1.

##  2 The system of equations

We consider the dynamical system

$\ddot{q}^{a}=-\Gamma^{a}_{bc}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$ (2)

where $\Gamma^{a}_{bc}$ are the Riemannian connection coefficients determined
by the kinetic metric $\gamma_{ab}$ (kinetic energy) of the system and
$-\omega(t)Q^{a}(q)$ are the time-dependent generalized forces. Einstein
summation convention is assumed and the metric $\gamma_{ab}$ is used for
lowering and raising the indices.

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

$I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ (3)

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

We demand $I$ be a FI of (2) by imposing the condition

$\frac{dI}{dt}=0.$ (4)

Using the dynamical equations (2) to replace $\ddot{q}^{a}$ whenever it
appears we find222Round brackets indicate symmetrization of the enclosed
indices. A comma indicates partial derivative and a semicolon Riemannian
covariant derivative. the system of equations

$\displaystyle K_{(ab;c)}$ $\displaystyle=$ $\displaystyle 0$ (5)
$\displaystyle K_{ab,t}+K_{(a;b)}$ $\displaystyle=$ $\displaystyle 0$ (6)
$\displaystyle-2\omega K_{ab}Q^{b}+K_{a,t}+K_{,a}$ $\displaystyle=$
$\displaystyle 0$ (7) $\displaystyle K_{,t}-\omega K_{a}Q^{a}$
$\displaystyle=$ $\displaystyle 0$ (8) $\displaystyle
K_{a,tt}+\omega\left(K_{b}Q^{b}\right)_{,a}-2\omega_{,t}K_{ab}Q^{b}-2\omega
K_{ab,t}Q^{b}$ $\displaystyle=$ $\displaystyle 0$ (9) $\displaystyle
K_{[a;b],t}-2\omega\left(K_{[a|c|}Q^{c}\right)_{;b]}$ $\displaystyle=$
$\displaystyle 0$ (10)

where the last two equations333 These equations come from the conditions
$K_{,[at]}=0$ and $K_{,[ab]}=0$ respectively. (9), (10) express the
integrability conditions for the scalar $K$.

Equation (5) implies that $K_{ab}$ is a KT of order 2 (possibly zero) of the
kinetic metric $\gamma_{ab}$.

The solution of the system 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 FIs.

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 $K_{ab}$ as above both the
function $\omega(t)$ and the quantities $Q^{a}(q)$ stay unspecified and can
act as constraints.

##  3 The point Noether FIs of the time-dependent dynamical system (2)

We consider the simplest choice

$K_{ab}=N(t)\gamma_{ab}$ (11)

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
equations (5) - (10) become (eq. (5) vanishes trivially)

$\displaystyle N_{,t}\gamma_{ab}+K_{(a;b)}$ $\displaystyle=$ $\displaystyle 0$
(12) $\displaystyle-2\omega NQ_{a}+K_{a,t}+K_{,a}$ $\displaystyle=$
$\displaystyle 0$ (13) $\displaystyle K_{,t}-\omega K_{a}Q^{a}$
$\displaystyle=$ $\displaystyle 0$ (14) $\displaystyle
K_{a,tt}+\omega\left(K_{b}Q^{b}\right)_{,a}-2\omega_{,t}NQ_{a}-2\omega
N_{,t}Q_{a}$ $\displaystyle=$ $\displaystyle 0$ (15) $\displaystyle
K_{[a;b],t}-2\omega NQ_{[a;b]}$ $\displaystyle=$ $\displaystyle 0.$ (16)

We consider the following cases.

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

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

Equation (12) gives

$N_{,t}=-\psi\implies N=-\psi t+c$

where $c$ is an arbitrary constant.

Equation (16) 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 (13) 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 (14) we get

$\omega K_{a}V^{,a}-2\omega_{,t}(-\psi t+c)V+2\omega\psi V-M_{,t}=0.$ (17)

The remaining condition (15) is just the partial derivative of (17), and hence
is satisfied trivially.

Moreover, since $\omega\neq 0$, equation (17) can be written in the form

$K_{a}V^{,a}-2(\ln\omega)_{,t}(-\psi t+c)V+2\psi V-\frac{M_{,t}}{\omega}=0$
(18)

which implies that

$\displaystyle 2(\ln\omega)_{,t}(-\psi t+c)$ $\displaystyle=$ $\displaystyle
c_{1}$ (19) $\displaystyle M_{,t}$ $\displaystyle=$ $\displaystyle
c_{2}\omega$ (20)

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

Therefore equation (18) becomes

$K_{a}V^{,a}+(2\psi-c_{1})V-c_{2}=0.$ (21)

The QFI is

$I_{1}=(-\psi
t+c)\gamma_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}(q)\dot{q}^{a}+2\omega(-\psi
t+c)V+M(t)$ (22)

where $Q_{a}=V_{,a}$ and the quantities $\omega(t),M(t),V(q),K_{a}(q)$ satisfy
the conditions (19) - (21).

###  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) implies $N_{,t}=\psi M$.

From equation (16) we find that there exists a potential function $V(q)$ such
that $Q_{a}=V_{,a}$.

Replacing the above results in (13) 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 (14) 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

$\displaystyle\frac{2(\ln\omega)_{,t}N}{M}$ $\displaystyle=$ $\displaystyle
d_{1}$ (23) $\displaystyle\frac{M_{,tt}}{\omega M}$ $\displaystyle=$
$\displaystyle m$ (24) $\displaystyle\frac{C_{,t}}{\omega M}$ $\displaystyle=$
$\displaystyle k$ (25) $\displaystyle S_{,a}V^{,a}+(2\psi+d_{1})V+mS+k$
$\displaystyle=$ $\displaystyle 0$ (26)

where $d_{1},m,k$ are arbitrary constants. The remaining condition (15) is
satisfied identically.

The QFI is

$I_{2}=N\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-MS_{,a}\dot{q}^{a}+2\omega
NV+M_{,t}S+C(t)$ (27)

where $Q_{a}=V_{,a}$, $N_{,t}=\psi M$ and the conditions (23) - (26) must be
satisfied.

###  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) implies $N_{,t}=\psi M$ where $\psi$ is the homothety factor of
$V_{,a}$.

From equation (13) 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 (14) 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

$\displaystyle\frac{M_{,tt}}{\omega M}+\frac{2(\ln\omega)_{,t}N}{M}$
$\displaystyle=$ $\displaystyle d_{2}$ (28) $\displaystyle\frac{C_{,t}}{\omega
M}$ $\displaystyle=$ $\displaystyle k$ (29) $\displaystyle
V_{,a}V^{,a}+(2\psi+d_{2})V+k$ $\displaystyle=$ $\displaystyle 0$ (30)

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

The QFI is

$I_{3}=N\gamma_{ab}\dot{q}^{a}\dot{q}^{b}-MV_{,a}\dot{q}^{a}+\left(2\omega
N+M_{,t}\right)V+C$ (31)

where $Q_{a}=V_{,a}$, $N_{,t}=\psi M$ and the conditions (28) - (30) must be
satisfied.

The above results reproduce Theorem 2 of [12] 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 (22), (27), (31) produced by point
Noether symmetries can be also produced by generalized (gauged) Noether
symmetries using the inverse Noether theorem. This proves that a Noether FI is
not associated with a unique Noether symmetry!

##  4 The polynomial method for computing the QFIs

In the polynomial approach one assumes a polynomial form in $t$ of the KT
$K_{ab}(t,q)$ and the vector $K_{a}(t,q)$ and solves the resulting system for
given $\omega(t),Q^{a}(q)$. One application of this method can be found in
[27] where a general theorem is given which allows the finding of the QFIs of
an autonomous conservative dynamical system. In the present work we generalize
the considerations made in [27] and assume that the quantity $K_{ab}(t,q)$ has
the form

$K_{ab}(t,q)=C_{(0)ab}(q)+\sum_{N=1}^{n}C_{(N)ab}(q)\frac{t^{N}}{N}$ (32)

where $C_{(N)ab}$, $N=0,1,...,n$, is a sequence of arbitrary KTs of order 2 of
the kinetic metric $\gamma_{ab}$.

This choice of $K_{ab}$ and equation (6) indicate that we set

$K_{a}(t,q)=\sum^{m}_{M=0}L_{(M)a}(q)t^{M}$ (33)

where $L_{(M)a}(q)$, $M=0,1,...,m$, are arbitrary vectors.

We note that both powers $n$, $m$ in the above polynomial expressions may be
infinite.

Substituting (32), (33) in the system of equations (5) - (10) (equation (5) is
identically zero since $C_{(N)ab}$ are KTs) we obtain the system of equations

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
C_{(1)ab}+C_{(2)ab}t+...+C_{(n)ab}t^{n-1}+L_{(0)(a;b)}+L_{(1)(a;b)}t+...+L_{(m)(a;b)}t^{m}$
(34) $\displaystyle 0$ $\displaystyle=$ $\displaystyle-2\omega
C_{(0)ab}Q^{b}-2\omega C_{(1)ab}Q^{b}t-...-2\omega
C_{(n)ab}Q^{b}\frac{t^{n}}{n}+L_{(1)a}+2L_{(2)a}t+...+mL_{(m)a}t^{m-1}+K_{,a}$
(35) $\displaystyle 0$ $\displaystyle=$ $\displaystyle K_{,t}-\omega
L_{(0)a}Q^{a}-\omega L_{(1)a}Q^{a}t-...-\omega L_{(m)a}Q^{a}t^{m}$ (36)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(-2C_{(0)ab}Q^{b}-2C_{(1)ab}Q^{b}t-...-2C_{(n)ab}Q^{b}\frac{t^{n}}{n}\right)\omega_{,t}-2\omega
C_{(1)ab}Q^{b}-2\omega C_{(2)ab}Q^{b}t-...-2\omega C_{(n)ab}Q^{b}t^{n-1}+$
(37)
$\displaystyle+2L_{(2)a}+6L_{(3)a}t+...+m(m-1)L_{(m)a}t^{m-2}+\omega\left(L_{(0)b}Q^{b}\right)_{,a}+\omega\left(L_{(1)b}Q^{b}\right)_{,a}t+...+\omega\left(L_{(m)b}Q^{b}\right)_{,a}t^{m}$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\omega\left(C_{(0)[a\left|c\right|}Q^{c}\right)_{;b]}+2\omega\left(C_{(1)[a\left|c\right|}Q^{c}\right)_{;b]}t+...+2\omega\left(C_{(n)[a\left|c\right|}Q^{c}\right)_{;b]}\frac{t^{n}}{n}-L_{(1)\left[a;b\right]}-$
(38)
$\displaystyle-2L_{(2)\left[a;b\right]}t-...-mL_{(m)\left[a;b\right]}t^{m-1}.$

In this system of PDEs the pairs $\omega(t),Q^{a}(q)$ are not specified. As we
explained in the introduction 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.

