# Infinitesimal time reparametrisation and its applications

José F<EMAIL_ADDRESS>Eduardo
<EMAIL_ADDRESS>and Miguel C. Muñoz-
<EMAIL_ADDRESS>
1Departamento de Física Teórica and IUMA, Universidad de Zaragoza,
Pedro Cerbuna 12, E-50009 Zaragoza, Spain
2Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza,
Pedro Cerbuna 12, E-50009 Zaragoza, Spain
3Departament de Matemátiques, Campus Nord U.P.C., Ed. C-3
C/ Jordi Girona 1. E-08034 Barcelona, Spain

###### Abstract

A geometric approach to Sundman infinitesimal time-reparametrisation is given
and some of its applications are used to illustrate the general theory.
Special emphasis is put on geodesic motions and systems described by
mechanical type Lagrangians. The Jacobi metric appears as a particular case of
a Sundman transformation.

Mathematics Subject Classifications (2010):  34A34, 37N05, 53C15, 70F16

PACS numbers:  02.30.Hq, 02.40.Yy, 02.40.Hw, 02.40.Ky, 45.10.Na

#### Keywords:

Sundman transformation, Tangent bundle, regularisation, Jacobi metrics

##  1 Introduction

An infinitesimal time reparametrisation, usually called Sundman transformation
[1], was introduced when looking for an analytic solution to the three-body
problem. Such transformation is very intriguing, at least from a geometric
perspective, but it allowed to find solutions for many different problems in
the theory of differential equations and related physical problems. For
instance it has been very useful in problems of linearisation of differential
equations or to regularise some equations of motion and avoid collision
singularities. More generally, to transform a given equation into another of
some appropriate form.

Even if in the early days of the beginning of the nineteenth century the
methods developed for studying differential equations were of an ad hoc
character, after the pioneer work by Lie on symmetry methods, systematic
approaches have been developed, most of them based on a geometric theory of
differential equations and dynamical systems. The geometric approach is
intrinsic and the results do not depend on a particular choice of coordinates
and may be generalised to infinite dimensional systems (with some topological
difficulties).

Within this approach, an autonomous system of first order differential
equations is replaced by a vector field $X$ on a differentiable manifold $M$,
the system being used to compute, in a local coordinate system, the integral
curves of the vector field. But the time is not explicitly appearing in the
expression of the vector field and only appears as the parameter of such
integral curves, and therefore the geometric interpretation of an
infinitesimal change of time is not clear. It will be shown that the
infinitesimal time reparametrisation can then simply be understood as a change
of the dynamical vector field $X$, replacing it by a conformally related one,
$f\,X$, where $f$ is a nonvanishing real function on the manifold $M$.

The existence of compatible geometric structures on $M$ provided by special
tensor fields has been shown to be very efficient to establish and study
hidden properties of the given system of differential equations and its
solutions. This is the main reason for the usefulness of Sundman
transformations, because the tensor fields invariant under $f\,X$ are not, in
general, invariant under $X$.

The aim of this paper is to investigate from a geometric perspective the
meaning of such infinitesimal time reparametrisation, to relate it with
changes in the vector field describing the dynamical system, as well as to
point out many of its applications. Section 2 is devoted to first recall the
classical Sundman transformation and, as an example of its applications, to
show its use in the linearisation of the Kepler problem, and then to
introduce, from a geometric perspective, a concept of generalised Sundman
transformation for systems of first-order differential equations. Section 3
points out some possible applications in mathematics and classical mechanics.

The corresponding generalisation of Sundman transformation for systems of
second-order differential equations is not so easy and we restrict ourselves
in this article to the case of those derivable from a variational principle.
Therefore, as we also aim to study this generalised Sundman transformation in
the framework of Riemannian manifolds, in order to the paper be self
contained, we recall in Section 4 the main definitions and properties to be
used in the geometric study of such transformation, and in particular the
conformal equivalence relation in the set of Riemanniant metrics is explicitly
given. A brief overview of symplectic geometry and Lagrangian formalism is
presented to introduce some additional notation.

The rest of the paper is devoted to different applications in geometry and
mechanical systems. In Section 5, we describe geodesics and free motions in a
Riemannian manifold and the relation with geodesic vector fields and the
effect of a Sundman transformation on the set of geodesic curves, while
Section 6 is devoted to mechanical and Newtonian systems in a Riemannian
manifold. In both cases we solve the question of the relation between two
vector fields whose integral curves are related by a time reparametrisation.

In Section 7 we take a different approach. Instead of changing the vector
field, we change the metric on the manifold to a conformally related one in
order to obtain some specific properties of the vector field of a mechanical
system or its trajectories. In particular we obtain in a new fashion, related
to a Sundman transformation, the well known Jacobi metric. Different relations
between the Jacobi metric and the Hamilton-Jacobi equation for a Newtonian
system are briefly commented in Section 8.

As usually in this kind of geometric papers, all the manifolds and mappings
are assumed to be of ${\mathcal{C}}^{\infty}$ class.

##  2 Sundman transformation

The classical Sundman transformation [1] introduced to regularise the
equations of motion and avoid collision singularities (see also [2]), which
had previously been used by Levi-Civita [3, 4], was shown to be useful in many
other situations and can be generalised and extended to other more general
cases. For instance the classical paper by Bohlin [5] used a similar relation
to define the Keplerian anomaly. Moreover, Sundman transformation can be used
in the study of linearisation of differential equations [6] and in numerical
solution of systems of differential equations (see e.g. [7, 8, 9, 10, 11]).

The classical Sundman transformation is an infinitesimal scaling of time from
the time $t$ to a new fictitious time $\tau$ given by

$dt=r\,d\tau,$ (2.1)

where $r$ is the radial distance in the plane, which was later on generalised
to $dt=c\,r^{\alpha}\,d\tau$, where $c\in\mathbb{R}$ and $\alpha$ is a
positive constant [9], or more generally to $dt=f(r)\,d\tau$ [12, 13, 14].

###  2.1 An illustrative example

We next present an explicit example of application of the transformation (2.1)
in the Kepler problem. It is well known that the motion of a particle under a
central force takes place in a plane and we can restrict our study to such a
plane. Using the standard polar coordinates the Lagrange function for a $m=1$
particle is given by

$L(r,\theta)=\frac{1}{2}(\dot{r}^{2}+r^{2}\,\dot{\theta}^{2})-V(r),\quad r>0,$

and as the angular variable $\theta$ is cyclic, the Euler–Lagrange equation
for such variable shows that the corresponding (angular) momentum
$\ell=r^{2}\,\dot{\theta}$ is constant. The other Euler–Lagrange equation,
$\ddot{r}=r\,\dot{\theta}^{2}-V^{\prime}(r)$, reduces, using the constant of
motion $\ell$, to the equation of motion of the particle in $(0,\infty)$ under
the action of the reduced potential $\mathcal{V}(r)=V(r)+\ell^{2}/(2r^{2})$,
i.e. in the case of Coulomb–Kepler problem for which $V(r)=-k/r$, the radial
equation of motion is

$\ddot{r}=\frac{\ell^{2}}{r^{3}}-V^{\prime}(r)=-\mathcal{V}^{\prime}(r)=\frac{\ell^{2}}{r^{3}}-\frac{k}{r^{2}}.$
(2.2)

The conserved energy is given by

$E=\frac{1}{2}\dot{r}^{2}+\frac{\ell^{2}}{2r^{2}}-\frac{k}{r},$ (2.3)

and when we introduce the eccentric anomaly parameter $\tau$ by the classical
Sundman transformation (2.1), then we get

$\frac{d}{dt}=\frac{1}{r}\frac{d}{d\tau},\qquad\frac{d^{2}}{dt^{2}}=\frac{1}{r^{2}}\frac{d^{2}}{d\tau^{2}}-\frac{r^{\prime}}{r^{3}}\frac{d}{d\tau},$
(2.4)

where derivatives with respect to $\tau$ of a function $f$ are denoted by
$f^{\prime}$ instead of $\dot{f}$. The differential equation (2.2) becomes

$\frac{1}{r^{2}}r^{\prime\prime}-\frac{1}{r^{3}}r^{\prime
2}=-\mathcal{V}^{\prime}(r)=\frac{\ell^{2}}{r^{3}}-\frac{k}{r^{2}},$ (2.5)

and simplifying

$r^{\prime\prime}=\frac{1}{r}r^{\prime 2}+\frac{\ell^{2}}{r}-k.$ (2.6)

But the expression of the energy (2.3) can now be rewritten as

$2r^{2}E={r^{\prime}}^{2}+{\ell^{2}}-2k{r},$ (2.7)

and therefore we see that for motions with a fixed energy $E$, the equation
(2.6) reduces to

$r^{\prime\prime}=2rE+k,$ (2.8)

which is an inhomogeneous linear second order differential equation with
constant coefficients, whose general solution is easily found. This shows that
Sundman transformation (2.1) provides a linearisation of the motion equation
for the Kepler problem with a fixed energy $E$.

For instance, when the energy is negative and the angular momentum is
different from zero, the solutions are ellipses given by [15]

$r(\tau)=A(1-e\,\cos(\omega\tau)),\quad\omega=\sqrt{2|E|},$ (2.9)

where $A$ is the major semiaxis and

$t=A\left(\tau-\frac{e}{\omega}\,\sin(\omega\tau)\right).$ (2.10)

###  2.2 Generalised Sundman transformation and geometric approach

The important point is that even if this classical Sundman transformation
(2.1) was carried out for the second order differential equations of motion,
it admits a generalisation to the case of systems of first-order differential
equations and, moreover, its geometric interpretation is more clear. Recall
that a second-order differential equation vector field in $Q$ can be seen (see
Subsection 4.3) as a particular case of vector fields on $TQ$, and therefore
it is enough to consider, for the time being, Sundman transformations for
autonomous systems of first-order differential equations.

As indicated in [16], given an autonomous system of first order differential
equations

$\frac{dx^{i}}{dt}=X^{i}(x^{1},\ldots,x^{n}),\quad i=1,\ldots,n,$ (2.11)

we can consider the generalisation of Sundman transformation defined by

$dt=f(x^{1},\ldots,x^{n})\ d\tau,\quad f(x^{1},\ldots,x^{n})>0,$ (2.12)

and then (2.11) becomes

$\frac{dx^{i}}{d\tau}=f(x^{1},\ldots,x^{n})\ X^{i}(x),\quad i=1,\ldots,n.$
(2.13)

Let us first remark that when each one of the integral curves of a vector
field $X$ is arbitrarily reparametrised we obtain a new family of curves which
may be, or not, the integral curves of a vector field $Y$. In the affirmative
case, as the two vector fields have the same local constants of motion, they
generate the same 1-dimensional distribution and, at least locally, there
exists a nonvanishing function $h$ such that $Y=h\,X$. Let us prove that this
is the case if we consider the reparametrisation defined by a Sundman
transformation and then $h$ coincides with the the function $f$ defining the
transformation (2.12).

In fact, if $\gamma(t)$ is a given curve and we carry out the
reparametrisation for which the new parameter $\tau$ is defined by the
relation (2.12) written as

$\frac{d\tau}{dt}=\frac{1}{f(\gamma(t))},$ (2.14)

we obtain the reparametrised curve $\bar{\gamma}(\tau)$ such that
$\bar{\gamma}(\tau(t))=\gamma(t)$ and then

$\frac{d\gamma}{dt}=\frac{d\bar{\gamma}}{d\tau}\,\frac{d\tau}{dt}=\frac{1}{f(\gamma(t))}\frac{d\bar{\gamma}}{d\tau},$

and consequently, if the curve $\gamma(t)$ is an integral curve of $X$, i.e.
$d\gamma/dt=X_{\gamma}$, then

$\frac{d\bar{\gamma}}{d\tau}=f(\gamma(t))\frac{d\gamma}{dt}=f(\gamma(t))\,X_{\gamma(t)}=(f\,X)_{\gamma(t)}=(f\,X)_{\bar{\gamma}(\tau)}\,,$

i.e. the curve $\bar{\gamma}$ is an integral curve of the vector field $f\,X$.

From the geometric viewpoint the solutions of the system (2.11) provide us the
integral curves of the vector field
$X=X^{i}(x^{1},\ldots,x^{n})\partial/\partial{x^{i}}$ and then the solutions
of the new system (2.13) provide the integral curves of the vector field
$f\,X$. In other words, the effect of the generalised Sundman transformation
(2.12) is the replacement by the vector field $f\,X$ instead of $X$ [17]. But
the integral curves of the vector fields $X\in\mathfrak{X}(M)$ and
$f\,\,X\in\mathfrak{X}(M)$ coincide up to respective reparametrisations,
because both vector fields have the same local constants of motion.

Equivalently, the reparametrisation defined by Sundman transformation (2.14)
should be carried out for each orbit, i.e. as pointed out in [18], if $x(t)$
is a solution of (2.11), then we consider the reparametrisation defined by

$\tau(t)=\int_{0}^{t}\frac{1}{f(x(\zeta))}\ d\zeta,$

and the inverse expression $t=\varphi(\tau)$, and then $x(\varphi(\tau))$ is a
solution of (2.13).

It is noteworthy that the ‘velocity’ with respect to the new time is different
and so the new velocity $\bar{v}$ is related to the old one by
$\bar{v}^{i}=f\,v^{i}$, as a consequence of (2.14).

The reinterpretation of this ‘infinitesimal time scaling’ was used in [19] to
deal with the theory described by Bond and Janin in [20] in satellite theory.
By appropriately selecting the function $f$, a problem with singular solutions
can be transformed into a related one with globally defined solutions in terms
of the new time variable.

##  3 Applications in mathematics and classical mechanics

Let us first remark that even if the vector fields $X$ and $f\,X$ on a
manifold $M$, with $f$ a nonvanishing function, usually called conformally
related vector fields, have the same local constants of motion, the same
property does not hold for general tensor fields $T$, because
$\mathcal{L}_{f\,X}T\neq f\,\mathcal{L}_{X}T$ and this fact is quite important
because it provides us a way for obtaining, from a vector field $X$ that does
not preserve $T$, vector fields under which a given tensor field $T$ is
invariant.

We have shown before how the classical Sundman transformation (2.1) provides a
method to linearise the equations of motion for a given energy in the
Kepler–Coulomb problem. Recall that a manifold $M$ can be endowed with a
linear structure (see [21], Chapter 3) if there exists a complete vector field
$\Delta$, playing the rôle of Liouville vector field, with only one non-
degenerate critical point and such that
$\mathcal{F}_{\Delta}^{(0)}=\mathbb{R}$ and $\mathcal{F}_{\Delta}^{(1)}$
separates derivations, where $\mathcal{F}_{\Delta}^{(k)}$ denotes the set of
functions on the manifold $M$ defined by

$\mathcal{F}_{\Delta}^{(k)}=\\{f\in\mathcal{F}(M)\mid\Delta f=k\,f\\},\quad
k\in\mathbb{N}.$

Linear functions (with respect to such a linear structure) are those
$f\in\mathcal{F}(M)$ satisfying that $\mathcal{L}_{\Delta}f=f$, that is, the
elements of $\mathcal{F}_{\Delta}^{(1)}$, and linear vector fields
$X\in\mathfrak{X}(M)$ are those preserving the $\mathbb{R}$-linear subspace of
linear functions, that is, satisfying
$\mathcal{L}_{X}(\mathcal{F}_{\Delta}^{(1)})\subset\mathcal{F}_{\Delta}^{(1)}$,
or what is equivalent, $\mathcal{L}_{X}\Delta=[X,\Delta]=0$.

Then, given a nonlinear vector field $X$ such that $\mathcal{L}_{X}\Delta\neq
0$, it may exists a positive function $f\in C^{\infty}(M)$ such that
$\mathcal{L}_{f\,X}\Delta=0$, because
$\mathcal{L}_{f\,X}\Delta=[f\,X,\Delta]=f\,[X,\Delta]-\Delta(f)\,X$. This
happens when there exists a function $h$ such that $[X,\Delta]=h\,X$ and we
choose the function $f$ in such a way that $h=\Delta(\log f)$.

In the case of $M$ being a $n$-dimensional manifold, a chart $(U,\varphi)$ of
$M$ identifies $U$ with an open set of $\mathbb{R}^{n}$, and this one, as a
linear space, is endowed with the complete vector field
$\Delta={\displaystyle\sum_{i=1}^{n}}x^{i}\,\partial/\partial x^{i}$, where
$x^{i}$ are the usual coordinates, and therefore, if
$X={\displaystyle\sum_{i=1}^{n}}X^{i}(x)\,\partial/\partial x^{i}$ is the
coordinate expression of the vector field $X$ in the mentioned chart, then we
can say that the vector field $X$ is linear in this chart when $[\Delta,X]=0$,
and taking into account that

$[\Delta,X]=\left[\sum_{i=1}^{n}x^{i}\frac{\partial}{\partial
x^{i}},\sum_{j=1}^{n}X^{j}\frac{\partial}{\partial
x^{j}}\right]=\sum_{k=1}^{n}\left(\sum_{i=1}^{n}x^{i}\frac{\partial
X^{k}}{\partial x^{i}}-X^{k}\right)\frac{\partial}{\partial x^{k}},$ (3.1)

we see from $[\Delta,X]=0$ that the components of the vector field $X$ in the
given chart are homogeneous of order one functions, i.e. linear functions,
$X^{i}(x)={\displaystyle\sum_{j=1}^{n}}A^{i}\,_{j}\,x^{j}$, where
$A^{i}\,_{j}\in\mathbb{R}$, and then their integral curves are given by the
solutions of the linear system
$\dot{x}^{i}={\displaystyle\sum_{j=1}^{n}}A^{i}\,_{j}\,x^{j}$, $i=1,\ldots,n$.
The vector fields $X$ whose integral curves are given by solutions of
inhomogeneous linear systems
$\dot{x}^{i}={\displaystyle\sum_{j=1}^{n}}A^{i}\,_{j}\,x^{j}+B^{i}$,
$i=1,\ldots,n$, where $B^{i}\in\mathbb{R}$, are the sum of a linear vector
field $X_{0}$ and another one $X_{-1}$ with components that are homogeneous of
degree zero, i.e. constants. Of course as $[\Delta,X_{0}]=0$ and
$[\Delta,X_{-1}]=-X_{-1}$ we have that

$[\Delta,X]=[\Delta,X_{0}+X_{-1}]=[\Delta,X_{0}]+[\Delta,X_{-1}]=-X_{-1},$

and hence,

$[\Delta,[\Delta,X]]=-[\Delta,X_{-1}]=X_{-1}=-[\Delta,X].$

Conversely, if this last relation holds, then having in mind (3.1) we see that

$[\Delta,[\Delta,X]]=\sum_{i=1}^{n}\left(\Delta(\Delta(X^{i})-X^{i})-\Delta(X^{i})-X^{i}\right)\frac{\partial}{\partial
x^{i}},$

and therefore, $[\Delta,[\Delta,X]]=-[\Delta,X]$ if and only if, for each
index $i=1,\ldots,n$, $\Delta^{2}(X^{i})-\Delta(X^{i})=0$, i.e.
$\Delta(X^{i})-X^{i}$ is a homogeneous function of order zero, because there
exist constants $B^{i}$ such

$\sum_{j=1}^{n}x^{j}\frac{\partial X^{i}}{\partial x^{j}}-X^{i}=-B^{i},$

from where we find that
$X^{i}={\displaystyle\sum_{j=1}^{n}}A^{i}\,_{j}x^{j}+B^{i}$, for each index
$i=1,\ldots,n$. Without using local coordinates, we observe that
$[\Delta,[\Delta,X]]=-[\Delta,X]$ implies that the components of the vector
field $[\Delta,X]$ are homogeneous functions of order zero, i.e. is a vector
field with constant components, i.e., $X$ is if the form $X=X_{0}+X_{-1}$.

We can summarise the preceding results as follows:

###### Theorem 3.1

Let $M$ be a differentiable manifold endowed with a linear structure $\Delta$.
If $X$ is a vector field in $M$, then:

  1. 1.

$\mathcal{L}_{\Delta}X=h\,X$, for some function $h$, if and only if there
exists a nonvanishing positive function $f$ such that the vector field $fX$
satisfies $\mathcal{L}_{\Delta}(fX)=0$.

  2. 2.

$\mathcal{L}_{\Delta}X=0$ if and only if for any coordinate system
$(U,\varphi)$, where $\varphi=(x^{1},\ldots,x^{n})$, such that
$\left.\Delta\right|_{U}={\displaystyle\sum_{i=1}^{n}x^{i}\frac{\partial}{\partial
x^{i}}}$, the local expression of $X$ is linear in $x^{i}$, that is,

$\left.X\right|_{U}=\sum_{i,j=1}^{n}A^{i}\,_{j}\,x^{j}\frac{\partial}{\partial
x^{i}},\quad A^{i}\,_{j}\in\mathbb{R}.$

  3. 3.

$\mathcal{L}_{\Delta}^{2}X=-\mathcal{L}_{\Delta}X$ if and only if for any
coordinate system $(U,\varphi=(x^{1},\ldots,x^{n}))$ such that
$\left.\Delta\right|_{U}={\displaystyle\sum_{i=1}^{n}}x^{i}\frac{\partial}{\partial
x^{i}}$, the local expressión of $X$ is affine in $x^{i}$, that is,

$\left.X\right|_{U}=\sum_{i=1}^{n}\left(\sum_{j=1}^{n}A^{i}\,_{j}\,x^{j}+B^{i}\right)\frac{\partial}{\partial
x^{i}},\quad A^{i}\,_{j},B^{i}\in\mathbb{R}.$

$\Box$

Comment: This theorem gives us a characterisation of the vector fields $X$
that are linearisable with respect to a general linear structure $\Delta$ in a
manifold $M$, by means of a general Sundman transformation.

As another instance of application of Sundman transformations, if $(M,\Omega)$
is an oriented manifold, i.e. $\Omega$ is a volume form in $M$, and a vector
field $X\in\mathfrak{X}(M)$ is such that $\mathcal{L}_{X}\Omega\neq 0$, there
may be a positive function $f\in C^{\infty}(M)$ such that
$\mathcal{L}_{f\,X}\Omega=0$: these functions are called Jacobi multipliers
[22] and play a relevant rôle in the process of integrability by quadratures.
Note that as $\mathcal{L}_{f\,X}\Omega=\mathcal{L}_{X}(f\,\Omega)$, the search
of such a Jacobi multiplier is equivalent to the determination of a
$X$-invariant volume form $f\,\Omega$. For examples of applications of Jacobi
multipliers in integrability and the inverse problem of mechanics see e.g. the
recent review paper [23] and references therein.

Another possibility would be to consider a Riemannian structure on the
manifold $M$ given by a non-degenerate symmetric two times covariant tensor
field $g$ and then, as $\mathcal{L}_{f\,X}g\neq f\,\mathcal{L}_{X}g$, some
non-Killing vector fields $X$, i.e such that $\mathcal{L}_{X}g\neq 0$, can
give rise to Killing ones by just multiplication by a convenient function $f$.

Similarly, given a closed 2-form on a manifold, for instance a symplectic form
$\omega$, that may be not invariant under the vector field $X$, i.e.
$\mathcal{L}_{X}\omega\neq 0$, if there is a positive function $f$ such that
$\mathcal{L}_{f\,X}\omega=0$, i.e. $f\,i(X)\omega$ is a closed form, then,
when the non-degeneracy condition $df\wedge\omega=0$ is satisfied, the 2-form
$f\,\omega$ is symplectic and the vector field $X$ is locally-Hamiltonian with
respect to such symplectic form $f\,\omega$. Recall that in the case of $X$
being also locally-Hamiltonian with respect to the original symplectic form
$\omega$, such function $f$ must be a constant of motion for $X$, $Xf=0$ (see
[24]).

The same can be said about skew-symmetric two times contravariant tensor
fields, i.e. bivector fields $\Lambda$, and a particularly interesting case is
that of Poisson structures, i.e. such that $[\Lambda,\Lambda]_{\textrm{S}}=0$,
where $[\cdot,\cdot]_{\textrm{S}}$ denotes the Schouten bracket [25, 26].
Recall that in this case

$\\{f,g\\}=\Lambda(df,dg),\quad f,g\in C^{\infty}(M),$

defines a Poisson bracket and vector fields $X_{f}$ on $M$ of the form
$X_{f}g=\\{g,f\\}$ are said to be Hamiltonian vector fields. Then given a
vector field $X$ and a Poisson structure $\Lambda$ sometimes there exists a
function $f$ such that $fX$ is Hamiltonian with respect to the Poisson
structure $\Lambda$. In this way an infinitesimal Sundman time-
reparametrisation can transform a given vector field into a Hamiltonian one.
This is usually called Hamiltonization process [27, 28].