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

We assume that

$\omega(t)=b_{0}+b_{1}t+...+b_{\ell}t^{\ell},\enskip b_{\ell}\neq
0,\enskip\ell\geq 1$ (39)

where $\ell$ is the degree of the polynomial. Substituting the function (39)
in the system of equations (34) - (38) we find444The proof of Theorem 1 is in
the appendix. that there are two independent QFIs as given in Theorem 1.

###### Theorem 1

The independent QFIs of the time-dependent dynamical system (2) 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.

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

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

$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$ (40)

$s$ is an arbitrary constant defined by the condition

$L_{(n)a}Q^{a}=s$ (41)

and the following conditions are satisfied

$\sum_{s=0}^{\ell-1}\left[-\frac{2(r+s)b_{(r+s\leq\ell)}}{n-s}C_{(n-s\geq
0)ab}Q^{b}-2b_{(r+s\leq\ell)}C_{(n-s>0)ab}Q^{b}+b_{(r+s\leq\ell)}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right]=0,\enskip r=1,2,...,\ell$ (42)
$-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{n-s}C_{(n-s\geq
0)ab}Q^{b}\right]+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(n-s>0)ab}Q^{b}+b_{s}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right]=0$ (43)
$k(k-1)L_{(k)a}-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{k-s-1}C_{(k-s-1\geq
0)ab}Q^{b}\right]+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(k-s-1>0)ab}Q^{b}+b_{s}\left(L_{(k-s-2\geq
0)b}Q^{b}\right)_{,a}\right]=0$ (44)

with $k=2,3,...n$.

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 $\ell=1$.

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

The parameter $n$ in the case Integral 1 of Theorem 1 runs over all positive
integers, i.e. $n=0,1,2,...$. This results in a sequence of QFIs
$I_{0},I_{1},I_{2},...$, 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 $n$.

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

For $n=0$ we have

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

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

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

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

For $n=1$ the conditions (41) - (44) become

$\displaystyle L_{(1)a}Q^{a}$ $\displaystyle=$ $\displaystyle s$ (45)
$\displaystyle\left(L_{(0)b}Q^{b}\right)_{,a}$ $\displaystyle=$
$\displaystyle-2(\ell+1)L_{(0)(a;b)}Q^{b}$ (46) $\displaystyle
kb_{k}C_{(0)ab}Q^{b}$ $\displaystyle=$
$\displaystyle-(\ell-k+1)b_{k-1}L_{(0)(a;b)}Q^{b},\enskip k=1,...,\ell.$ (47)

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 $\ell\geq 2$. From these equations,
using mathematical induction, we prove after successive substitutions that

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

The QFI is ($I_{0}$ is a subcase of $I_{1}$)

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

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

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 $C_{(0)ab}$, $L_{(0)(a;b)}$ 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 $C_{(0)ab}$, $L_{(0)(a;b)}$ 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

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

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

##  7 The basis method for computing QFIs

As it has been explained in the introduction, in the basis method instead of
considering the KT $K_{ab}$ to be given as a polynomial in $t$ with
coefficients arbitrary KTs (see equation (32) ) one defines the KT
$K_{ab}(t,q)$ by the requirement

$K_{ab}(t,q)=\sum_{N=1}^{m}\alpha_{N}(t)C_{(N)ab}(q)$ (48)

where $\alpha_{N}(t)$ are arbitrary smooth functions and the $m$ linearly
independent KTs $C_{(N)ab}(q)$ constitute a basis of the 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 (5) - (10).

The basis method has been used previously by Katzin and Levine in [17] 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 $E^{3}$.

##  8 The geometric quantities of $E^{3}$

In $E^{3}$ the general KT of order 2 has independent components

$\displaystyle C_{11}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}y^{2}+\frac{a_{1}}{2}z^{2}+a_{4}yz+a_{5}y+a_{2}z+a_{3}$
$\displaystyle C_{12}$ $\displaystyle=$
$\displaystyle\frac{a_{10}}{2}z^{2}-\frac{a_{6}}{2}xy-\frac{a_{4}}{2}xz-\frac{a_{14}}{2}yz-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17}$
$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{a_{14}}{2}y^{2}-\frac{a_{4}}{2}xy-\frac{a_{1}}{2}xz-\frac{a_{10}}{2}yz-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19}$
(49) $\displaystyle C_{22}$ $\displaystyle=$
$\displaystyle\frac{a_{6}}{2}x^{2}+\frac{a_{7}}{2}z^{2}+a_{14}xz+a_{15}x+a_{12}z+a_{13}$
$\displaystyle C_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{4}}{2}x^{2}-\frac{a_{14}}{2}xy-\frac{a_{10}}{2}xz-\frac{a_{7}}{2}yz-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20}$
$\displaystyle C_{33}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}x^{2}+\frac{a_{7}}{2}y^{2}+a_{10}xy+a_{11}x+a_{8}y+a_{9}$

where $a_{I}$ with $I=1,2,...,20$ are arbitrary real constants.

The vector $L^{a}$ generating the KT $C_{ab}=L_{(a;b)}$ is

$L_{a}=\left(\begin{array}[]{c}-a_{15}y^{2}-a_{11}z^{2}+a_{5}xy+a_{2}xz+2(a_{16}+a_{18})yz+a_{3}x+2a_{4}y+2a_{1}z+a_{6}\\\
-a_{5}x^{2}-a_{8}z^{2}+a_{15}xy-2a_{18}xz+a_{12}yz+2(a_{17}-a_{4})x+a_{13}y+2a_{7}z+a_{14}\\\
-a_{2}x^{2}-a_{12}y^{2}-2a_{16}xy+a_{11}xz+a_{8}yz+2(a_{19}-a_{1})x+2(a_{20}-a_{7})y+a_{9}z+a_{10}\end{array}\right)$
(50)

and the generated KT is

$C_{ab}=\left(\begin{array}[]{ccc}a_{5}y+a_{2}z+a_{3}&-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17}&-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19}\\\
-\frac{a_{5}}{2}x-\frac{a_{15}}{2}y+a_{16}z+a_{17}&a_{15}x+a_{12}z+a_{13}&-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20}\\\
-\frac{a_{2}}{2}x+a_{18}y-\frac{a_{11}}{2}z+a_{19}&-(a_{16}+a_{18})x-\frac{a_{12}}{2}y-\frac{a_{8}}{2}z+a_{20}&a_{11}x+a_{8}y+a_{9}\end{array}\right)$
(51)

which is a subcase of the general KT (49) for
$a_{1}=a_{4}=a_{6}=a_{7}=a_{10}=a_{14}=0$.

We note that the covariant expression of the most general KT $M_{ij}$ of order
2 of $E^{3}$ is (see [32], [33])

$M_{ij}=(\varepsilon_{ikm}\varepsilon_{jln}+\varepsilon_{jkm}\varepsilon_{iln})A^{mn}q^{k}q^{l}+(B_{(i}^{l}\varepsilon_{j)kl}+\lambda_{(i}\delta_{j)k}-\delta_{ij}\lambda_{k})q^{k}+D_{ij}$
(52)

where $A^{mn},B_{i}^{l},D_{ij}$ are constant tensors all being symmetric and
$B_{i}^{l}$ also being traceless; $\lambda^{k}$ is a constant vector; and
$\varepsilon_{ijk}$ is the 3d Levi-Civita symbol. This result is obtained from
the solution of the Killing tensor equation in Euclidean space.

Observe that $A^{mn}$, $D_{ij}$ have each 6 independent components;
$B_{i}^{l}$ has 5 independent components; and $\lambda^{k}$ has 3 independent
components. Therefore $M_{ij}$ depends on $6+6+5+3=20$ arbitrary real
constants, a result which is in accordance with the one given above in
equation (49).

##  9 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 [8], [10], [13], [14], [15] for $\nu=-2$, the
time-dependent Kepler potential [12], [16], [17], [18] for $\nu=1$ and the
Newton-Cotes potential for $\nu=2$ [31]. 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

$L=\frac{1}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})+\frac{\omega(t)}{r^{\nu}}$
(53)

and the corresponding Euler-Lagrange equations are

$\ddot{x}=-\frac{\nu\omega(t)}{r^{\nu+2}}x,\enskip\ddot{y}=-\frac{\nu\omega(t)}{r^{\nu+2}}y,\enskip\ddot{z}=-\frac{\nu\omega(t)}{r^{\nu+2}}z.$
(54)

For this 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 those $\omega(t)$
for which the resulting FIs are not combinations of the angular momentum.

The LFIs and the QFIs of the autonomous generalized Kepler potential, that is,
$\omega(t)=k=const$, have been determined in [27] using the direct method and
are listed in Table LABEL:T1.

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

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.

Using $Q^{a}=\frac{\nu q^{a}}{r^{\nu+2}}$ conditions (5) - (10) become (see
[17])

$\displaystyle K_{(ab;c)}$ $\displaystyle=$ $\displaystyle 0$ (55)
$\displaystyle K_{(a;b)}+K_{ab,t}$ $\displaystyle=$ $\displaystyle 0$ (56)
$\displaystyle K_{,a}-\frac{2\nu\omega}{r^{\nu+2}}K_{ab}q^{b}+K_{a,t}$
$\displaystyle=$ $\displaystyle 0$ (57) $\displaystyle
K_{,t}-\frac{\nu\omega}{r^{\nu+2}}K_{a}q^{a}$ $\displaystyle=$ $\displaystyle
0$ (58) $\displaystyle
K_{a,tt}+\nu\omega\left(\frac{K_{b}q^{b}}{r^{\nu+2}}\right)_{,a}-\frac{2\nu\omega_{,t}}{r^{\nu+2}}K_{ab}q^{b}-\frac{2\nu\omega}{r^{\nu+2}}K_{ab,t}q^{b}$
$\displaystyle=$ $\displaystyle 0$ (59) $\displaystyle
K_{[a;b],t}-2\nu\omega\left(\frac{K_{[a|c|}q^{c}}{r^{\nu+2}}\right)_{;b]}$
$\displaystyle=$ $\displaystyle 0.$ (60)

From the Lagrangian (53) 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 (55) is the KT given
by (49) but the 20 arbitrary constants $a_{I}$ are assumed to be time-
dependent functions $a_{I}(t)$.

Condition (56) gives

$K_{a,b}+K_{b,a}=-2K_{ab,t}\implies$ $\displaystyle K_{1,1}$ $\displaystyle=$
$\displaystyle-K_{11,t}$ (61) $\displaystyle K_{2,2}$ $\displaystyle=$
$\displaystyle-K_{22,t}$ (62) $\displaystyle K_{3,3}$ $\displaystyle=$
$\displaystyle-K_{33,t}$ (63) $\displaystyle K_{1,2}+K_{2,1}$ $\displaystyle=$
$\displaystyle-2K_{12,t}$ (64) $\displaystyle K_{1,3}+K_{3,1}$
$\displaystyle=$ $\displaystyle-2K_{13,t}$ (65) $\displaystyle
K_{2,3}+K_{3,2}$ $\displaystyle=$ $\displaystyle-2K_{23,t}.$ (66)

From the first three conditions (61) - (63) 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 (64) - (66) we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{10}z^{2}-3\dot{a}_{6}xy-2\dot{a}_{4}xz-2\dot{a}_{14}yz-2\dot{a}_{5}x-2\dot{a}_{15}y+2\dot{a}_{16}z+2\dot{a}_{17}+A_{,2}+B_{,1}$
(67) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{14}y^{2}-2\dot{a}_{4}xy-3\dot{a}_{1}xz-2\dot{a}_{10}yz-2\dot{a}_{2}x+2\dot{a}_{18}y-2\dot{a}_{11}z+2\dot{a}_{19}+A_{,3}+C_{,1}$
(68) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{4}x^{2}-2\dot{a}_{14}xy-2\dot{a}_{10}xz-3\dot{a}_{7}yz-2(\dot{a}_{16}+\dot{a}_{18})x-2\dot{a}_{12}y-2\dot{a}_{8}z+2\dot{a}_{20}+B_{,3}+C_{,2}.$
(69)

By taking the second partial derivatives of (67) with respect to (wrt) $x,y$,
of (68) wrt $x,z$ and of (69) wrt $y,z$ we find that

$a_{1}=c_{1},\enskip a_{6}=c_{2},\enskip a_{7}=c_{3}$

are arbitrary constants.

Then equations (67) - (69) become

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{10}z^{2}-2\dot{a}_{4}xz-2\dot{a}_{14}yz-2\dot{a}_{5}x-2\dot{a}_{15}y+2\dot{a}_{16}z+2\dot{a}_{17}+A_{,2}+B_{,1}$
(70) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{14}y^{2}-2\dot{a}_{4}xy-2\dot{a}_{10}yz-2\dot{a}_{2}x+2\dot{a}_{18}y-2\dot{a}_{11}z+2\dot{a}_{19}+A_{,3}+C_{,1}$
(71) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\dot{a}_{4}x^{2}-2\dot{a}_{14}xy-2\dot{a}_{10}xz-2(\dot{a}_{16}+\dot{a}_{18})x-2\dot{a}_{12}y-2\dot{a}_{8}z+2\dot{a}_{20}+B_{,3}+C_{,2}.$
(72)

By suitable differentiations of the above equations we obtain

$\displaystyle A_{,22}$ $\displaystyle=$ $\displaystyle
2\dot{a}_{14}z+2\dot{a}_{15}$ $\displaystyle A_{,33}$ $\displaystyle=$
$\displaystyle 2\dot{a}_{10}y+2\dot{a}_{11}$ $\displaystyle B_{,11}$
$\displaystyle=$ $\displaystyle 2\dot{a}_{4}z+2\dot{a}_{5}$ $\displaystyle
B_{,33}$ $\displaystyle=$ $\displaystyle 2\dot{a}_{10}x+2\dot{a}_{8}$
$\displaystyle C_{,11}$ $\displaystyle=$ $\displaystyle
2\dot{a}_{4}y+2\dot{a}_{2}$ $\displaystyle C_{,22}$ $\displaystyle=$
$\displaystyle 2\dot{a}_{14}x+2\dot{a}_{12}.$

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 in (70) - (72) 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

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

where $c_{4},c_{5},c_{6}$ are arbitrary constants.

Therefore the KT $K_{ab}$ is

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

and the vector $K_{a}$ is

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

Replacing the above results in the constraint (60) we find the following set
of equations:

$a_{2}=a_{12},\enskip a_{5}=a_{8},\enskip a_{11}=a_{15},\enskip
a_{16}=a_{18}=0$ (75)
$(\nu-1)a_{2}=0,\enskip(\nu-1)a_{5}=0,\enskip(\nu-1)a_{11}=0$ (76)
$(\nu+2)a_{17}=0,\enskip(\nu+2)a_{19}=0,\enskip(\nu+2)a_{20}=0,\enskip(\nu+2)(a_{3}-a_{9})=0,\enskip(\nu+2)(a_{3}-a_{13})=0$
(77)
$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0,\enskip\dot{\sigma}_{2}=-\ddot{a}_{17},\enskip\dot{\sigma}_{3}=-\ddot{a}_{19},\enskip\dot{\tau}_{3}=-\ddot{a}_{20}.$
(78)

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.

##  10 The general case

This case holds for any value of $\nu$ and conditions (75) - (78) give

$a_{2}=a_{5}=a_{8}=a_{11}=a_{12}=a_{15}=a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=0,$
$a_{3}=a_{9}=a_{13},\enskip\sigma_{2}=c_{7},\enskip\sigma_{3}=c_{8},\enskip\tau_{3}=c_{9}$

where $c_{7},c_{8},c_{9}$ are arbitrary constants.

Substituting in the constraint (59) we find that

$\dddot{a}_{3}=0,\enskip(\nu-2)\omega\dot{a}_{3}-2\dot{\omega}a_{3}=0$ (79)
$\ddot{\sigma}_{4}=\ddot{\tau}_{4}=\ddot{\eta}_{4}=0,\enskip\omega\sigma_{4}=\omega\tau_{4}=\omega\eta_{4}=0\implies\sigma_{4}=\tau_{4}=\eta_{4}=0.$

Therefore the KT $K_{ab}$ becomes

$K_{ab}=\left(\begin{array}[]{ccc}\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{3}&\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz\\\
\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz&\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{3}&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz\\\
\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz&\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz&\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{3}\\\
\end{array}\right)$ (80)

and the vector

$K_{a}=\left(\begin{array}[]{c}-\dot{a}_{3}x+c_{7}y+c_{8}z\\\
-c_{7}x-\dot{a}_{3}y+c_{9}z\\\ -c_{8}x-c_{9}y-\dot{a}_{3}z\\\
\end{array}\right).$ (81)

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.

###  10.1 $a_{3}(t)=0$

In this case the conditions (79) are satisfied identically leaving the
function $\omega(t)$ free to be any function.

Therefore the KT (80) 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 (81) 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 (58) implies that (since $K_{a}q^{a}=0$) $K=G(x,y,z)$
which when replaced in (57) 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. We note that $I$ 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 result.

###### Proposition 2

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 $L_{i}$.

###  10.2 $c_{A}=0$ where $A=1,2,...,9$

In this case the conditions (79) imply that $a_{3}(t)=b_{0}+b_{1}t+b_{2}t^{2}$
and

$\omega_{(\nu)}(t)=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$
(82)

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 (80) and (81) become

$K_{ab}=a_{3}\delta_{ab},\enskip K_{a}=-\dot{a}_{3}q_{a}.$

Substituting in the remaining constraints (57) and (58) 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

$J_{\nu}=(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k(b_{0}+b_{1}t+b_{2}t^{2})^{\frac{\nu-2}{2}}}{r^{\nu}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}.$
(83)

We note that the resulting time-dependent generalized Kepler potential

$V=-\frac{\omega_{\nu}(t)}{r^{\nu}},\enskip\omega_{\nu}=k\left(b_{0}+b_{1}t+b_{2}t^{2}\right)^{\frac{\nu-2}{2}}$
(84)

is a subcase of the Case III potential of [18] if we set the function

$U\left(\frac{r}{\phi}\right)=k_{1}\frac{r^{2}}{\phi^{2}}-\frac{k\phi^{\nu}}{r^{\nu}}$

with

$\phi=\sqrt{b_{0}+b_{1}t+b_{2}t^{2}},\enskip
k_{1}=\frac{b_{0}b_{2}}{2}-\frac{b_{1}^{2}}{8}.$

Then the associated QFI (3.13) of [18] (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 subsection 11.2 below).

\- $\nu=2$ (Newton-Cotes potential [31]).

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

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

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
[27] (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 (111) found below for
$a_{3}(t)=b_{0}+b_{1}t+b_{2}t^{2}$. Substituting this $a_{3}(t)$ in (110) and
(111) 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

$I_{ij}=\Lambda_{ij}(a_{3}=b_{0}+b_{1}t+b_{2}t^{2})=(b_{0}+b_{1}t+b_{2}t^{2})\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-(b_{1}+2b_{2}t)q_{(i}\dot{q}_{j)}+b_{2}q_{i}q_{j}.$ (85)

Therefore the trace $Tr[I_{ij}]=I_{11}+I_{22}+I_{33}=2J_{-2}$. 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 3 (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 (83).

##  11 The time-dependent Kepler potential

In this case $\nu=1$ and conditions (75) - (78) give

$a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=0,\enskip a_{5}=a_{8},\enskip
a_{2}=a_{12},\enskip a_{3}=a_{9}=a_{13},\enskip a_{11}=a_{15}$
$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0$
$\sigma_{2}=c_{7},\enskip\sigma_{3}=c_{8},\enskip\tau_{3}=c_{9}.$

Then constraint (59) gives

$\dddot{a}_{3}=0,\enskip\sigma_{4}=\tau_{4}=\eta_{4}=0$

and

$a_{3}\omega^{2}=c_{10},\enskip a_{2}\omega=c_{11},\enskip
a_{5}\omega=c_{12},\enskip a_{11}\omega=c_{13}$

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

$\ddot{a}_{2}=\ddot{a}_{5}=\ddot{a}_{11}=0,\enskip\dddot{a}_{3}=0,\enskip
a_{3}\omega^{2}=c_{10},\enskip a_{2}\omega=c_{11},\enskip
a_{5}\omega=c_{12},\enskip a_{11}\omega=c_{13}.$ (86)

From the last conditions follow that in order QFIs to be admitted 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], [17], [18]. We note
that the time-dependent Kepler potential $V=-\frac{\omega_{2K}(t)}{r}$ is a
subcase of the Case II potential of [18] 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 [18] (see subsection 10.2).