##  4 Digression on some geometric complements

The applications of the Sundman transformation for systems of second-order
differential equations we will develop in next sections cover different
topics, going from minimal length curves to systems of a mechanical type and
from the geodesic vector fields to the Jacobi metric. This is why we need to
extend our standpoint and formulate the Sundman transformation in a different
framework, that of a Riemannian manifold.

We begin with a short review of the concepts of Riemannian geometry we will
need, some comments on conformally related metrics and some ideas on the
tangent bundle and second-order differential equation vector fields in the
case of Lagrangian systems and systems of a mechanical type. The remaining
sections are devoted to a detailed study of different applications using these
topics.

###  4.1 A quick survey of Riemannian geometry

In order to the paper be self-contained, this Subsection is devoted to recall
well known concepts and properties in geometry and also to establish the
notation to be used. The results can be found in many classical books on
Riemann geometry (see e.g. [29] and [30] for more details).

A (pseudo-)Riemann manifold $(M,g)$ is a pair given by a differentiable
manifold $M$, $\dim M=n$, and a non-degenerate symmetric two times covariant
tensor field $g$ on $M$. Nondegeneracy means that the map $\widehat{g}:TM\to
T^{*}M$ from the tangent bundle, $\tau_{M}:TM\to M$, to the cotangent bundle,
$\pi_{M}:T^{*}M\to M$, defined by $\langle\widehat{g}(v),w\rangle=g(v,w)$,
where $v,w\in T_{x}M$, is a regular vector bundle map, that is, a vector
bundle isomorphism. As the map $\,\widehat{g}\,$ is a fibred map over the
identity on $M$, it induces the corresponding map between the
$C^{\infty}(M)$-modules of sections of the tangent and cotangent bundles, to
be denoted by the same letter
$\widehat{g}:\mathfrak{X}(M)\to\bigwedge^{1}(M)$:
$\langle\widehat{g}(X),Y\rangle=g(X,Y)$. Given a local chart
$(U,q^{1},\ldots,q^{n})$ on $M$ we can consider the coordinate basis of
$\mathfrak{X}(U)$ usually denoted $\\{\partial/\partial q^{j}\mid
j=1,\ldots,n\\}$ and its dual basis for $\bigwedge^{1}(U)$, $\\{dq^{j}\mid
j=1,\ldots,n\\}$. Then a vector $v$ in a point $q\in U$ is
$v={\displaystyle\sum_{j=1}^{n}}v^{j}\,(\partial/\partial q^{j})_{|q}$ and a
covector $\zeta$ in such a point is
$\zeta={\displaystyle\sum_{j=1}^{n}}p_{j}\,(dq^{j})_{|q}$, with $v^{j}=\langle
dq^{j},v\rangle$ and $p_{j}=\langle\zeta,\partial/\partial q^{j}\rangle$ being
the usual velocities and momenta. The local expression for $g$ in the open set
$U$ is

$g=\sum_{i,j=1}^{n}g_{ij}(q)\,dq^{i}\otimes dq^{j},\quad
g_{ij}(q)=g\left(\frac{\partial}{\partial q^{i}},\frac{\partial}{\partial
q^{j}}\right),\quad i,j=1,\ldots,\dim M,$ (4.1)

and therefore the arc-length of a curve $\gamma$ in $M$,
$\gamma(t)=(q^{1}(t),\ldots,q^{n}(t))$, between the points $\gamma(t_{1})$ and
$\gamma(t_{2})$ is given by

$s(\gamma)=\int_{t_{1}}^{t_{2}}\sqrt{g(\dot{\gamma},\dot{\gamma})}\,dt=\int_{t_{1}}^{t_{2}}\sqrt{{\displaystyle\sum_{i,j=1}^{n}}g_{ij}(q)\dot{q}^{i}\dot{q}^{j}}\
dt,$ (4.2)

i.e. the classical local expression for the arc-length $ds$ is given by

$ds^{2}=\sum_{i,j=1}^{n}g_{ij}(q)\,dq^{i}dq^{j}.$ (4.3)

Given $q_{1},q_{2}\in M$, the extremal length curves in $(M,g)$ from $q_{1}$
to $q_{2}$ are the curves $\gamma:[t_{1},t_{2}]\to
M,\gamma(t_{1})=q_{1},\gamma(t_{2})=q_{2}$, such that the integral (4.2) is
extremal among all the curves going from $q_{1}$ to $q_{2}$. Such curves will
be studied in Subsection 5.1.

Recall that a linear connection $\nabla$ in a manifold $M$ is a map
$\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$, such that if
$X,Y,Z\in\mathfrak{X}(M)$, $f\in C^{\infty}(M)$ and $\nabla_{X}Y$ denotes
$\nabla_{X}Y=\nabla(X,Y)$:

$\begin{array}[]{rl}{\text{i)}}&\nabla_{X+Y}Z=\nabla_{X}Z+\nabla_{Y}Z,\\\
{\text{ii)}}&\nabla_{X}(Y+Z)=\nabla_{X}Y+\nabla_{X}Z,\\\
{\text{iii)}}&\nabla_{fX}Y=f\,\nabla_{X}Y,\\\
{\text{iv)}}&\nabla_{X}(fY)=(Xf)Y+f\,\nabla_{X}Y.\end{array}$

In a local coordinate system the connection symbols $\Gamma^{i}_{jk}$ are
defined by

$\nabla_{\frac{\partial}{\partial x^{i}}}\left(\frac{\partial}{\partial
x^{j}}\right)=\sum_{l=1}^{n}\Gamma^{l}_{ji}\,\frac{\partial}{\partial x^{l}},$
(4.4)

while $\nabla_{X}Y$ is given by

$\nabla_{X}Y=\nabla_{\sum_{i=1}^{n}X^{i}\frac{\partial}{\partial
x^{i}}}\left(\sum_{j=1}^{n}Y^{j}\frac{\partial}{\partial
x^{j}}\right)=\sum_{i,k=1}^{n}X^{i}\left(\frac{\partial Y^{k}}{\partial
x^{i}}+\sum_{j=1}^{n}Y^{j}\Gamma^{k}_{ji}\right)\frac{\partial}{\partial
x^{k}}.$ (4.5)

Moreover, each linear connection has associated a (1,2) tensor $\mathbb{T}$,
called torsion tensor, which is skewsymmetric in the two last indices and it
is defined as follows

$\mathbb{T}(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y],$ (4.6)

and a (1,3) tensor field called curvature tensor defined by

$R(X,Y)Z=\nabla_{X}(\nabla_{Y}Z)-\nabla_{Y}(\nabla_{X}Z)-\nabla_{[X,Y]}Z.$
(4.7)

A Riemann manifold $(M,g)$ is endowed with a uniquely determined torsionless
metric connection $\nabla$, called the Levi–Civita connection. By metric we
understand that if $X,Y,Z\in\mathfrak{X}(M)$, then

$\mathcal{L}_{X}(g(Y,Z))=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z),$ (4.8)

and by torsionless we mean that the corresponding torsion tensor $\mathbb{T}$
is null, hence

$\nabla_{X}Y-\nabla_{Y}X=[X,Y].$ (4.9)

Such a Levi-Civita connection is given by the so-called Koszul formula:

$\begin{array}[]{rcl}2g(\nabla_{X}Y,Z)&=&\mathcal{L}_{X}(g(Y,Z))+\mathcal{L}_{Y}(g(Z,X))-\mathcal{L}_{Z}(g(X,Y))\\\
&&\\\ &-&g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]),\end{array}$ (4.10)

for every $X,Y,Z\in\mathfrak{X}(M)$. In terms of local coordinates on $M$
Koszul formula gives

$2g\left(\nabla_{\partial/\partial x^{i}}\left(\frac{\partial}{\partial
x^{j}}\right),\frac{\partial}{\partial x^{k}}\right)=\frac{\partial
g_{jk}}{\partial x^{i}}+\frac{\partial g_{ik}}{\partial x^{j}}-\frac{\partial
g_{ij}}{\partial x^{k}},$

and using the definition of connection symbols (usually called second class
Christoffel symbols)

$g\left(\nabla_{\partial/\partial x^{i}}\left(\frac{\partial}{\partial
x^{j}}\right),\frac{\partial}{\partial
x^{k}}\right)=\sum_{l=1}^{n}\Gamma_{ji}^{l}\,g_{lk}.$

Hence, the second class Christoffel symbols are given by

$\Gamma^{i}_{jk}=\frac{1}{2}\sum_{l=1}^{n}g^{il}\left(\frac{\partial
g_{lj}}{\partial q^{k}}+\frac{\partial g_{lk}}{\partial q^{j}}-\frac{\partial
g_{jk}}{\partial q^{l}}\right),$ (4.11)

where ${\displaystyle\sum_{j=1}^{n}}g^{ij}g_{jk}=\delta^{i}_{k}$.

On the other hand, the vanishing of the torsion tensor is equivalent to the
following symmetry property of Christoffel symbols:
$\Gamma^{i}_{jk}=\Gamma^{i}_{kj}$.

As indicated above, these symbols $\Gamma^{i}_{jk}$ locally determine
$\nabla_{X}Y$ for every $X,Y,$ by making use of (4.5). The remarkable fact
(see e.g. [29, 30] for details) is that as given a vector field
$Y\in\mathfrak{X}(M)$ the value of $\nabla_{X}Y$ at a point $x\in M$,
$(\nabla_{X}Y)(x)$, only depends, with respect to $X$, on the value of $X$ in
such a point, then for each $v\in T_{x}M$ we can define
$\nabla_{v}Y=(\nabla_{X}Y)(x)$, where $X$ is any vector field
$X\in\mathfrak{X}(M)$ such that $X(x)=v$, and then this allows us to introduce
the concept of parallelism of a vector field along a curve as follows: A
vector field along a curve $\gamma:I\to M$, $Y\in\mathfrak{X}(\gamma)$, is
parallel along $\gamma$ if $\nabla_{\dot{\gamma}(t)}\bar{Y}=0$, for all $t\in
I$, where $\bar{Y}\in\mathfrak{X}(M)$ is an arbitrary extension to $M$ of $Y$.

An example of a vector field along a curve $\gamma$ in $M$ is given by its
velocity vector field $\dot{\gamma}\in\mathfrak{X}(\gamma)$, and the curves
whose velocity vector field is parallel along the curve,
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0$, are called geodesics. In
particular in a Riemann manifold $(M,g)$ the connection to be considered is
its Levi-Civita connection and using this connection $\nabla$, the geodesics
in $(M,g)$ are the curves $\gamma$ in $M$ satisfying the equation
$\nabla_{\dot{\gamma}}\,\dot{\gamma}=0$, where $\dot{\gamma}$ is the vector
field along $\gamma$ given by the velocity of the curve at each point. They
are uniquely defined from the Riemann metric $g$ through the Levi-Civita
connection $\nabla$, which is uniquely defined by $g$. Their geometric
properties can be studied through the so called second order geodesic vector
field defined by $g$ and we will see below the relation with extremal length
curves through an appropriate Sundman transformation.

For a curve parametrised by its arc-length $s$,
$\gamma(s)=(q^{1}(s),\ldots,q^{n}(s))$, as we have

$\nabla_{\dot{\gamma}}\,\dot{\gamma}=\sum_{l=1}^{n}\left(\sum_{j=1}^{n}\dot{q}^{j}\frac{\partial\dot{q}^{l}}{\partial
q^{j}}+\sum_{j,k=1}^{n}\Gamma^{l}_{jk}\dot{q}^{j}\dot{q}^{k}\right)\frac{\partial}{\partial
q^{l}}\circ{\gamma}=\sum_{l=1}^{n}\left(\ddot{q}^{l}+\sum_{j,k=1}^{n}\Gamma^{l}_{jk}\dot{q}^{j}\dot{q}^{k}\right)\frac{\partial}{\partial
q^{l}}\circ\gamma\,,$

the local equation of the geodesic lines,
$\nabla_{\dot{\gamma}}\,\dot{\gamma}=0$, is the second order differential
equation

$\ddot{q}^{i}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\dot{q}^{j}\dot{q}^{k}=0\,,\quad
l=1,\ldots,n\,.$ (4.12)

See Subsection 5.1 for another approach to this equation.

Consequently, the geodesic curves are the projection on the base manifold $M$
of the integral curves of the second order geodesic vector field
$\Gamma\in\mathfrak{X}(TM)$ whose local expression is

$\Gamma=\sum_{i=1}^{n}\left(v^{i}\frac{\partial}{\partial
q^{i}}-\left(\sum_{j,k=1}^{n}\Gamma^{i}_{jk}v^{j}v^{k}\right)\frac{\partial}{\partial
v^{i}}\right).$ (4.13)

###  4.2 Conformal metrics

We have seen in Section 2.2 that the vector fields $X$ and $fX$ on a manifold
$M$ have integral curves related by a Sundman transformation. Other
interesting problems related to Sundman transformation for second-order
differential equation vector fields use conformal Riemannian metrics to study
dynamical systems. Two metrics, $g$ and $\bar{g}$, are conformally related if
there exists a function $\varphi$ such that $\bar{g}=\exp(2\varphi)\,g$. This
establishes an equivalence relation in the set of metrics. A conformal
structure is an equivalence class of metrics. The covariant derivatives with
respect to both metrics $g$ and $\bar{g}$ are related by:

$\begin{array}[]{rcl}\bar{\nabla}_{X}Y&=&\nabla_{X}Y+(\mathcal{L}_{X}\varphi)Y+(\mathcal{L}_{Y}\varphi)X-g(X,Y)\,{\textrm{grad}}_{g}\varphi\\\
&&\\\
&=&\nabla_{X}Y+(\mathcal{L}_{X}\varphi)Y+(\mathcal{L}_{Y}\varphi)X-\bar{g}(X,Y)\,{\textrm{grad}}_{\bar{g}}\varphi\,.\end{array}$
(4.14)

To obtain these formulas, consider the Koszul formula for the connection
$\bar{\nabla}$ that gives

$\begin{array}[]{rcl}e^{2\varphi}g(\bar{\nabla}_{X}Y,Z)&=&e^{2\varphi}\left[\mathcal{L}_{X}g(Y,Z)+\mathcal{L}_{Y}g(Z,X)-\mathcal{L}_{Z}g(X,Y)+2g(Y,Z)X(\varphi)+2g(Z,X)Y(\varphi)\right.\\\
&&\\\
&-&\left.2g(X,Y)Z(\varphi)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y])\right],\end{array}$
(4.15)

and simplifying the factor $e^{2\varphi}$, it may be rewritten as

$\begin{array}[]{rcl}2g(\bar{\nabla}_{X}Y,Z)&=&\mathcal{L}_{X}g(Y,Z)+\mathcal{L}_{Y}g(Z,X)-\mathcal{L}_{Z}g(X,Y)+2g(Y,Z)X(\varphi)+2g(Z,X)Y(\varphi)\\\
&&\\\ &-&2g(X,Y)Z(\varphi)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]),\end{array}$
(4.16)

from where, once again by simplification, we obtain

$g(\bar{\nabla}_{X}Y,Z)=g(\nabla_{X}Y,Z)+g(Y,Z)X(\varphi)+g(Z,X)Y(\varphi)-g(X,Y)Z(\varphi).$
(4.17)

Finally, by making use of

$g(X,Y)\mathcal{L}_{Z}\varphi=g(X,Y)d\varphi(Z)=g(X,Y)g(\textrm{grad}_{g}\varphi,Z)=g(g(X,Y)\textrm{grad}_{g}\varphi,Z),$

we find

$\bar{\nabla}_{X}Y=\nabla_{X}Y+(\mathcal{L}_{X}\varphi)Y+(\mathcal{L}_{Y}\varphi)X-g(X,Y)\,{\textrm{grad}}_{g}\varphi,$
(4.18)

where ${\textrm{grad}}_{g}$ is the gradient with respect to the metric $g$.
The above relation can also be rewritten as

$\bar{\nabla}_{X}Y=\nabla_{X}Y+(\mathcal{L}_{X}\varphi)Y+(\mathcal{L}_{Y}\varphi)X-\bar{g}(X,Y)\,{\textrm{grad}}_{\bar{g}}\varphi.$
(4.19)

Remark that the relation $\bar{g}=\exp(2\varphi)\,g$ implies that
${\textrm{grad}}_{\bar{g}}\varphi=\exp(-2\varphi)\,{\textrm{grad}}_{g}\varphi$.

In the case of $X=Y$ the expression (4.18) reduces to

$\bar{\nabla}_{X}X=\nabla_{X}X+2(\mathcal{L}_{X}\varphi)X-g(X,X){\textrm{grad}}_{g}\varphi\,.$
(4.20)

It is also interesting to study the relationship between the corresponding
Christoffel symbols to both metrics. The new Christoffel symbols of the second
kind are [31]:

$\bar{\Gamma}^{i}_{jk}(q)=\Gamma^{i}_{jk}(q)+\delta^{i}_{j}\frac{\partial\varphi}{\partial
q^{k}}+\delta^{i}_{k}\frac{\partial\varphi}{\partial
q^{j}}-\sum_{l=1}^{n}g_{jk}\,g^{il}\frac{\partial\varphi}{\partial
q^{l}},\quad i,j,k=1,\ldots,\dim M,$ (4.21)

where $\delta^{i}_{k}$ denotes Kronecker delta symbol, because

$\bar{\Gamma}^{i}_{jk}=\frac{1}{2}e^{-2\varphi}\sum_{l=1}^{n}g^{il}\left(\frac{\partial}{\partial
q^{k}}(e^{2\varphi}g_{lj})+\frac{\partial}{\partial
q^{j}}(e^{2\varphi}g_{lk})-\frac{\partial}{\partial
q^{l}}(e^{2\varphi}g_{jk})\right),\quad i,j,k=1,\ldots,\dim M,$

and using Leibniz rule for derivatives and simplifying terms we arrive to
(4.21).

There is also an interesting equivalence relation in the set of symmetric
linear connections (see e.g. [31] and references therein): two such
connections are said to be projectively equivalent when their geodesics differ
only by a parametrisation. Each equivalence class is characterised by its
Thomas symbol. A projective structure is compatible with a conformal structure
if there exists a metric $g$ in its conformal class such the associated Levi-
Civita connection is in the equivalence class defining the projective
structure. Necessary and sufficient conditions for local compatibility are
given in [31].

###  4.3 Lagrangian systems

Given a regular Lagrangian (see later on), $L:TM\to\mathbb{R}$, $L(q,v)$,
defined on the velocity phase space, that is, a function depending on the
positions $q$ and velocities $v$, we can obtain the associated dynamical
equations in two different ways. In the traditional one we consider the
Hamilton principle, that is, we try to determine the curves $q(t)$ with fixed
end-points making extremal the action defined by:

$S(q(t))=\int_{t_{1}}^{t_{2}}{L}(q(t),\dot{q}(t))\,dt.$

Then, applying the classical variational calculus, the curves we are looking
for are solutions to the Euler-Lagrange equations associated to the Lagrangian
$L$:

$\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}^{i}}\right)-\frac{\partial L}{\partial q^{i}}=0,\quad
i=1,\ldots,n\,.$ (4.22)

But the theory of the above systems can be presented more geometrically by
means of the symplectic approach to Lagrangian formalism (see e.g [21, 32,
33]). In this approach, from the Lagrangian function $L$ we construct the
Cartan 1-form $\theta_{L}$ given by $\theta_{L}=d{L}\circ S$, where $S$ is the
vertical endomorphism [32, 33] in the tangent bundle $TM$ of the configuration
manifold $M$. Associated to the 1-form $\theta_{L}$, we have the 2-form
$\omega_{L}=-d\theta_{L}$. These forms are locally given by

$\theta_{L}=\sum_{j=1}^{n}\frac{\partial L}{\partial v^{j}}dq^{j}\,,\qquad
d\theta_{L}=\sum_{j=1}^{n}d\left(\frac{\partial L}{\partial
v^{j}}\right)\wedge dq^{j}=\sum_{i,j=1}^{n}\left(\frac{\partial^{2}L}{\partial
q^{i}\partial v^{j}}dq^{i}\wedge dq^{j}+\frac{\partial^{2}L}{\partial
v^{i}\partial v^{j}}dv^{i}\wedge dq^{j}\right)$

If $\omega_{L}$ is symplectic, we say that the Lagrangian $L$ is regular.
Otherwise $L$ is called singular. In the sequel we suppose that $L$ is
regular.

The 2-form $\omega_{L}$ has the usual form

$\widehat{\omega}_{L}=\begin{pmatrix}A&-W\\\
W&0\end{pmatrix},\qquad\widehat{\omega}_{L}^{-1}=\begin{pmatrix}0&W^{-1}\\\
-W^{-1}&W^{-1}AW^{-1}\end{pmatrix}\,,$ (4.23)

with the matrices $A$ and $W$ being given by

$A_{ij}=\frac{\partial^{2}L}{\partial q^{i}\partial
v^{j}}-\frac{\partial^{2}L}{\partial v^{i}\partial q^{j}},\qquad
W_{ij}=\frac{\partial^{2}L}{\partial v^{i}\partial v^{j}},\quad
i,j=1,\ldots,n.$

The regularity of the Lagrangian depends on the regularity of
$\widehat{\omega}_{L}$ and, consequently, on the regularity of the matrix $W$.

On the other side, recall that the energy of the Lagrangian system determined
by $L$ is defined by $E_{L}=\Delta L-L$, where $\Delta\in\mathfrak{X}(TM)$ is
the Liouville vector field, generator of dilations along the fibres of $TM$,
given by

$\Delta
f(q,v)=\frac{d}{dt}f(q,e^{t}v)|_{t=0},\qquad\Delta=\sum_{i=1}^{n}v^{i}\frac{\partial}{\partial
v^{i}},$ (4.24)

for all $(q,v)\in TM$ and $f\in C^{\infty}(TM)$. Hence, as
$\Delta({L})={\displaystyle\sum_{j=1}^{n}}v^{j}{\partial L}/{\partial v^{j}}$,
the total energy is $E_{L}={\displaystyle\sum_{j=1}^{n}}v^{j}{\partial
L}/{\partial v^{j}}-L$. The corresponding associated dynamics when $L$ is
regular is given by the unique dynamical vector field
$\Gamma_{L}\in\mathfrak{X}(TM)$ defined by

$i(\Gamma_{L})\omega_{L}=dE_{L},$ (4.25)

and since we know the expression of $E_{L}$, we see that $\Gamma_{L}$ is of
the form

$\Gamma_{L}=\sum_{i=1}^{n}\left(v^{i}\frac{\partial}{\partial
q^{i}}+F^{i}(q,v)\,\frac{\partial}{\partial v^{i}}\right),$ (4.26)

where

$F^{i}(q,v)=-\sum_{j,l=1}^{n}W^{ij}A_{lj}v^{l}+\sum_{j=1}^{n}W^{ij}\frac{\partial
L}{\partial q^{j}}\,,$ (4.27)

with ${\displaystyle\sum_{j=1}^{n}}W^{ij}W_{jk}=\delta^{i}_{k}$.

The uniqueness of $\Gamma_{L}$ satisfying equation (4.25) is a consequence of
the assumed regularity of the 2-form $\omega_{L}$.

The dynamical vector field $\Gamma_{L}$ is a second order differential
equation and the corresponding system of differential equations is

$\left\\{\begin{array}[]{rcl}\dfrac{dq^{j}}{dt}&=&v^{j}\\\
\dfrac{dv^{j}}{dt}&=&F^{j}=-{\displaystyle\sum_{i,l=1}^{n}W^{ij}A_{li}v^{l}+\sum_{i=1}^{n}W^{ij}\frac{\partial
L}{\partial q^{i}}}\end{array}\right.$ (4.28)

which gives the second order differential equation

$\frac{d^{2}q^{j}}{dt^{2}}=F^{j}=-\sum_{i,l=1}^{n}W^{ij}A_{li}\frac{dq^{l}}{dt}+\sum_{i=1}^{n}W^{ij}\frac{\partial
L}{\partial q^{i}}.$

In the case of a mechanical type Lagrangian system in the Riemanian manifold
$(M,g)$, the Lagrangian is given by $L=T_{g}-V$ where $T_{g}$ is the kinetic
energy defined by the metric and $V:M\to\mathbb{R}$ is the potential energy.
The local coordinate expression for the kinetic energy $T_{g}$ is

$T_{g}(v)=\frac{1}{2}\sum_{i,j=1}^{n}\,g_{ij}(\tau_{M}(v))\,v^{i}v^{j}\,.$
(4.29)

To understand $V$ as a function defined in $TM$ we need to write
$\tau_{M}^{*}V$. By simplicity we continue writing simply $V$.