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 10.1.

###  11.1 $\omega(t)=\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$,
$c_{11}b_{1}\neq 0$

In that case conditions (86) give

$a_{2}=b_{0}+b_{1}t,\enskip
a_{3}=\frac{c_{10}}{c_{11}^{2}}(b_{0}+b_{1}t)^{2},\enskip
a_{5}=\frac{c_{12}}{c_{11}}(b_{0}+b_{1}t),\enskip
a_{11}=\frac{c_{13}}{c_{11}}(b_{0}+b_{1}t).$

Substituting the resulting vector $K_{a}$ and the KT $K_{ab}$ in (58) we find
the solution

$K(q,t)=-\frac{2c_{10}b_{1}t}{c_{11}r}+G(q).$

Replacing this solution in the remaining constraint (57) 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

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

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

$\displaystyle L_{i}$ $\displaystyle\equiv$ $\displaystyle
q_{i+1}\dot{q}_{i+2}-q_{i+2}\dot{q}_{i+1}$ (87) $\displaystyle E_{2}$
$\displaystyle\equiv$
$\displaystyle(b_{0}+b_{1}t)^{2}\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{c_{11}}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)q^{i}\dot{q}_{i}+\frac{b_{1}^{2}r^{2}}{2}$
(88) $\displaystyle\tilde{R}_{i}$ $\displaystyle\equiv$
$\displaystyle(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{c_{11}}{r(b_{0}+b_{1}t)}q_{i}$
(89) $\displaystyle A_{i}$ $\displaystyle\equiv$
$\displaystyle(b_{0}+b_{1}t)\tilde{R}_{i}+b_{1}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right).$
(90)

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

$x=q_{1}=q_{4}=q_{7}=...,\enskip y=q_{2}=q_{5}=q_{8}=...,\enskip
z=q_{3}=q_{6}=q_{9}.$

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 (88) - (90) are written compactly as follows

$\displaystyle E_{2}$ $\displaystyle\equiv$ $\displaystyle
c_{11}^{2}\left[\frac{1}{\omega_{2K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{2K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{2K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{2K}}\right)^{2}\frac{r^{2}}{4}\right]$
(91) $\displaystyle\tilde{R}_{i}$ $\displaystyle\equiv$
$\displaystyle(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{\omega_{2K}}{r}q_{i}$
(92) $\displaystyle A_{i}$ $\displaystyle\equiv$ $\displaystyle
c_{11}\left[\frac{1}{\omega_{2K}}\tilde{R}_{i}-\frac{(\ln\omega_{2K})^{\cdot}}{\omega_{2K}}\left(q_{i+2}L_{i+1}-q_{i+1}L_{i+2}\right)\right]$
(93)

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

$\mathbf{A}\cdot\mathbf{L}=0,\enskip
2E_{2}\mathbf{L}^{2}+c_{11}^{2}=\mathbf{A}^{2}.$ (94)

For $b_{1}=0$, $b_{0}\neq 0$ 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 (94) reduces to the
well-known relation

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

###  11.2
$\omega(t)=\omega_{3K}(t)=\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$, $k\neq
0$, $b_{1}^{2}-4b_{2}b_{0}\neq 0$

In that case conditions (86) give555Observe that if $b_{1}^{2}-4b_{2}b_{0}=0$
this case reduces to the case of the subsection 11.1 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 $b_{2}(t-t_{0})^{2}$.

$a_{2}=a_{5}=a_{11}=0,\enskip c_{11}=c_{12}=c_{13}=0,\enskip
a_{3}=\frac{c_{10}}{k^{2}}(b_{0}+b_{1}t+b_{2}t^{2}).$

Substituting the $K_{a}$ and $K_{ab}$ of that case in (58) we find the
solution

$K(q,t)=-\frac{2c_{10}}{r\omega_{3K}}+G(q).$

When this solution is introduced in the remaining constraint (57) 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

$E_{3}\equiv(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}$
(95)

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

$E_{3}=k^{2}\left[\frac{1}{\omega_{3K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{3K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{3K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{3K}}\right)^{2}\frac{r^{2}}{4}\right].$
(96)

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 (88), (95) can be written compactly as (see eq.
(2.86) in [17])

$E_{\mu}=k^{2}\left[\frac{1}{\omega_{\mu
K}^{2}}\left(\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{\omega_{\mu
K}}{r}\right)-\frac{1}{2}\frac{d}{dt}\left(\frac{1}{\omega_{\mu
K}}\right)^{2}q^{i}\dot{q}_{i}+\frac{d^{2}}{dt^{2}}\left(\frac{1}{\omega_{\mu
K}}\right)^{2}\frac{r^{2}}{4}\right]$ (97)

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 4 (Time-dependent Kepler potentials which admit additional
FIs [17])

The time-dependent Kepler potential $V(t,q)=-\frac{\omega(t)}{r}$ for the
function $\omega_{2K}(t)=\frac{c_{11}}{b_{0}+b_{1}t}$ , $c_{11}b_{1}\neq 0$
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 (88), (90) and (95) respectively.

##  12 The 3d time-dependent oscillator

In this case $\nu=-2$ and conditions (75) - (78) give

$a_{2}=a_{5}=a_{8}=a_{11}=a_{12}=a_{15}=a_{16}=a_{18}=0$

and

$\enskip\dot{\sigma}_{2}=-\ddot{a}_{17},\enskip\dot{\sigma}_{3}=-\ddot{a}_{19},\enskip\dot{\tau}_{3}=-\ddot{a}_{20}.$
(98)

Then the constraint (59) implies that

$\ddot{\sigma}_{4}-2\omega\sigma_{4}=0,\enskip\ddot{\tau}_{4}-2\omega\tau_{4}=0,\enskip\ddot{\eta}_{4}-2\omega\eta_{4}=0,$
(99)
$\dddot{a}_{3}-8\omega\dot{a}_{3}-4\dot{\omega}a_{3}=0,\enskip\dddot{a}_{9}-8\omega\dot{a}_{9}-4\dot{\omega}a_{9}=0,\enskip\dddot{a}_{13}-8\omega\dot{a}_{13}-4\dot{\omega}a_{13}=0,$
(100)
$\dddot{a}_{17}-8\omega\dot{a}_{17}-4\dot{\omega}a_{17}=0,\enskip\dddot{a}_{19}-8\omega\dot{a}_{19}-4\dot{\omega}a_{19}=0,\enskip\dddot{a}_{20}-8\omega\dot{a}_{20}-4\dot{\omega}a_{20}=0.$
(101)

Therefore

$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}y^{2}+\frac{c_{1}}{2}z^{2}+c_{6}yz+a_{3}$
$\displaystyle K_{12}$ $\displaystyle=$
$\displaystyle\frac{c_{4}}{2}z^{2}-\frac{c_{2}}{2}xy-\frac{c_{6}}{2}xz-\frac{c_{5}}{2}yz+a_{17}$
$\displaystyle K_{13}$ $\displaystyle=$
$\displaystyle\frac{c_{5}}{2}y^{2}-\frac{c_{6}}{2}xy-\frac{c_{1}}{2}xz-\frac{c_{4}}{2}yz+a_{19}$
$\displaystyle K_{22}$ $\displaystyle=$
$\displaystyle\frac{c_{2}}{2}x^{2}+\frac{c_{3}}{2}z^{2}+c_{5}xz+a_{13}$ (102)
$\displaystyle K_{23}$ $\displaystyle=$
$\displaystyle\frac{c_{6}}{2}x^{2}-\frac{c_{5}}{2}xy-\frac{c_{4}}{2}xz-\frac{c_{3}}{2}yz+a_{20}$
$\displaystyle K_{33}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{2}x^{2}+\frac{c_{3}}{2}y^{2}+c_{4}xy+a_{9}$

and

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

Before we proceed with considering various subcases it is important that we
discuss the ordinary differential equations (ODEs) (100) and (101).

###  12.1 The Lewis invariant

Equations of the form

$\dddot{a}-8\omega\dot{a}-4\dot{\omega}a=0$ (104)

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

$a\ddot{a}-\frac{1}{2}\dot{a}^{2}-4\omega a^{2}=c_{0}=const.$ (105)

By putting $a=-\rho^{2}$ where $\rho=\rho(t)$ equation (105) becomes

$\ddot{\rho}-2\omega\rho-\frac{c_{0}}{2\rho^{3}}=0.$ (106)

For $2\omega(t)=-\psi^{2}(t)$ equation (106) is written

$\ddot{\rho}+\psi^{2}\rho-\frac{c_{0}}{2\rho^{3}}=0.$ (107)

Equation (107) is the auxiliary equation (see [8], [34], [35]) that should be
introduced in order to derive the Lewis invariant for the one-dimensional (1d)
time-dependent oscillator

$\ddot{x}+\psi^{2}x=0.$ (108)

By eliminating the $\psi^{2}$ using (108) and multiplying with the factor
$x\dot{\rho}-\rho\dot{x}$ equation (107) gives

$\ddot{\rho}-\frac{\rho}{x}\ddot{x}-\frac{c_{0}}{2\rho^{3}}=0\implies\left[\frac{1}{2}\left(x\dot{\rho}-\rho\dot{x}\right)^{2}+\frac{c_{0}}{4}\left(\frac{x}{\rho}\right)^{2}\right]^{\cdot}=0\implies$
$I\equiv\frac{1}{2}\left(x\dot{\rho}-\rho\dot{x}\right)^{2}+\frac{c_{0}}{4}\left(\frac{x}{\rho}\right)^{2}=const$
(109)

which is the well-known Lewis invariant for the 1d time-dependent harmonic
oscillator or, equivalently, a FI for the two-dimensional (2d) time-dependent
system with equations of motion (107) and (108).