For such kind of systems the Cartan 1-form $\theta_{T_{g}}$ given by
$\theta_{T_{g}}=d{L}\circ S=d{T_{g}}\circ S$, reduces to
$\theta_{L}(v)=\widehat{g}(v)=g(v,\cdot)$, which in local coordinates looks as

$\theta_{L}(q,v)=\sum_{i,j=1}^{n}g_{ij}(q)\,v^{j}\,dq^{i},$ (4.30)

with associated symplectic form $\omega_{L}$ given by

$\omega_{L}=\sum_{i,j=1}^{n}g_{ij}\,dq^{i}\wedge
dv^{j}+\frac{1}{2}\sum_{i,j,k=1}^{n}\left(\frac{\partial g_{ij}}{\partial
q^{k}}v^{j}-\frac{\partial g_{kj}}{\partial q^{i}}v^{j}\right)dq^{i}\wedge
dq^{k}.$ (4.31)

Here this 2-form $\omega_{L}$ is symplectic and consequently the Lagrangian
$L=T_{g}-V$ is a regular Lagrangian as defined above. The matrix of
$\omega_{L}$ is of the form (4.23), but now with

$A_{ij}=\sum_{k=1}^{n}\left(\frac{\partial g_{kj}}{\partial
q^{i}}-\frac{\partial g_{ki}}{\partial q^{j}}\right)v^{k},\qquad
W_{ij}=g_{ij}.$

The total energy is $E_{L}=T_{g}+V$, hence the dynamical vector field defined
as in (4.25) is given by (4.26) with

$\displaystyle F^{i}(q,v)$ $\displaystyle=$
$\displaystyle-\sum_{j,l=1}^{n}W^{ij}A_{lj}v^{l}+\sum_{j=1}^{n}W^{ij}\frac{\partial
L}{\partial q^{j}}-\sum_{j=1}^{n}g^{ji}\frac{\partial V}{\partial q^{j}}$
(4.32) $\displaystyle=$
$\displaystyle-\sum_{j,k,l=1}^{n}g^{ij}\left(\frac{\partial g_{kj}}{\partial
q^{l}}-\frac{\partial g_{kl}}{\partial
q^{j}}\right)v^{k}v^{l}-\sum_{j=1}^{n}g^{ji}\frac{\partial V}{\partial q^{j}}$
(4.33) $\displaystyle=$
$\displaystyle-\sum_{k,l=1}^{n}\Gamma_{kl}^{i}v^{k}v^{l}-\sum_{j=1}^{n}g^{ji}\frac{\partial
V}{\partial q^{j}}\,,$ (4.34)

where use has been made of definition (4.11).

Once again the vector field $\Gamma_{L}$ corresponds to a second order
differential equation we can write as

$\ddot{q}^{i}=-\sum_{k,l=1}^{n}\Gamma_{kl}^{i}\dot{q}^{k}\dot{q}^{l}-\sum_{j=1}^{n}g^{ji}\frac{\partial
V}{\partial q^{j}}\,,\quad i=1,\ldots,n,$ (4.35)

where the last term,
$X_{V}=-{\displaystyle\sum_{i,j=1}^{n}g^{ji}\frac{\partial V}{\partial
q^{i}}\frac{\partial}{\partial v^{j}}}=-(\mathrm{grad}\;V)^{v}$, comes from
the external force. Here $Y^{v}$ denotes the vertical lift of the vector field
$Y\in\mathfrak{X}(M)$ (see e.g. [32, 33]). We are in the usually called case
of conservative systems or potential forces.

If $\mathrm{grad}\;V=0$, the external force vanishes, we say we have a free
motion, $L=T_{g}$, and then

$\Gamma_{T_{g}}=\sum_{i=1}^{n}\left(v^{i}\frac{\partial}{\partial
q^{i}}+F^{i}(q,v)\,\frac{\partial}{\partial v^{i}}\right)\,,\quad
F^{i}(q,v)=-\sum_{k,l=1}^{n}\Gamma_{kl}^{i}v^{k}v^{l}.$ (4.36)

###  4.4 General forces and Newtonian mechanical systems

We can consider a more general situation where the external forces are not
potential ones. This is the case when the force is given by a non-exact
semibasic 1-form $F\in\bigwedge^{1}(TM)$; then the dynamics
$\Gamma\in\mathfrak{X}(TM)$ is given by

$i(\Gamma)\omega_{T_{g}}-dE_{T_{g}}=F\,,$ (4.37)

instead of a free motion whose dynamics $\Gamma_{T_{g}}$ is given by
$i(\Gamma_{T_{g}})\omega_{T_{g}}=dE_{T_{g}}$ as we know.

The 1-form $F$ is called the force form or work form and, being semibasic in
$TM$, its local expression is
$F={\displaystyle\sum_{i=1}^{n}}F_{i}(q,v)\,dq^{i}$. When the 1-form $F$ is
basic, that is $F={\displaystyle\sum_{i=1}^{n}}F_{i}(q)dq^{i}$, it does not
depend on the velocities, and then the system is said Newtonian. In this last
case, given the force form $F$, then
$X_{F}={\displaystyle\sum_{i,j=1}^{n}g^{ij}F_{i}\frac{\partial}{\partial
v^{j}}}$ is the local expression of the vector field of force which is the
vertical lift to $TM$ of the vector field in $M$, $Z_{F}=\widehat{g}^{-1}F$,
i.e. $X_{F}=Z_{F}^{v}$. The above case of potential forces, with $L=T_{g}-V$,
is given by $F=-\tau^{*}(dV)$ with $V:M\to\mathbb{R}$.

Conversely, we can give a vector field of forces, $Z\in\mathfrak{X}(M)$, with
$Z={\displaystyle\sum_{i,j=1}^{n}}Z^{i}\partial/{\partial q^{i}}$, then
$F_{Z}=i(Z)g={\displaystyle\sum_{i,j=1}^{n}}g_{ij}Z^{i}dq^{j}$ is a 1-form in
$M$ and $\tau^{*}F_{Z}$ is a basic 1-form in $TM$, the force form associated
to the force vector field $Z$. In this approach the dynamical vector field
$\Gamma$ is given by

$i(\Gamma)\omega_{T_{g}}-dE_{T_{g}}=F_{Z}\,,$ (4.38)

where we write $F_{Z}$ instead of $\tau^{*}F_{Z}$ for simplicity. The
corresponding second order differential equation for its integral curves is

$\ddot{q}^{i}=-\sum_{k,l=1}^{n}\Gamma_{kl}^{i}\dot{q}^{k}\dot{q}^{l}+Z^{i}\,,\quad
i=1,\ldots,n.$ (4.39)

Furthermore, we can consider the case where the vector field of forces is
depending on the velocities
$Z={\displaystyle\sum_{i=1}^{n}}Z^{i}(q,v){\partial}/{\partial q^{i}}$, that
is, it is a vector field along the projection $\tau_{M}:TM\to M$. In this case
we proceed as above obtaining the 1-form of forces
$F_{Z}=i(Z)g={\displaystyle\sum_{i,j=1}^{n}}g_{ij}Z^{i}dq^{j}$ and
thecorresponding expression for the dynamical vector field $\Gamma$.

In all these situations the dynamical vector field $\Gamma$ is a second order
differential equation with a similar local expression as above.

##  5 Application to geodesics, free motion and geodesic fields

###  5.1 Curves with extremal length

As defined in (4.2), the extremal length curves in $(M,g)$ are those
corresponding to the action defined by the Lagrangian $\ell(v)=\sqrt{g(v,v)}$,
even if we have to restrict ourselves to the open submanifold $T_{0}M=\\{v\in
TM\mid v\neq 0\\}$ in order to preserve the differentiability. Remark that the
length does not depend on the parametrisation of the curve and consequently a
reparametrisation of an extremal length curve leads to another extremal length
curve.

Consider, therefore, the Lagrangian $\ell(v)=\sqrt{2\,{T_{g}}(v)}$, where
$T_{g}\in C^{\infty}(TM)$ is the function defined in (4.29)

$T_{g}(v)=\frac{1}{2}\,g(v,v),\qquad v\in TM,$ (5.1)

usualy called kinetic energy in the context of mechanical systems.

This function $\ell(v)\in C^{\infty}(T_{0}M)$ is a singular Lagrangian that
has been studied in [34] in a problem with applications in a geometric
approach to optics. This singular character of $\ell$ is responsible of the
fact that a reparametrisation of an extremal length curve gives rise to
another extremal length curve.

Being $\ell$ a singular Lagrangian, the symplectic approach is not directly
applicable but we can use the first method and obtain the Euler-Lagrange
equations. As $\ell(q,\dot{q})=\sqrt{g_{ij}(q)\dot{q}^{i}\dot{q}^{j}}$, then

$\frac{\partial\ell}{\partial\dot{q}^{i}}=\sum_{j=1}^{n}\frac{g_{ij}(q)\dot{q}^{j}}{\ell},\qquad\frac{\partial\ell}{\partial
q^{i}}=\frac{1}{2\ell}\sum_{j,k=1}^{n}\left(\frac{\partial g_{jk}}{\partial
q^{i}}\dot{q}^{j}\,\dot{q}^{k}\right),\quad i=1\ldots,n,$

and the Euler-Lagrange equations of the Lagrangian $\ell$ are:

$\frac{d}{dt}\left(\sum_{j=1}^{n}\frac{g_{ij}(q)\,\dot{q}^{j}}{\ell}\right)=\frac{1}{2\ell}\left(\sum_{j=1}^{n}\frac{\partial
g_{jk}}{\partial q^{i}}\dot{q}^{j}\,\dot{q}^{k}\right),\quad i=1\ldots,n,$

where $t$ is the parameter of the solution curves.

If we parametrise the curves by the arc-lenght $s$, that is we apply a Sundman
transformation, then as $ds/dt=\ell$, $\dot{q}^{i}=\ell\,q^{\prime
i}=\ell\,dq^{i}/ds$, we have that the previous system becomes

$\frac{d}{ds}\left(\sum_{j=1}^{n}g_{ij}(q)\,q^{\prime
j}\right)=\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial g_{jk}}{\partial
q^{i}}\,q^{\prime j}\,q^{\prime k},$ (5.2)

i.e.

$q^{\prime\prime i}=\sum_{j,k,l=1}^{n}g^{il}(q)\left(\frac{1}{2}\frac{\partial
g_{jk}}{\partial q^{l}}-\frac{\partial g_{lj}}{\partial
q^{k}}\right)\,q^{\prime j}\,q^{\prime k},$ (5.3)

or in another form,

$q^{\prime\prime i}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}(q)\,q^{\prime
j}\,q^{\prime k}=0,$ (5.4)

where $\Gamma_{jk}^{i}$ are the Christoffel symbols of the second kind defined
by the metric $g$.

Equation (5.4), which is the same equation as (4.12), is called geodesics’
equation obtained from Euler-Lagrange equations for $\ell\,$ by the above
Sundman transformation. It is to be remarked that the same can be said for a
Sundman reparametrisation $d\tau/dt=a\,\ell$, where $a\in\mathbb{R}$. This
corresponds to the fact that the parameter of geodesic curves is defined up to
an affine transformation $s\mapsto a\,s+b$. See Subsection 5.3 for a
reciprocal of this result and some comments on it.

###  5.2 Free motions on a Riemann manifold

Free motion on a Riemann manifold $(M,g)$ is described by a regular Lagrangian
given by the function $T_{g}\in C^{\infty}(TM)$, as defined in (4.29), which
is the kinetic energy defined by the metric.

We can follow the geometric approach with the Lagrangian $L=T_{g}$, which
reduces to the kinetic energy and is a regular Lagrangian. The Cartan 1-form
$\theta_{T_{g}}$ and the symplectic 2-form are given by

$\begin{array}[]{rcl}\theta_{T_{g}}(q,v)&=&{\displaystyle\sum_{i,j=1}^{n}}g_{ij}(q)\,v^{j}\,dq^{i},\\\
\omega_{T_{g}}&=&{\displaystyle\sum_{i,j=1}^{n}}g_{ij}\,dq^{i}\wedge
dv^{j}+{\displaystyle\frac{1}{2}\sum_{i,j,k=1}^{n}\left(\frac{\partial
g_{ij}}{\partial q^{k}}v^{j}-\frac{\partial g_{kj}}{\partial
q^{i}}v^{j}\right)}dq^{i}\wedge dq^{k}.\end{array}$ (5.5)

The total energy is $E_{T_{g}}=T_{g}$ and the dynamical vector field solution
of $i(\Gamma_{T_{g}})\omega_{T_{g}}=dE_{T_{g}}$ is

$\Gamma_{T_{g}}=\sum_{i=1}^{n}\left(v^{i}\frac{\partial}{\partial
q^{i}}+F^{i}(q,v)\,\frac{\partial}{\partial v^{i}}\right),\quad
F^{i}(q,v)=-\sum_{j,k=1}^{n}\Gamma^{i}_{jk}(q)v^{j}v^{k},$ (5.6)

where, as above, $\Gamma_{jk}^{i}$ denote the second class Christoffel symbols
defined by the metric.

The projected curves on the base manifold of the integral curves of the vector
field $\Gamma_{T_{g}}$ are solutions of the system of second order
differential equations

$\ddot{q}^{i}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}(q)\,\dot{q}^{j}\,\dot{q}^{k}=0,$
(5.7)

which is the local expression for $\nabla_{\dot{\gamma}}\,\dot{\gamma}=0$,
where $\gamma$ is a curve in $M$ and $\dot{\gamma}$ the vector field along
$\gamma$ given by the velocities of the curve at each point. This shows that
the above integral curves are geodesics of the Riemannian structure, and
therefore, the geodesic curves $\gamma$ are such that their tangent
prolongations $\dot{\gamma}$ are integral curves of $\Gamma_{T_{g}}$.

In particular, as indicated above, an extremal length curve parametrised by
its arc-length is a solution of

$\frac{d^{2}q^{i}}{ds^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\,\frac{dq^{j}}{ds}\,\frac{dq^{k}}{ds}=0,$
(5.8)

and therefore in terms of such parametrisations the extremal length curves are
geodesics of the corresponding metric.

As we will see in Subsection 5.3, if a curve $\gamma(t)$ satisfies
$\nabla_{\dot{\gamma}}\dot{\gamma}=0$, then $g(\dot{\gamma},\dot{\gamma})$ is
constant along $\gamma$, hence the parameter $t$ is a real affine function of
the arc-length, that is $t=as+b$ with $a,b\in\mathbb{R}$.

###  5.3 Geodesic fields and Sundman transformation

There are distinguished classes of vector fields on a Riemann manifold
$(M,g)$. For instance, Killing vector fields $X$ are those such that
$\mathcal{L}_{X}g=0$, i.e. vector fields whose local flows are preserving $g$.
Another relevant class is that of autoparallel vector fields, characterised by
the property $\nabla_{X}X=0$, where $\nabla$ is the corresponding Levi–Civita
connection.

Remark that $X\in\mathfrak{X}(M)$ is an autoparallel vector field if and only
if its integral curves $\gamma$ are geodesics, i.e. they satisfy the equation
$\nabla_{\dot{\gamma}}\,\dot{\gamma}=0$, and then when lifted to the tangent
bundle $TM$ are integral curves of $\Gamma_{T_{g}}$, the second order geodesic
vector field given in (4.13) or (5.6). In fact, suppose that $\nabla_{X}X=0$
and let $\gamma:I\subset\mathbb{R}\to M$ be an integral curve of $X$, that is
$\dot{\gamma}=X\circ\gamma$. Then:

$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=\nabla_{(X\circ\gamma)(t)}(X\circ\gamma)(t)=(\nabla_{X}X)(\gamma(t))=0,$

and therefore $\gamma$ is a geodesic curve in $M$. If, on the contrary, we
suppose that every integral curve of $X$ satisfies the equation
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0$, let $p\in M$ and
$\gamma:I\subset\mathbb{R}\to M$ be the integral curve of $X$ with initial
condition $p$, we have

$(\nabla_{X}X)(p)=\nabla_{X(p)}X=\nabla_{\dot{\gamma}(0)}\dot{\gamma}(t)=(\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t))_{|t=0}=0.$

But as this is true for every $p\in M$, we have $\nabla_{X}X=0$.

Because of this property the autoparallel vector fields are also called
geodesic vector fields [35]. It is to be remarked that for each real number
$a$, if $X$ is a geodesic vector field, then $a\,X$ is geodesic too.

There exists an even more general class of generalised geodesic vector fields,
usually also called pregeodesic vector fields, which are those such that there
exists a function $f$ satisfying $\nabla_{X}X=f\,X$.

Before going to the application of Sundman transform to these vector fields,
we recall some known results with slightly new proofs.

In [36], it is proven, Lemma 3, and in [37] is used, that a Killing vector
field $X\in\mathfrak{X}(M)$ has constant length if and only if every integral
curve of $X$ is a geodesic, that is, $X$ is a geodesic vector field. We can
give a little more general proof:

a) Using that the connection $\nabla$ is metric and torsionless, we have on
the one hand that for arbitrary vector fields $X,Y\in\mathfrak{X}(M)$,

$\begin{array}[]{rcl}\mathcal{L}_{X}(g(X,Y))&=&(\mathcal{L}_{X}g)(X,Y)+g(\mathcal{L}_{X}X,Y)+g(X,\mathcal{L}_{X}Y)\\\
&&\\\ &=&(\mathcal{L}_{X}g)(X,Y)+g(X,\nabla_{X}Y)-g(X,\nabla_{Y}X)\\\ &&\\\
&=&(\mathcal{L}_{X}g)(X,Y)+g(X,\nabla_{X}Y)-{\displaystyle\frac{1}{2}}\mathcal{L}_{Y}(g(X,X)),\end{array}$

and on the other hand,

$\mathcal{L}_{X}(g(X,Y))=g(\nabla_{X}X,Y)+g(X,\nabla_{X}Y)\,.$ (5.9)

Comparing both expressions we have that

$g(\nabla_{X}X,Y)=(\mathcal{L}_{X}g)(X,Y)-\frac{1}{2}\mathcal{L}_{Y}(g(X,X)).$

b) If $X=Y$, the expression (5.9) reduces to
$\mathcal{L}_{X}(g(X,X))=2g(\nabla_{X}X,X)$ and this relation shows that if
$X$ is a geodesic vector field, $\nabla_{X}X=0$, then
$\mathcal{L}_{X}(g(X,X))=0$, that is, the norm of $X$ is constant along the
integral curves of $X$.

c) Moreover, if $X$ is a Killing vector field, we have that
$g(\nabla_{X}X,Y)=-(1/2)\mathcal{L}_{Y}(g(X,X))$, and, consequently, as $Y$ is
an arbitrary vector field, we see that then $X$ is a geodesic vector field,
i.e. $\nabla_{X}X=0$, if and only if $g(X,X)$ is a real constant as we wanted.

A remarkable similar property was proved in [38]: a generalised geodesic
vector field $X$ of constant length on a Riemannian manifold is a geodesic
vector field, because if $\varphi$ is the function defined by
$\varphi=\frac{1}{2}g(X,X)$, as $\nabla_{X}X=f\,X$, we have that

$\mathcal{L}_{X}(\varphi)=\frac{1}{2}g(\nabla_{X}X,X)+\frac{1}{2}g(X,\nabla_{X}X)=2f\varphi,$

and equivalently, $2f=\mathcal{L}_{X}(\log\varphi)$, which shows that as
$\varphi$ is constant, then $f=0$, and therefore $X$ is a geodesic vector
field.

As a time-reparametrisation of an extremal length curve gives rise to another
extremal length curve, we can study the reparametrisation of geodesics curves,
which we know are parametrised by the arc-length. If we consider a
reparametrisation of a solution of (5.8) with a generic parameter $\tau$
defined, as a given function of $s$, by

$\frac{ds}{d\tau}=\xi(\tau),$ (5.10)

we have

$\frac{d}{ds}=\frac{1}{\xi}\frac{d}{d\tau}\Longrightarrow\frac{d^{2}}{ds^{2}}=\frac{1}{\xi}\frac{d}{d\tau}\frac{1}{\xi}\frac{d}{d\tau}=-\frac{d\xi/d\tau}{\xi^{3}}\frac{d}{d\tau}+\frac{1}{\xi^{2}}\frac{d^{2}}{d\tau^{2}},$
(5.11)

and the corresponding differential equation (5.8) for geodesics in such a
generic parametrisation is

$\frac{d^{2}q^{i}}{d\tau^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\,\frac{dq^{j}}{d\tau}\,\frac{dq^{k}}{d\tau}=\lambda(\tau)\frac{dq^{i}}{d\tau},$
(5.12)

with

$\lambda(\tau)=\frac{d^{2}s}{d\tau^{2}}\left(\frac{ds}{d\tau}\right)^{-1},$
(5.13)

that can also be written as

$\lambda(s)=-\frac{d^{2}\tau}{ds^{2}}\left(\frac{ds}{d\tau}\right)^{2}.$

Note that when $\xi(\tau)$ is a constant $a$, that corresponds to an affine
change of parameter, we obtain from (5.13) that $\lambda=0$, i.e. (5.12)
reduces to the original equation. This corresponds to the case $\tau=as+b$,
wth $a,b\in\mathbb{R}$, that is, ${\displaystyle\frac{d^{2}\tau}{ds^{2}}=0}$,
as we commented at the end of Subsection 5.1.

Conversely, a solution of the differential equation (5.12) corresponds to a
solution of (5.8) with a reparametrisation defined by (5.10) with
$\xi(\tau)=\exp\left({\displaystyle\int^{\tau}}\lambda(\zeta)\,d\zeta\right)$.

Instead of a reparametrisation (5.10) we can consider a classical Sundman
transformation

$\frac{ds}{d\tau}=f(q),$ (5.14)

and then differential equation (5.8) for geodesics becomes

$\frac{d^{2}q^{i}}{d\tau^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\,\frac{dq^{j}}{d\tau}\,\frac{dq^{k}}{d\tau}=\frac{1}{f}\left(\sum_{k=1}^{n}\frac{\partial
f}{\partial
q^{k}}\frac{dq^{k}}{d\tau}\right)\frac{dq^{i}}{d\tau}=\frac{d}{d\tau}(\log
f)\,\frac{dq^{i}}{d\tau}.$ (5.15)

Now we try to obtain the relation between geodesic fields and Sundman
transformation in a more geometric way.

If we carry out a Sundman transformation (2.12) in the description of the
geodesics of a Riemann manifold $(M,g)$ as integral curves of geodesic vector
fields, a geodesic vector field $X$ should be replaced by the vector field
$Y=f\,X$, and then its integral curves are not geodesics because their
velocities are not covariantly constant. Actually, from

$\nabla_{f\,X}(f\,X)=f\,\nabla_{X}(f\,X)=f^{2}\,\nabla_{X}X+f\,(Xf)X,$

we see that $\nabla_{X}X=0$ implies that $\nabla_{f\,X}(f\,X)=(Xf)\,(f\,X)$,
i.e. $\nabla_{Y}Y=(Xf)Y$. Conversely, the integral curves of a vector field
$X\in\mathfrak{X}(M)$ can be reparametrised to be geodesic curves if and only
if there exists a function $f:M\to\mathbb{R},\,f>0$, such that
$\nabla_{X}X=f\,X$, because in this case, for any function
$\lambda:M\to\mathbb{R},\,\lambda>0$, we have

$\nabla_{\lambda X}(\lambda
X)=(\lambda\mathcal{L}_{X}\lambda)X+\lambda^{2}\nabla
XX=(\lambda\mathcal{L}_{X}\lambda)X+\lambda^{2}fX=(\lambda\,X(\lambda)+\lambda^{2}f)X,$

and taking as $\lambda$ any solution of the differential equation
$\mathcal{L}_{X}(\log\lambda)=-f$, we have that $\nabla_{\lambda
X}(\lambda\,X)=0$, and then $\lambda X$ is a geodesic field in $M$ and its
integral curves are geodesic ones.