###  12.2 The system of equations (98) - (101)

The conditions (99) are not involved into the conditions (98), (100) and
(101). 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.

####  12.2.1 $a_{3}\neq 0$, $\sigma_{4}=\tau_{4}=\eta_{4}=0$

Because the ODEs (100), (101) are independent (i.e. each one leads to a
different FI) and are of the same form without loss of generality we assume

$a_{9}=k_{1}a_{3},\enskip a_{13}=k_{2}a_{3},\enskip a_{17}=k_{3}a_{3},\enskip
a_{19}=k_{4}a_{3},\enskip a_{20}=k_{5}a_{3}$

where $k_{1},k_{2},k_{3},k_{4},k_{5}$ are arbitrary constants.

From the discussion of subsection 12.1 and the assumption $a_{3}\neq 0$
condition (100) concerning $a_{3}(t)$ becomes (see eq. (9.2) in [8])

$\dddot{a}_{3}-8\omega\dot{a}_{3}-4\dot{\omega}a_{3}=0\implies
a_{3}\ddot{a}_{3}-\frac{1}{2}\dot{a}_{3}^{2}-4\omega
a_{3}^{2}=c_{0}\implies\omega(t)=\frac{\ddot{a}_{3}}{4a_{3}}-\frac{1}{8}\left(\frac{\dot{a}_{3}}{a_{3}}\right)^{2}-\frac{c_{0}}{4a_{3}^{2}}$
(110)

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

Moreover, conditions (98) become

$\sigma_{2}=-\dot{a}_{17},\enskip\sigma_{3}=-\dot{a}_{19},\enskip\tau_{3}=-\dot{a}_{20}$

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 (102) and the vector (103) become666We 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 (57) and (58) 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 (110) 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 [8])

$\Lambda_{ij}=a_{3}\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{a}_{3}q_{(i}\dot{q}_{j)}+\frac{\ddot{a}_{3}}{2}q_{i}q_{j}.$
(111)

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

If we make the transformation (see subsection 12.1) $a_{3}(t)=-\rho^{2}(t)$
and $2\omega(t)=-\psi^{2}(t)$, equation (54) becomes

$\ddot{q}^{a}-2\omega q^{a}=0\implies\ddot{q}^{a}+\psi^{2}q^{a}=0$ (112)

and the QFIs (111) give

$\Lambda_{ij}=-\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\left(\rho\dot{q}_{j}-\dot{\rho}q_{j}\right)-\frac{c_{0}}{2}\rho^{-2}q_{i}q_{j}$
(113)

where the condition (110) takes the form (107).

The symmetric tensor (113) may be thought of as a 3d generalization of the 1d
Lewis invariant (109). Moreover equation (113) coincides with eq. (8) in [14]
and eq. (1.4) in [8] when $c_{0}=2$.

####  12.2.2 $a_{3}=a_{9}=a_{13}=a_{17}=a_{19}=a_{20}=0$, $\sigma_{4}\neq 0$

In this case the conditions (100), (101) vanish identically; and the
conditions (98) imply that $\sigma_{2}=c_{7}$, $\sigma_{3}=c_{8}$ and
$\tau_{3}=c_{9}$.

Since the remaining ODEs (99) are all independent (i.e. each one generates an
independent FI) and of the same form without loss of generality we assume

$\tau_{4}=k_{1}\sigma_{4},\enskip\eta_{4}=k_{2}\sigma_{4}$

where $k_{1},k_{2}$ are arbitrary constants.

From (99) for $\sigma_{4}\neq 0$ we get

$\omega(t)=\frac{\ddot{\sigma}_{4}}{2\sigma_{4}}.$ (114)

The parameters $c_{A}$ where $A=1,2,...,9$ produce the FIs of the angular
momentum and we fix them to zero. Therefore

$K_{ab}=0,\enskip K_{a}=\sigma_{4}\left(1,k_{1},k_{2}\right).$

Substituting in the remaining constraints (57) and (58) 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 [8])

$I_{4i}=f\dot{q}_{i}-\dot{f}q_{i}$ (115)

where $f(t)$ is an arbitrary non-zero function satisfying (114). We note that
the LFIs (115) 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 5 (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 $a_{3}(t)\neq 0$, $c_{0}$ 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 (111) and the
LFIs (115) respectively.

##  13 A special class of time-dependent oscillators

In proposition 5 it has been shown that the time-dependent oscillator
($\nu=-2$) for the frequency

$\omega_{1O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}$
(116)

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

$\Lambda_{ij}=f(t)\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{f}q_{(i}\dot{q}_{j)}+\frac{\ddot{f}}{2}q_{i}q_{j}$
(117)

and for the frequency

$\omega_{2O}(t)=\frac{\ddot{g}}{2g(t)}$ (118)

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

$I_{4i}=g(t)\dot{q}_{i}-\dot{g}q_{i}.$ (119)

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
$f(t),g(t)$ as follows

$\omega_{3O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}=\frac{\ddot{g}}{2g(t)}.$
(120)

It can be easily proved that

$g=f^{1/2}\cos\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{f(t)}$
(121)

and

$g=f^{1/2}\sin\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{f(t)}$
(122)

satisfy the requirement (120) for any non-zero function $f(t)$. In other words
all the time-dependent oscillators with frequency

$\omega_{3O}(t)=\frac{\ddot{f}}{4f(t)}-\frac{1}{8}\left(\frac{\dot{f}}{f}\right)^{2}-\frac{c_{0}}{4f^{2}}$
(123)

admit the six QFIs

$\Lambda_{ij}=f(t)\left(\dot{q}_{i}\dot{q}_{j}-2\omega
q_{i}q_{j}\right)-\dot{f}q_{(i}\dot{q}_{j)}+\frac{\ddot{f}}{2}q_{i}q_{j}$
(124)

and the six LFIs

$\displaystyle I_{41i}$ $\displaystyle=$
$\displaystyle\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\sin\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\cos\theta$
(125) $\displaystyle I_{42i}$ $\displaystyle=$
$\displaystyle-\left(\frac{c_{0}}{2}\right)^{1/2}f^{-1/2}q_{i}\cos\theta+\left(f^{1/2}\dot{q}_{i}-\frac{\dot{f}}{2}f^{-1/2}q_{i}\right)\sin\theta.$
(126)

These are the LFIs $J^{k}_{3}$, $J^{k}_{4}$ derived in eqs. (44), (45) in [10]
using Noether point symmetries and Noether’s theorem.

We note that

$\frac{dI_{42i}}{d\theta}=I_{41i}$ (127)

and

$\Lambda_{ij}=I_{41i}I_{41j}+I_{42i}I_{42j}.$ (128)

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

$L_{ij}=q_{i}\dot{q}_{j}-q_{j}\dot{q}_{i}=\varepsilon_{ijk}L^{k}$ (129)

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 [10])

$L_{ij}=\left(\frac{2}{c_{0}}\right)^{1/2}\left(I_{41i}I_{42j}-I_{41j}I_{42i}\right).$
(130)

###### Proposition 6

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 (123), the only independent FIs are
the LFIs $I_{41i},I_{42i}$.

In order to recover the results of [10], we assume a time-dependent oscillator
with $\omega_{3O}(t)$ given by (123) and we write the non-zero function $f(t)$
in the form $f(t)=\rho^{2}(t)$. Then equation (123) becomes

$\omega_{3O}(t)=\frac{\ddot{\rho}}{2\rho}-\frac{c_{0}}{4\rho^{4}}.$ (131)

The relations (121), (122) become

$g=\rho\cos\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-2}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{\rho^{2}}$
(132)
$g=\rho\sin\theta,\enskip\dot{\theta}=\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-2}\implies\theta(t)=\left(\frac{c_{0}}{2}\right)^{1/2}\int\frac{dt}{\rho^{2}}$
(133)

and the LFIs (125), (126) take the form

$\displaystyle I_{41i}$ $\displaystyle=$
$\displaystyle\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-1}q_{i}\sin\theta+\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\cos\theta$
(134) $\displaystyle I_{42i}$ $\displaystyle=$
$\displaystyle-\left(\frac{c_{0}}{2}\right)^{1/2}\rho^{-1}q_{i}\cos\theta+\left(\rho\dot{q}_{i}-\dot{\rho}q_{i}\right)\sin\theta.$
(135)

These latter expressions for $c_{0}=2$ coincide with the independent LFIs (44)
and (45) found in [10].

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 [27] can be
written in terms of $I_{41i},I_{42j}$. Indeed we have
$I_{3i\pm}(k>0)=I_{41i}\mp iI_{42i}$ and $I_{3i\pm}(k<0)=I_{41i}\pm iI_{42i}$.

##  14 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 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 $\tilde{R}_{i}=(\dot{q}^{j}\dot{q}_{j})q_{i}-(\dot{q}^{j}q_{j})\dot{q}_{i}-\frac{k}{r(b_{0}+b_{1}t)}q_{i}$
$\frac{k}{(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}$ | $E_{3}=(b_{0}+b_{1}t+b_{2}t^{2})\left[\frac{\dot{q}^{i}\dot{q}_{i}}{2}-\frac{k}{r(b_{0}+b_{1}t+b_{2}t^{2})^{1/2}}\right]-\frac{b_{1}+2b_{2}t}{2}q^{i}\dot{q}_{i}+\frac{b_{2}r^{2}}{2}$
$2$ | $k$ |  $I_{1}=-H_{2}t^{2}+t(\dot{q}^{i}q_{i})-\frac{r^{2}}{2}$, $I_{2}=-H_{2}t+\frac{1}{2}(\dot{q}^{i}q_{i})$

##  15 Integrating the equations

In this section we use the independent LFIs $I_{41i},I_{42i}$ to integrate the
equations of the special class of 3d time-dependent oscillators ($\nu=-2$)
defined in section 13 with $\omega(t)$ given by (123). 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
subsection 11.1).

###  15.1 The 3d time-dependent oscillator with $\omega(t)$ given by (123)

Using the LFIs (125) and (126) we find

$q_{i}(t)=\left(\frac{2}{c_{0}}\right)^{1/2}f^{1/2}\Big{(}I_{41i}\sin\theta-
I_{42i}\cos\theta\Big{)}$ (136)

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

The solution (136) coincides with the solution (52) in [10].