Observe that if $\nabla_{X}X=fX$, i.e. $X$ is a generalised geodesic vector
field, then $Y=\lambda X$ with $\mathcal{L}_{X}(\log\lambda)=-f$ is a geodesic
vector field.

Summarising, we have proved:

###### Proposition 5.1

The integral curves of a vector field $X\in\mathfrak{X}(M)$ can be transformed
by a Sundman transformation to be geodesic curves if, and only if, the vector
field $X$ satisfies $\nabla_{X}X=fX$ for some nonvanishing function
$f:M\to\mathbb{R}$.

##  6 Applications in mechanical systems

###  6.1 Newtonian systems and time reparametrisation

In this section we study a mechanical Newtonian system determined by a vector
field of forces $Z\in\mathfrak{X}(M)$ in the Riemannian manifold $(M,g)$. We
know from (4.38) that the corresponding dynamical vector field $\Gamma$
satisfies the equation

$i(\Gamma)\omega_{T_{g}}-dE_{T_{g}}=F_{Z}\,,$ (6.1)

where $F_{Z}=i(Z)g$. The integral curves of $\Gamma$ satisfy the second order
differential equation

$\ddot{q}^{i}=-\sum_{k,l=1}^{n}\Gamma_{kl}^{i}\dot{q}^{k}\dot{q}^{l}+Z^{i}\,,\quad
i=1,\ldots,n.$ (6.2)

But a curve $\gamma:I\subset\mathbb{R}\to M$ satisfies this equation if and
only if it satisfies the following one

$\nabla_{\dot{\gamma}}\dot{\gamma}=Z\circ\gamma,$ (6.3)

as we can prove expressing both equations in coordinates.

We are now interested in studying the properties of vector fields
$X\in\mathfrak{X}(M)$ playing the rôle of the geodesic vector fields but in
presence of a non-null vector field of forces $Z\in\mathfrak{X}(M)$.

The main result, to be compared with a previous one for the case $Z=0$, is the
following:

###### Proposition 6.1

Given $Z\in\mathfrak{X}(M)$, the integral curves $\gamma$ of
$X\in\mathfrak{X}(M)$ satisfy the equation
$\nabla_{\dot{\gamma}}{\dot{\gamma}}=Z\circ{\gamma}$, if and only if the
vector field $X$ satisfies the equation $\nabla_{X}X=Z$.

Proof.- Suppose that the integral curves $\gamma$ of $X\in\mathfrak{X}(M)$
satisfy the equation $\nabla_{\dot{\gamma}}{\dot{\gamma}}=Z\circ{\gamma}$. If
$p$ is a point of $M$ and $\gamma$ is the integral curve of $X$ starting from
$p$, we have:

$(\nabla_{X}X)(p)=\nabla_{X(p)}X=\nabla_{\dot{\gamma}(0)}(X\circ\gamma)(t)=\nabla_{\dot{\gamma}(0)}{\dot{\gamma}(0)}(t)=Z(\gamma(0))=Z(p),$

and as this is valid for any $p\in M$, then $\nabla_{X}X=Z$.

Conversely, if $\nabla_{X}X=Z$ and $\gamma$ is an integral curve of $X$, that
is $\dot{\gamma}=X\circ\gamma$, then $(\nabla_{X}X)(\gamma(t))=Z(\gamma(t))$,
which gives $\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=Z(\gamma(t))$.

$\hfill\Box$

This result can be particularized to the case $Z=-\textrm{grad\,}V$
corresponding to a simple mechanical type system.

The relation with Sundman transformation is the following:

Suppose that the vector field $X\in\mathfrak{X}(M)$ satisfies the equation
$\nabla_{X}X=Z$ and consider a Sundman reparametrisation of its integral
curves, that is, consider the vector field $Y=h\,X$, with $h$ a function,
$h:M\to\mathbb{R},\,h>0$. Then the corresponding equation for the vector field
$Y=h\,X$ is:

$\nabla_{Y}Y=\nabla_{h\,X}(h\,X)=h\,(\mathcal{L}_{X}h)X+h^{2}\nabla_{X}X=(\mathcal{L}_{Y}(\log
h))Y+h^{2}\,Z=\lambda Y+h^{2}Z,$

where $\lambda=\mathcal{L}_{Y}(\log
h)={\displaystyle\frac{1}{h}}\mathcal{L}_{Y}h$.

This last equation, $\nabla_{Y}Y=\lambda Y+h^{2}\,Z$, is the general
expression of time reparametrisation of the equation $\nabla_{X}X=Z$, when
changing the vector field $X$ by $Y=h\,X$ in the Newtonian system
$\nabla_{X}X=Z$.

This result can be summarised as:

###### Proposition 6.2

The integral curves of a given vector field $Y\in\mathfrak{X}(M)$ can be
reparametrised to be solutions of the Newtonian equation
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=Z(\gamma(t))$ if, and only if, there
exists a function $h:M\to\mathbb{R},\,h>0$, which satisfies the equation

$\nabla_{Y}Y=\lambda Y+h^{2}Z,$

where $\lambda=\mathcal{L}_{Y}(\log
h)={\displaystyle\frac{1}{h}}\mathcal{L}_{Y}h$. $\hfill\Box$

Remark that, as a consequence of this last Proposition, unless $Z=0$ or
$Z=f\,X$ with $\nabla_{X}X=0$, we cannot find a Sundman reparametrisation of
the curves solution to $\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=Z(\gamma(t))$
to obtain geodesics of the metric $g$, that is curves satisfying equation
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0$, and therefore, in order to
understand such curves as geodesics we need to change the Riemannian metric
$g$ to eliminate the external force with an adequate connection and, as we
will see in the sequel, we cannot do this in the whole configuration manifold
$M$. Later on we will show how to proceed in the case of a mechanical type
Lagrangian system, leading to the so called Jacobi metric.

###  6.2 Mechanical type systems on a Riemann manifold

We consider now Lagrangian systems of simple mechanical type, that is,
Lagrangians of the form $L=T_{g}-\tau_{M}^{*}V$, defined by a Riemann metric
$g$ on $M$ and a function $V:M\to\mathbb{R}$. Its energy function is then
$E=T_{g}+\tau_{M}^{*}V$. We have shown in (4.35) that the second order
differential equation of the dynamical trajectories is

$\ddot{q}^{i}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}(q)\,\dot{q}^{j}\,\dot{q}^{k}=-\sum_{l=1}^{n}g^{il}\frac{\partial
V}{\partial q^{l}},\quad i=1,\ldots,\dim M.$ (6.4)

Which equation is the corresponding one when we apply a time
reparametrisation?

When expressed in terms of a generic parameter $\tau$ defined as a given
function of $t$, as in (5.10),

$\frac{dt}{d\tau}=\xi(\tau),$ (6.5)

we obtain as in (5.11) that

$\frac{d}{dt}=\frac{1}{\xi}\frac{d}{d\tau}\Longrightarrow\frac{d^{2}}{dt^{2}}=\frac{1}{\xi}\frac{d}{d\tau}\frac{1}{\xi}\frac{d}{d\tau}=-\frac{d\xi/d\tau}{\xi^{3}}\frac{d}{d\tau}+\frac{1}{\xi^{2}}\frac{d^{2}}{d\tau^{2}},$
(6.6)

and the differential equation for those curves in such a generic
parametrisation will be

$\frac{d^{2}q^{i}}{d\tau^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\,\frac{dq^{j}}{d\tau}\,\frac{dq^{k}}{d\tau}-\lambda(\tau)\frac{dq^{i}}{d\tau}=-\xi^{2}(\tau)\sum_{l=1}^{n}\,g^{il}\frac{\partial
V}{\partial q^{l}},\quad i=1,\ldots,\dim M,$ (6.7)

with

$\lambda(\tau)=\frac{1}{\xi}\frac{d\xi}{d\tau}.$ (6.8)

In this way we have obtained the differential equation of the trajectories of
the given mechanical system in a general parametrisation.

##  7 Conformal metrics and Sundman transformation. The Jacobi metric

###  7.1 Sundman transformation in free motions. Changes in the Riemannian
metric

In this Subsection and the next one we go to a more geometric approach to the
same problem of reparametrisation. We will try to change the metric $g$ in
order to modify the geometric properties of the dynamical trajectories of the
system.

Let us remark that if we carry out the Sundman transformation (2.12) in the
framework of Lagrangian formulation, with $f$ being a basic function, the new
velocities $\bar{v}$ are related to the previous ones $v$ by $\bar{v}=f\,v$,
and, moreover, if a system was defined by a Lagrangian $L$, in order to the
action be well defined and preserved, the system must be described in terms of
the new time $\tau$ by a new Lagrangian

$\bar{L}(q,\bar{v})=f\,L\left(q,\frac{\bar{v}}{f}\right),$

and then ${\displaystyle\int L\,dt=\int\bar{L}\,d\tau}$.

The new dynamical vector field will be given by

$\Gamma_{\bar{L}}(q,\bar{v})=\sum_{i=1}^{n}\left(\bar{v}^{i}\frac{\partial}{\partial
q^{i}}+\bar{F}^{i}(q,\bar{v})\,\frac{\partial}{\partial{\bar{v}}^{i}}\right),$

for appropriate functions $\bar{F}^{i}$ as given by (4.27).

In the particular case of a free motion on a Riemann manifold $(M,g)$, the new
Lagrangian will be the kinetic energy defined by the metric $(1/f)\,g$,
because $g$ is quadratic in velocities and therefore
$f\,L(q,\bar{v}/f)=(1/f)L(q,\bar{v})$.

On the other side, the considered Sundman transformation for a mechanical type
system amounts to change not only the Riemann structure from $g$ to the
conformally related one $\bar{g}=(1/f)g$ but also the potential function $V$
to $\bar{V}=f\,V$, a fact that has recently been used in the corresponding
Hamiltonian formalism in the search for a new superintegrable system [39]
according to the metod developed in [40]. Recall that a mechanical type system
for which there is a coordinate system such that the potential function is a
sum $V(q)=V_{1}(q_{1})+\ldots+V_{n}(q_{n})$ and the local expression of the
Riemann structure is diagonal, i.e.

$L(q,v)=\frac{1}{2}\sum_{i=1}^{n}a_{i}(q_{i})v_{i}^{2}-\sum_{i=1}^{n}V_{i}(q_{i}),$
(7.1)

is separable as a sum of one-dimensional systems and hence integrable by
quadratures. A generalisation of such system is due to Liouville [40] and
consists on the Hamiltonian

$H(q,v)=\frac{1}{2W(q)}\sum_{i=1}^{n}a_{i}(q_{i})v_{i}^{2}+\frac{1}{W(q)}\sum_{i=1}^{n}V_{i}(q_{i}),$
(7.2)

where $W(q)=W_{1}(q_{1})+\ldots+W_{n}(q_{n})$. These systems are called
Liouville systems [41, 42] and one can check that the $n$ functions

$F_{i}=\frac{1}{2}a_{i}(q)\,p_{i}^{2}+V_{i}(q)-W_{i}\,H,\quad i=1,\ldots,n,$

are constants of motion $\\{H,F_{i}\\}=0$, but they are not independent
because $\sum_{i=1}^{n}F_{i}=0$. Therefore, starting from an appropriate
Hamiltonian $H$ one can look for possible functions $f$ such that the new
Hamiltonian $fH$ satisfies the required properties of superintegrability.

This method of deriving from a given set of involutive functions a new one
recalls very much the Stäckel transforms [43] and the coupling constant
metamorphosis [44] where also infinitesimal time reparametrisations are used
(see also [45] where reciprocal transformations of different times are used in
the study of generalised Stäckel transforms).