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 (54) and
the defining relation (123) for $\omega(t)$ become

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle-\psi^{2}x$ (137)
$\displaystyle\ddot{\rho}$ $\displaystyle=$
$\displaystyle-\psi^{2}\rho+\rho^{-3}.$ (138)

The LFIs (134) and (135) become

$\displaystyle I_{41}$ $\displaystyle=$
$\displaystyle\rho^{-1}x\sin\theta+\left(\rho\dot{x}-x\dot{\rho}\right)\cos\theta$
(139) $\displaystyle I_{42}$ $\displaystyle=$
$\displaystyle-\rho^{-1}x\cos\theta+\left(\rho\dot{x}-x\dot{\rho}\right)\sin\theta.$
(140)

The general solution (136) is

$x(t)=\rho(t)\Big{(}I_{41}\sin\theta-I_{42}\cos\theta\Big{)}$ (141)

where $\dot{\theta}=\rho^{-2}$ and $\rho(t)$ is a given non-zero function
which defines $\psi(t)$ through (138). This is the 1d solution (9) in [10].

###  15.2 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 subsection 11.1 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

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

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
$x=r\cos\theta$, $y=r\sin\theta$. 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

$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}t)^{2}\left[\frac{\dot{r}^{2}}{2}+\frac{L_{3}^{2}}{2r^{2}}-\frac{k}{r(b_{0}+b_{1}t)}\right]-b_{1}(b_{0}+b_{1}t)r\dot{r}+\frac{b_{1}^{2}r^{2}}{2}$
(142) $\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\sin\theta+\Big{[}(b_{0}+b_{1}t)\frac{L_{3}^{2}}{r}-k\Big{]}\cos\theta$
(143) $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle-
L_{3}\Big{[}(b_{0}+b_{1}t)\dot{r}-b_{1}r\Big{]}\cos\theta+\Big{[}(b_{0}+b_{1}t)\frac{L_{3}^{2}}{r}-k\Big{]}\sin\theta.$
(144)

By multiplying equation (143) with $\cos\theta$ and (144) with $\sin\theta$ we
find that

$\frac{1}{r}=\frac{k}{L_{3}^{2}(b_{0}+b_{1}t)}\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)\implies
r=\frac{L_{3}^{2}(b_{0}+b_{1}t)}{k\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)}$
(145)

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 (145) is written (see also section 5 in
[17])

$\frac{1}{r}=\frac{\omega_{2K}}{L_{3}^{2}}\Big{[}1+\alpha\cos\left(\theta-\beta\right)\Big{]}\implies
r=\frac{L_{3}^{2}\omega_{2K}^{-1}}{1+\alpha\cos\left(\theta-\beta\right)}$
(146)

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 (94) becomes

$2E_{2}L_{3}^{2}+k^{2}=\alpha^{2}k^{2}\implies
2E_{2}L_{3}^{2}=k^{2}(\alpha^{2}-1).$

Moreover, equation (142) 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 (54) for $\nu=1$ become

$\displaystyle\ddot{r}-r\dot{\theta}^{2}+\frac{\omega_{2K}}{r^{2}}$
$\displaystyle=$ $\displaystyle 0$ (147) $\displaystyle
r\ddot{\theta}+2\dot{r}\dot{\theta}$ $\displaystyle=$ $\displaystyle 0.$ (148)

Equation (148) implies the FI of the angular momentum
$L_{3}=r^{2}\dot{\theta}$. It can be easily checked that the solution (145)
satisfies equation (147) by replacing $\ddot{\theta}$ from (148) and
$\dot{\theta}$ with $\frac{L_{3}}{r^{2}}$. The solution (145) into the FI
$L_{3}$ gives

$\int\frac{k^{2}dt}{L_{3}^{3}(b_{0}+b_{1}t)^{2}}=\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}\implies\frac{k}{L_{3}^{2}(b_{0}+b_{1}t)}=-\frac{b_{1}L_{3}}{k}\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}.$
(149)

Substituting (149) in (145) we obtain

$\frac{1}{r}=-\frac{b_{1}L_{3}}{k}\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)\int\frac{d\theta}{\left(1+k_{1}\cos\theta+k_{2}\sin\theta\right)^{2}}$
(150)

which coincides with eq. (5.17) in [17].

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

In this section we use the well-known result [12] that the non-linear
dynamical system

$\ddot{q}^{a}=-\Gamma^{a}_{bc}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)+\phi(t)\dot{q}^{a}$
(151)

is equivalent to the linear dynamical system (without damping term)

$\frac{d^{2}q^{a}}{ds^{2}}=-\Gamma^{a}_{bc}\frac{dq^{b}}{ds}\frac{dq^{c}}{ds}-\bar{\omega}(s)Q^{a}(q)$
(152)

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

$s(t)=\int
e^{\int\phi(t)dt}dt,\enskip\bar{\omega}(s)=\omega(t(s))\left(\frac{dt}{ds}\right)^{2}\iff\omega(t)=\bar{\omega}(s(t))e^{2\int\phi(t)dt}.$
(153)

We apply this result to the following problem:

_Consider the second order differential equation_

$\ddot{x}=-\omega(t)x^{\mu}+\phi(t)\dot{x}$ (154)

_where the constant_ $\mu\neq-1$_and determine the relation between the
functions_ $\omega(t),\phi(t)$_for which the equation admits a QFI, therefore
it is integrable._

This problem has been considered previously in [36], [37] (see eq. (28a) in
[36] and eq. (17) in [37]) and has been answered partially using different
methods. In [36] the author used the Hamiltonian formalism where one looks for
a canonical transformation to bring the Hamiltonian in a time-separable form.
In [37] the author used a direct method for constructing FIs by multiplying
the equation with an integrating factor. In [37] it is shown that both methods
are equivalent and that the results of [37] generalize those of [36]. In the
following we shall generalize the results of [37]; in addition we discuss a
number of applications.

Equation (154) is equivalent to the equation

$\frac{d^{2}x}{ds^{2}}=-\bar{\omega}(s)x^{\mu},\enskip\mu\neq-1$ (155)

where the function $\bar{\omega}(s)$ is given by (153).

Replacing with $Q^{1}=x^{\mu}$ in the system of equations (5) - (10) we find
that777 In 1d Euclidean space the KT condition (5) $K_{(ab;c)}=0$ becomes
$K_{11,1}=0$ $\implies K_{11}=K_{11}(s)$, that is, it is an arbitrary function
of $s$.  $K_{11}=K_{11}(s)$ and the following conditions

$\displaystyle K_{1}(s,x)$ $\displaystyle=$
$\displaystyle-\frac{dK_{11}}{ds}x+b_{1}(s)$ (156) $\displaystyle K(s,x)$
$\displaystyle=$ $\displaystyle
2\bar{\omega}K_{11}\frac{x^{\mu+1}}{\mu+1}+\frac{d^{2}K_{11}}{ds^{2}}\frac{x^{2}}{2}-\frac{db_{1}}{ds}x+b_{2}(s)$
(157) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\frac{2\frac{d\bar{\omega}}{ds}K_{11}}{\mu+1}+\frac{2\bar{\omega}\frac{dK_{11}}{ds}}{\mu+1}+\bar{\omega}\frac{dK_{11}}{ds}\right)x^{\mu+1}-\bar{\omega}b_{1}x^{\mu}+\frac{d^{3}K_{11}}{ds^{3}}\frac{x^{2}}{2}-\frac{d^{2}b_{1}}{ds^{2}}x+\frac{db_{2}}{ds}$
(158)

where $b_{1}(s),b_{2}(s)$ are arbitrary functions. Then the general QFI (3)
becomes

$I=K_{11}(s)\left(\frac{dx}{ds}\right)^{2}+K_{1}(s,x)\frac{dx}{ds}+K(s,x).$
(159)

We consider the solution of the system (156) - (158) for various values of
$\mu$.

As will be shown for $\mu\neq-1$ results a family of ‘frequencies’
$\bar{\omega}(s)$ parameterized with constants. However, 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

$I=K_{11}\left(\frac{dx}{ds}\right)^{2}-\frac{dK_{11}}{ds}x\frac{dx}{ds}+b_{1}(s)\frac{dx}{ds}+c_{3}x^{2}+2\bar{\omega}(s)K_{11}x-\frac{db_{1}}{ds}x+\int
b_{1}(s)\bar{\omega}(s)ds$ (160)

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

$\frac{d^{2}b_{1}}{ds^{2}}=2\frac{d\bar{\omega}}{ds}K_{11}+3\bar{\omega}\frac{dK_{11}}{ds}.$
(161)

Using the transformation (153) equations (160), (161) become

$\displaystyle I$ $\displaystyle=$ $\displaystyle\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}\dot{x}^{2}-\left[c_{2}+2c_{3}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}x\dot{x}+$ (162)
$\displaystyle+b_{1}(s(t))e^{-\int\phi(t)dt}\dot{x}+c_{3}x^{2}+2\omega(t)\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}x-$
$\displaystyle-\dot{b}_{1}e^{-\int\phi(t)dt}x+\int
b_{1}(s(t))\omega(t)e^{-\int\phi(t)dt}dt$

and

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

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 (153) we
deduce that the original equation

$\ddot{x}=-\omega(t)x+\phi(t)\dot{x}$ (164)

for the frequency

$\omega(t)=-\rho^{-1}\ddot{\rho}+\phi(\ln\rho)^{\cdot}+\rho^{-4}e^{2\int\phi(t)dt}$
(165)

admits the general solution

$x(t)=\rho(t)\left(A\sin\theta+B\cos\theta\right)$ (166)

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

$I=K_{11}(s)\left(\frac{dx}{ds}\right)^{2}-\frac{dK_{11}}{ds}x\frac{dx}{ds}+(c_{4}+c_{5}s)\frac{dx}{ds}+\frac{2}{3}K_{11}^{-3/2}x^{3}+\frac{d^{2}K_{11}}{ds^{2}}\frac{x^{2}}{2}-c_{5}x$
(167)

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

$\frac{d^{3}K_{11}}{ds^{3}}=2(c_{4}+c_{5}s)K_{11}^{-5/2}.$ (168)

Using the transformation (153) the above results become

$\omega(t)=K_{11}^{-5/2}e^{2\int\phi(t)dt}$ (169) $\displaystyle I$
$\displaystyle=$ $\displaystyle
K_{11}e^{-2\int\phi(t)dt}\dot{x}^{2}-\dot{K}_{11}e^{-2\int\phi(t)dt}x\dot{x}+\left[c_{4}+c_{5}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}\dot{x}+\frac{2}{3}K_{11}^{-3/2}x^{3}+$
(170)
$\displaystyle+\left(\ddot{K}_{11}-\phi\dot{K}_{11}\right)e^{-2\int\phi(t)dt}\frac{x^{2}}{2}-c_{5}x$

and

$\dddot{K}_{11}-3\phi\ddot{K}_{11}-\dot{\phi}\dot{K}_{11}+2\phi^{2}\dot{K}_{11}=2\left[c_{4}+c_{5}\int
e^{\int\phi(t)dt}dt\right]e^{3\int\phi(t)dt}K_{11}^{-5/2}$ (171)

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

We note that for $\mu=2$ equation (154), or to be more specific its equivalent
(155), 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. [38], [39], [40], [41], [42], [43]).

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 (159) becomes

$I=(c_{1}+c_{2}s+c_{3}s^{2})\left(\frac{dx}{ds}\right)^{2}-(c_{2}+2c_{3}s)x\frac{dx}{ds}+\frac{2}{\mu+1}(c_{1}+c_{2}s+c_{3}s^{2})^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}$
(172)

and the function

$\bar{\omega}(s)=(c_{1}+c_{2}s+c_{3}s^{2})^{-\frac{\mu+3}{2}}.$ (173)

It can be checked that (172), (173) for $\mu=0,1,2$ give results compatible
with the ones we found for these values of $\mu$.

Using the transformation (153) we deduce that the original system (154) is
integrable iff the functions $\omega(t),\phi(t)$ are related as follows

$\omega(t)=\left[c_{1}+c_{2}\int e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]^{-\frac{\mu+3}{2}}e^{2\int\phi(t)dt}.$
(174)

In this case the associated QFI (172) is

$\displaystyle I$ $\displaystyle=$ $\displaystyle\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]e^{-2\int\phi(t)dt}\dot{x}^{2}-\left[c_{2}+2c_{3}\int
e^{\int\phi(t)dt}dt\right]e^{-\int\phi(t)dt}x\dot{x}+$ (175)
$\displaystyle+\frac{2}{\mu+1}\left[c_{1}+c_{2}\int
e^{\int\phi(t)dt}dt+c_{3}\left(\int
e^{\int\phi(t)dt}dt\right)^{2}\right]^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}.$

These expressions generalize the ones given in [37]. Indeed if we introduce
the notation $\omega(t)\equiv\alpha(t)$, $\phi(t)\equiv-\beta(t)$, then
equations (174), (175) for $c_{3}=0$ become eqs. (25), (26) of [37].

###  16.1 The generalized Lane-Emden equation

Consider the 1d generalized Lane-Emden equation (see eq. (6) in [44])

$\ddot{x}=-\omega(t)x^{\mu}-\frac{k}{t}\dot{x}$ (176)

where $k$ is an arbitrary constant. This equation is well-known in the
literature because of its many applications in astrophysical problems (see
Refs. in [44]). In general, to find explicit analytic solutions of equation
(176) 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 (176) are guaranteed, if we find a way to determine
its FIs. We see that equation (176) is a subcase of the original equation
(154) for $\phi(t)=-\frac{k}{t}$, therefore we can apply the results found
earlier in section 16.

In what follows we discuss only the fourth case where $\mu\neq-1$ in order to
compare our results with those found in Table 1 of [44]. In particular, for
$\phi(t)=-\frac{k}{t}$ the function (174) and the associated QFI (175) become

$\omega(t)=t^{-2k}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)^{-\frac{\mu+3}{2}}$
(177)

and

$I=t^{2k}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)\dot{x}^{2}-t^{k}\left(c_{2}+2c_{3}M\right)x\dot{x}+\frac{2}{\mu+1}\left(c_{1}+c_{2}M+c_{3}M^{2}\right)^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}$
(178)

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 $M=\ln t$ and equations (177), (178) give

$\omega(t)=t^{-2}\left[c_{1}+c_{2}\ln t+c_{3}(\ln
t)^{2}\right]^{-\frac{\mu+3}{2}}$ (179)

and

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

We consider the following subcases:

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

Equations (179), (180) give 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 [44].

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

Equations (179), (180) give 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 [44].

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

Equations (181), (182) give 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 [44].

b) Case $k\neq 1$.

We have $M=\frac{t^{1-k}}{1-k}$ and equations (177), (178) give

$\omega(t)=t^{-2k}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]^{-\frac{\mu+3}{2}}$
(181)

and

$\displaystyle I$ $\displaystyle=$ $\displaystyle
t^{2k}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]\dot{x}^{2}-t^{k}\left(c_{2}+\frac{2c_{3}}{1-k}t^{1-k}\right)x\dot{x}+$
(182)
$\displaystyle+\frac{2}{\mu+1}\left[c_{1}+\frac{c_{2}}{1-k}t^{1-k}+\frac{c_{3}}{(1-k)^{2}}t^{2(1-k)}\right]^{-\frac{\mu+1}{2}}x^{\mu+1}+c_{3}x^{2}.$

We consider the following subcases:

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

Equations (181), (182) give 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 [44].

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

Equations (181), (182) give 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 [44].

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
[44] which is the Case 5.1 of [45].

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

Equations (181), (182) give 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 [44].

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
[44] which is the Case 5.2 of [45].

We conclude that the seven cases 1-7 found in Table 1 of [44] 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 $\mu=p$.

##  17 Conclusions

The purpose of the present work was to compute the QFIs of time-dependent
dynamical systems of the form
$\ddot{q}^{a}=-\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$, where
the connection coefficients are computed from the kinetic metric, using the
direct method instead of the Noether symmetries as it is usually done. In the
direct method one assumes that the QFI is of the form
$I=K_{ab}\dot{q}^{a}\dot{q}^{b}+K_{a}\dot{q}^{a}+K$ and demands that
$dI/dt=0$. This leads to a system of PDEs whose solution provides the QFIs.
One key result is that the tensor $K_{ab}$ is a KT of the kinetic metric.

We have discussed the solution of the system of equations at two levels. The
first level is purely geometric and concerns the KT $K_{ab}$; and the second
level is the physical one which concerns the quantities $\omega(t),Q^{a}(q)$
defining the dynamical system.

Concerning the first level we have applied two different methods:
a. The polynomial method in which one assumes a general polynomial form in the
variable $t$ both for the KT $K_{ab}$ and for the vector $K_{a}$.
b. The basis method where one computes first a basis of the KTs of order 2 of
the kinetic metric and then expresses $K_{ab}$ in this basis assuming that the
‘components’ are functions of $t$.
In both methods the key point is to compute the scalar $K$.

Concerning the dynamical quantities $\omega(t),Q^{a}(q)$ we have chosen to
work in two ways:
a. First we considered the polynomial method and assumed the function
$\omega(t)$ to be a polynomial leaving the quantities $Q^{a}$ unspecified. It
is found that in this case the resulting dynamical system admits two
independent QFIs whose explicit expression together with conditions involving
the quantities $Q^{a}$ and the collineations of the kinetic metric are given
in Theorem 1.
b. In the basis method we worked the other way. That is, we assumed the
quantities $Q^{a}(q)$ to be given by the time-dependent generalized Kepler
potential $V=-\frac{\omega(t)}{r^{\nu}}$ and determined the functions
$\omega(t)$ for which QFIs exist. The results of this detailed study are
displayed in Table LABEL:T2 for all values of $\nu$. For the values
$\nu=-2,1,2$ we recovered the known results concerning the time-dependent 3d
oscillator, the time-dependent Kepler potential and the Newton-Cotes potential
respectively. We note that these latter results have appeared over the years
in many works whereas in the present discussion occur as particular cases of a
single geometric approach.

The last part of our considerations concerns the well-known proposition that
under a reparameterization the linear damping $\phi(t)\dot{q}^{a}$ can be
absorbed to a time-dependent generalized force. We used this proposition in
the case of a 1d non-linear second order time-dependent differential equation,
we determined the condition that the time-dependent coefficients of the
equation must satisfy in order a QFI to exist and we computed this QFI. As an
application we studied the properties of the well-known generalized Lane-Emden
equation.

We note that one is possible to consider other dynamical quantities and/or
kinetic metric and compute the QFIs. What is the same in all cases is the
method of work which we hope we have presented adequately in the present work.

##  18 Appendix

Substituting the polynomial function $\omega(t)$ given by (39) in the system
of equations (34) - (38) we have the following cases.

I. Case $\mathbf{n=m}$ (both $n$, $m$ finite)

From equation (34) we obtain

$C_{(k)ab}=-L_{(k-1)(a;b)},\enskip k=1,...,n,\enskip L_{(n)(a;b)}=0.$ (183)

Therefore $L_{(n)a}$ is a KV of $\gamma_{ab}$.

Condition (37) gives

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\left(b_{1}+2b_{2}t+...+\ell
b_{\ell}t^{\ell-1}\right)\left(C_{(0)ab}Q^{b}+C_{(1)ab}Q^{b}t+...+C_{(n)ab}Q^{b}\frac{t^{n}}{n}\right)+2L_{(2)a}+6L_{(3)a}t+...+$
$\displaystyle+n(n-1)L_{(n)a}t^{n-2}-2\left(b_{0}+b_{1}t+...+b_{\ell}t^{\ell}\right)\left(C_{(1)ab}Q^{b}+C_{(2)ab}Q^{b}t+...+C_{(n)ab}Q^{b}t^{n-1}\right)+$
$\displaystyle+\left(b_{0}+b_{1}t+...+b_{\ell}t^{\ell}\right)\left[\left(L_{(0)b}Q^{b}\right)_{,a}+\left(L_{(1)b}Q^{b}\right)_{,a}t+...+\left(L_{(n-1)b}Q^{b}\right)_{,a}t^{n-1}+\left(L_{(n)b}Q^{b}\right)_{,a}t^{n}\right].$

This is a polynomial of the general form
$P_{(0)a}(q)+P_{(1)a}(q)t+...+P_{(n+\ell)a}(q)t^{n+\ell}=0$. The vanishing of
the coefficients $P_{(k)a}(q)$ in the last polynomial implies that

$L_{(n)a}Q^{a}=s=const$ (184)
$\sum_{s=0}^{\ell-1}\left[-\frac{2(k+s)b_{(k+s\leq\ell)}}{n-s}C_{(n-s\geq
0)ab}Q^{b}-2b_{(k+s\leq\ell)}C_{(n-s>0)ab}Q^{b}+b_{(k+s\leq\ell)}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right]=0$ (185)

where $k=1,2,...,\ell$,

$-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{n-s}C_{(n-s\geq
0)ab}Q^{b}\right]+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(n-s>0)ab}Q^{b}+b_{s}\left(L_{(n-s-1\geq
0)b}Q^{b}\right)_{,a}\right]=0$ (186)

and

$k(k-1)L_{(k)a}-\sum_{s=1}^{\ell}\left[\frac{2sb_{s}}{k-s-1}C_{(k-s-1\geq
0)ab}Q^{b}\right]+\sum_{s=0}^{\ell}\left[-2b_{s}C_{(k-s-1>0)ab}Q^{b}+b_{s}\left(L_{(k-s-2\geq
0)b}Q^{b}\right)_{,a}\right]=0$ (187)

where $k=2,3,...n$.

We note that in the $n+\ell+1$ formulae (185) - (187), when the undefined
quantity $\frac{C_{(0)ab}}{0}$ appears in the calculations, it must be
replaced by $C_{(0)ab}$ in order to have a consistent result.

We continue with the remaining constraints (35) and (36) in order to determine
the scalar coefficient $K(t,q)$.

The solution of (36) is

$\displaystyle K_{,t}$ $\displaystyle=$ $\displaystyle
L_{(0)a}Q^{a}\left(b_{0}+b_{1}t+...+b_{\ell}t^{\ell}\right)+L_{(1)a}Q^{a}\left(b_{0}t+b_{1}t^{2}+...+b_{\ell}t^{\ell+1}\right)+...+$
$\displaystyle+L_{(n-1)a}Q^{a}\left(b_{0}t^{n-1}+b_{1}t^{n}+...+b_{\ell}t^{n+\ell-1}\right)+s\left(b_{0}t^{n}+b_{1}t^{n+1}+...+b_{\ell}t^{n+\ell}\right)\implies$
$\displaystyle K$ $\displaystyle=$ $\displaystyle
L_{(0)a}Q^{a}\left(b_{0}t+b_{1}\frac{t^{2}}{2}+...+b_{\ell}\frac{t^{\ell+1}}{\ell+1}\right)+L_{(1)a}Q^{a}\left(b_{0}\frac{t^{2}}{2}+b_{1}\frac{t^{3}}{3}+...+b_{\ell}\frac{t^{\ell+2}}{\ell+2}\right)+...+$
$\displaystyle+L_{(n-1)a}Q^{a}\left(b_{0}\frac{t^{n}}{n}+b_{1}\frac{t^{n+1}}{n+1}+...+b_{\ell}\frac{t^{n+\ell}}{n+\ell}\right)+s\left(b_{0}\frac{t^{n+1}}{n+1}+b_{1}\frac{t^{n+2}}{n+2}+...+b_{\ell}\frac{t^{n+\ell+1}}{n+\ell+1}\right)+G(q).$

Replacing $K$ in (35) and using the conditions (184) - (187) we find that

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

Condition (38) is satisfied trivially from the above solutions.

The QFI is

$\displaystyle I$ $\displaystyle=$
$\displaystyle\left(\frac{t^{n}}{n}C_{(n)ab}+...+tC_{(1)ab}+C_{(0)ab}\right)\dot{q}^{a}\dot{q}^{b}+t^{n}L_{(n)a}\dot{q}^{a}+...+tL_{(1)a}\dot{q}^{a}+L_{(0)a}\dot{q}^{a}+$
$\displaystyle+L_{(0)a}Q^{a}\left(b_{0}t+b_{1}\frac{t^{2}}{2}+...+b_{\ell}\frac{t^{\ell+1}}{\ell+1}\right)+L_{(1)a}Q^{a}\left(b_{0}\frac{t^{2}}{2}+b_{1}\frac{t^{3}}{3}+...+b_{\ell}\frac{t^{\ell+2}}{\ell+2}\right)+...+$
$\displaystyle+L_{(n-1)a}Q^{a}\left(b_{0}\frac{t^{n}}{n}+b_{1}\frac{t^{n+1}}{n+1}+...+b_{\ell}\frac{t^{n+\ell}}{n+\ell}\right)+s\left(b_{0}\frac{t^{n+1}}{n+1}+b_{1}\frac{t^{n+2}}{n+2}+...+b_{\ell}\frac{t^{n+\ell+1}}{n+\ell+1}\right)+G(q)$

where $C_{(0)ab}$ is a KT, the KTs $C_{(k)ab}=-L_{(k-1)(a;b)}$ for
$k=1,...,n$, $L_{(n)a}$ is a KV such that $L_{(n)a}Q^{a}=s$,
$G_{,a}=2b_{0}C_{(0)ab}Q^{b}-L_{(1)a}$ and the conditions (185) - (187) are
satisfied.

II. Case $\mathbf{n\neq m}$. (one of $n$ or $m$ may be infinite)

We find QFIs that are subcases of those found in Case I and Case III which
follows.

III. Both $\mathbf{n}$, $\mathbf{m}$ are infinite.

In this case we consider the solution to have the form

$K_{ab}(t,q)=g(t)C_{ab}(q),\enskip K_{a}(t,q)=f(t)L_{a}(q)$

where the functions $g(t),f(t)$ are analytic so that they may be represented
by polynomial functions as follows

$g(t)=\sum^{n}_{k=0}c_{k}t^{k}=c_{0}+c_{1}t+...+c_{n}t^{n}$
$f(t)=\sum^{m}_{k=0}d_{k}t^{k}=d_{0}+d_{1}t+...+d_{m}t^{m}.$

In the above expressions the coefficients $c_{0},c_{1},...,c_{n}$ and
$d_{0},d_{1},...,d_{m}$ are arbitrary constants. We find that only the
following subcase gives a new independent FI. All other subcases give results
already found.

Subcase $\mathbf{(g=e^{\lambda t}}$, $\mathbf{f=e^{\mu t})}$,
$\mathbf{\lambda\mu\neq 0}$.

In this case the system of equations (34) - (37) becomes (equation (38) is
satisfied trivially from the solutions found below):

$\displaystyle\lambda e^{\lambda t}C_{ab}+e^{\mu t}L_{(a;b)}$ $\displaystyle=$
$\displaystyle 0$ (188)
$\displaystyle-2\left(b_{0}+b_{1}t+...+b_{\ell}t^{\ell}\right)e^{\lambda
t}C_{ab}Q^{b}+\mu e^{\mu t}L_{a}+K_{,a}$ $\displaystyle=$ $\displaystyle 0$
(189) $\displaystyle K_{,t}-(b_{0}+b_{1}t+...+b_{\ell}t^{\ell})e^{\mu
t}L_{a}Q^{a}$ $\displaystyle=$ $\displaystyle 0$ (190)
$\displaystyle-2\left(b_{1}+2b_{2}t+...+\ell
b_{\ell}t^{\ell-1}\right)e^{\lambda
t}C_{ab}Q^{b}-2\lambda(b_{0}+b_{1}t+...+b_{\ell}t^{\ell})e^{\lambda
t}C_{ab}Q^{b}+$ $\displaystyle+\mu^{2}e^{\mu
t}L_{a}+(b_{0}+b_{1}t+...+b_{\ell}t^{\ell})e^{\mu
t}\left(L_{b}Q^{b}\right)_{,a}$ $\displaystyle=$ $\displaystyle 0.$ (191)

We consider the following subcases.

a. For $\lambda\neq\mu$:

From (188) we have that $C_{ab}=0$ and $L_{a}$ is a KV.

From (191) we find that $L_{a}=0$.

Therefore, the QFI $I_{e}(\lambda\neq\mu)=const$ which is trivial.

b. For $\lambda=\mu$:

From (188) we have that $C_{ab}=-\frac{1}{\lambda}L_{(a;b)}$. Therefore
$L_{(a;b)}$ is a KT.

We consider two cases according to the degree $\ell$ of the polynomial
$\omega(t)$.

\- Case $\ell=1$.

From (191) we find that

$\displaystyle\left(L_{b}Q^{b}\right)_{,a}$ $\displaystyle=$ $\displaystyle
2\lambda C_{ab}Q^{b}$ (192)
$\displaystyle\lambda^{2}L_{a}+b_{0}\left(L_{b}Q^{b}\right)_{,a}-2(b_{1}+\lambda
b_{0})C_{ab}Q^{b}$ $\displaystyle=$ $\displaystyle 0.$ (193)

Replacing with $C_{ab}=-\frac{1}{\lambda}L_{(a;b)}$ and by substituting (192)
in (193) we obtain

$\displaystyle\left(L_{b}Q^{b}\right)_{,a}$ $\displaystyle=$
$\displaystyle-2L_{(a;b)}Q^{b}$ (194)
$\displaystyle\lambda^{3}L_{a}+2b_{1}L_{(a;b)}Q^{b}$ $\displaystyle=$
$\displaystyle 0.$ (195)

The solution of (190) is

$K=\left(\frac{b_{0}}{\lambda}-\frac{b_{1}}{\lambda^{2}}\right)e^{\lambda
t}L_{a}Q^{a}+\frac{b_{1}}{\lambda}te^{\lambda t}L_{a}Q^{a}+G(q)$

which when replaced in (189) gives $G_{,a}=0$, that is $G=const\equiv 0$.

The QFI is

$I_{e}(\ell=1)=-e^{\lambda t}L_{(a;b)}\dot{q}^{a}\dot{q}^{b}+\lambda
e^{\lambda
t}L_{a}\dot{q}^{a}+\left(b_{0}-\frac{b_{1}}{\lambda}\right)e^{\lambda
t}L_{a}Q^{a}+b_{1}te^{\lambda t}L_{a}Q^{a}$ (196)

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

\- Case $\ell>1$.

From (191) we find that $\left(L_{b}Q^{b}\right)_{,a}=2\lambda C_{ab}Q^{b}$,
$C_{ab}Q^{b}=0$ and $\lambda^{2}L_{a}=2b_{1}C_{ab}Q^{b}$.

Therefore $L_{a}=0$ and hence $C_{ab}=-\frac{1}{\lambda}L_{(a;b)}=0$. We end
up with a trivial FI $I_{e}=const$.

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