It is interesting to study the relationship with the case of a metric
$\bar{g}=e^{2\varphi}g$, which corresponds to $f=e^{-2\varphi}$. We have
proven in (4.21) that in this case the new Christoffel symbols of the second
kind are:

$\bar{\Gamma}^{i}_{jk}(q)=\Gamma^{i}_{jk}(q)+\delta^{i}_{j}\frac{\partial\varphi}{\partial
q^{k}}+\delta^{i}_{k}\frac{\partial\varphi}{\partial
q^{j}}-\sum_{l=1}^{n}g_{jk}\,g^{il}\frac{\partial\varphi}{\partial
q^{l}},\quad i,j,k=1,\ldots,\dim M.$ (7.3)

Then, instead of (5.7) the new equations of the motion for a mechanical system
defined by $\bar{g}$ and $V$ will be:

$\frac{d^{2}q^{i}}{d\tau^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\frac{dq^{j}}{d\tau}\frac{dq^{k}}{d\tau}+2\sum_{k=1}^{n}\frac{\partial\varphi}{\partial
q^{k}}\frac{dq^{k}}{d\tau}\frac{dq^{i}}{d\tau}-\sum_{j,k,l=1}^{n}g_{jk}\frac{dq^{j}}{d\tau}\frac{dq^{k}}{d\tau}\,g^{il}\frac{\partial\varphi}{\partial
q^{l}}=-\sum_{j}^{n}e^{-2\varphi}g^{ji}\frac{\partial V}{\partial q^{j}},$
(7.4)

for $i=1,\ldots,\dim M$.

Here we see that the two last terms of the left-hand side of (7.4) correspond
to the two last terms in (4.21).

If we reparametrise in (7.4) in a way similar to (6.5), i.e.

$\frac{d\tau}{d\eta}=\xi(\eta),$ (7.5)

using expressions corresponding to (6.6) such differential equation becomes

$\frac{d^{2}q^{i}}{d\eta^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\frac{dq^{j}}{d\eta}\frac{dq^{k}}{d\eta}-\lambda(\eta)\frac{dq^{i}}{d\eta}+2\sum_{k=1}^{n}\frac{\partial\varphi}{\partial
q^{k}}\frac{dq^{k}}{d\eta}\frac{dq^{i}}{d\eta}-\sum_{j,k,l=1}^{n}g_{jk}\frac{dq^{j}}{d\eta}\frac{dq^{k}}{d\eta}\,g^{il}\frac{\partial\varphi}{\partial
q^{l}}=-\xi(\eta)^{2}\sum_{j=1}^{n}e^{-2\varphi}g^{ji}\frac{\partial
V}{\partial q^{j}},\,\,$ (7.6)

for $i=1,\ldots,\dim M$, where

$\lambda(\eta)=\frac{1}{\xi}\frac{d\xi}{d\eta}.$ (7.7)

The expression (7.6) is the differential equation for the dynamical
trajectories of a simple mechanical control system in a general
parametrisation with respect to a metric $\bar{g}=e^{2\varphi}g$.

It is to be remarked that in the free (geodesic) case $V=0$ the equation of
geodesic curves for the Riemann structure $\bar{g}=e^{2\varphi}g$ in terms of
the new time $\tau$ such that $d\tau=e^{2\varphi}dt$ is

$\frac{d^{2}q^{i}}{d\tau^{2}}+\sum_{j,k=1}^{n}\Gamma^{i}_{jk}\frac{dq^{j}}{d\tau}\frac{dq^{k}}{d\tau}+2\sum_{k=1}^{n}\frac{\partial\varphi}{\partial
q^{k}}\frac{dq^{k}}{d\tau}\frac{dq^{i}}{d\tau}-\sum_{j,k,l=1}^{n}g_{jk}\frac{dq^{j}}{d\tau}\frac{dq^{k}}{d\tau}\,g^{il}\frac{\partial\varphi}{\partial
q^{l}}=0,$ (7.8)

which does not coincide with (5.15) for $f=e^{-2\varphi}$ because of the last
term on the leftt-hand side of (7.8).

###  7.2 Sundman transformation and the Jacobi metric

From a local viewpoint we can look for a possible choice of the function
$\varphi$ such that there exists a Sundman reparametrisation (5.14) of the
geodesics lines equation in $(M,g)$ given by (5.4) leading to the equation
(7.4). This choice is possible when we restrict our study to motions given by
(6.4) with a fixed energy $E$, i.e. when

$\sum_{j,k=1}^{n}g_{jk}\frac{dq^{j}}{d\eta}\frac{dq^{k}}{d\eta}=f^{2}.$ (7.9)

A simple comparison of (5.15) with (7.4) shows that both equations coincide
when

$\frac{d}{dt}(\log f)=-2\frac{d\varphi}{d\tau},$ (7.10)

i.e. $f=e^{-2\varphi}$, and

$f^{2}\frac{\partial\varphi}{\partial q^{i}}=e^{-2\varphi}\frac{\partial
V}{\partial q^{i}}.$ (7.11)

When the energy $E$ of the mechanical system defined by $\bar{g}$ and $V$ is
fixed,

$e^{2\varphi}\sum_{j,k=1}^{n}g_{jk}\frac{\partial
q^{j}}{\partial\tau}\frac{\partial q^{k}}{\partial\tau}=2(E-V),$

from where we obtain $f^{2}=2\,e^{-2\varphi}\,(E-V)$. Using this value for
$f^{2}$, (7.11) becomes

$2(E-V)\frac{\partial\varphi}{\partial q^{i}}=\frac{\partial V}{\partial
q^{i}},$

and from here we obtain that

$\varphi(q)=-\frac{1}{2}\log(E-V)\Longrightarrow e^{2\varphi}=(E-V)^{-1}.$
(7.12)

Consequently, we have recovered the well-known result about Jacobi metric: The
geodesics of the Jacobi metric $g_{E}=(E-V)g$ when using as parameter its arc-
length $s_{E}$, are the solutions with fixed energy $E$ of the mechanical
system determined by the Lagrangian $L_{g,V}$.

With a simple look at equations (5.8) and (6.4) or the corresponding equations
in terms of $\nabla_{X}$, one can try to relate both types of equations. From
the geometric viewpoint this suggests us the following question: Can we
conformally change the Riemannian metric $g$ to another one
$\bar{g}=\exp(2\varphi)\,g$, with an adequate $\varphi$, such that the vector
fields $X$ solution to the dynamical equation,
$\nabla_{X}X=-\textrm{grad\,}V$, are geodesic, or at least pregeodesic, vector
fields for the Riemannian metric $\bar{g}$?

We have the following result:

###### Theorem 7.1

Let $X\in\mathfrak{X}(M)$ be a solution to the equation $\nabla_{X}X=-{\rm
grad\,}V$, and suppose that the function $E(X):M\to\mathbb{R}$ defined on $M$
by $E(X)=E\circ X$, i.e. $E(X)=(T_{g}+V\circ\tau_{M})(X)$, takes a constant
value, $E=E_{0}\in\mathbb{R}$. Then, $X$ is a pregeodesic vector field for the
Riemannian metric $\bar{g}=\exp(2\varphi)\,g$ if, and only if, there exists a
function $f$ such that

$2(E_{0}-V)\,{\rm grad}_{g}\,\left(\varphi-\log(E_{0}-V)^{1/2}\right)=-f\,X.$

Moreover, if $f=0$ and $M$ is connected, then $\varphi=\log(E_{0}-V)^{1/2}+k$,
where $k$ is a real number, that is, $\bar{g}=e^{2k}(E_{0}-V)g$. The case
$k=0$ is usually called Jacobi metric: $\bar{g}=(E_{0}-V)g$.

Proof.- Let be $\bar{g}=\exp(2\varphi)\,g$, and recall the relation (4.18)
between the covariant derivatives with respect to $g$ and $\bar{g}$ and the
corresponding equation (4.20) for $X=Y$.

Therefore, the vector field $X$ is pregeodesic for $\bar{g}$ if, and only if,
there exists a function $h$ such that $\bar{\nabla}_{X}X=h\,X$, and hence if,
and only if, $\nabla_{X}X-g(X,X)\,{\textrm{grad}}_{g}\varphi=f\,X$ with
$f=h-2\mathcal{L}_{X}\varphi$.

But the condition $\nabla_{X}X=-{\textrm{grad}_{g}}V$ for the vector field $X$
is then equivalent to

$2(E_{0}-V)\,{\textrm{grad}}_{g}\left(\varphi-\log(E_{0}-V)^{1/2}\right)=-f\,X,$

where use has been made of $E(X)=E_{0}$ and

$\textrm{grad}_{g}\left(\log(E_{0}-V)^{1/2}\right)=-\frac{1}{2(E_{0}-V)}\,\textrm{grad}_{g}V.$

If $f=0$, then
${\textrm{grad}}_{g}\left(\varphi-\log(E_{0}-V)^{1/2}\right)=0$, and
consequently, when $M$ is connected, there exists a real number $k$ such that
$\varphi=\log(E_{0}-V)^{1/2}+k$, where $k$ is a real number.

We have obtained that $\bar{g}=\exp(2\varphi)g=e^{2k}(E_{0}-V)g$. Obviously
the factor $k$ is irrelevant for the characterization of the geodesics of the
family of metrics $\bar{g}$ and we can choose $k=0$ as in the classical Jacobi
metric.

$\hfill\Box$

The above Theorem is, at our knowledge, the most natural approximation to the
description of the Jacobi metric in a mechanical type system. Another approach
to Jacobi metric can be found in [46].

##  8 Comments on Jacobi metric and Hamilton-Jacobi equation

Remark that condition $E(X)=(T+V\circ\tau_{M})(X)=E_{0}\in\mathbb{R}$ is
equivalent to say that the image of $X:M\to TM$ is contained in the
hypersurface of $TM$

$\left\\{v\in TM\mid
E(v)=T(v)+V(\tau_{M}(v))=\frac{1}{2}g(v,v)+V(\tau_{M}(v))=E_{0}\right\\}\,.$

This is a natural condition because the total energy $E=T+V$ is a constant of
the motion for the dynamical Newtonian system. In fact, if
$X\in\mathfrak{X}(M)$ is a solution to the equation
$\nabla_{X}X=-\textrm{grad\,}V$, then:

$\mathcal{L}_{X}(E(X))=\mathcal{L}_{X}\left(\frac{1}{2}g(X,X)+V\right)=g(\nabla_{X}X,X)+\mathcal{L}_{X}V\,,$

and therefore, as

$g(\nabla_{X}X,X)=g(-\textrm{grad\,}V,X)=-dV(X)=-\mathcal{L}_{X}V\,,$

we obtain that $\mathcal{L}_{X}(E(X))=0$.

Hence for every vector field $X\in\mathfrak{X}(M)$ solution to
$\nabla_{X}X=-\textrm{grad\,}V$, we have that the image of every integral
curve of $X:M\to TM$ is contained in a constant energy surface, that is,
$E\circ\gamma:\mathbb{R}\to\mathbb{R}$ is a constant $E_{0}$. This constant is
generically different for every integral curve.

The above Theorem can also be stated for the curves solution to the mechanical
type systems, that is: a curve satisfying
$\nabla_{\dot{\gamma}}\dot{\gamma}=Z_{F}\circ\gamma=-(\textrm{grad\,}V)\circ\gamma$
is a pregeodesic of the metric $\bar{g}=e^{2k}\exp(2\varphi)g$, with
$k\in\mathbb{R}$, and in particular for the Jacobi metric determined by $k=0$.

For the general Newtonian case $\nabla_{X}X=Z_{F}$, where $F$ is an arbitrary
semi-basic 1-form on $TM$, the necessary and sufficient condition for $X$ to
be a pregeodesic vector field is the existence of a function $f$ such that
$\nabla_{X}X+g(X,X)X_{F}=f\,X$, but this situation is not easy to deal with
and we have not any constant of the motion to be chosen instead of the energy.
Observe that if $E(X)=E_{0}$ the family of integral curves of $X$ is contained
in the above hypersurface of $TM$.

The above results are related with Hamilton-Jacobi equation. In fact if we
suppose that the vector field $X$ satisfies the equation $d\,(i_{X}g)=0$, then
condition $\nabla_{X}X=-\textrm{grad\,}V$ is equivalent to $d\,(E(X))=0$,
which is the global expression of the classical Hamilton-Jacobi equation in
the Lagrangian form. See [47] for more details on this subject.

##  9 Conclusions and future work

We started with the classical definition of Sundman transformation for
autonomous systems of first-order differential equations and its applications
at the beginning of twentieth century and describe it in a geometric way as a
conformal change in the corresponding. dynamical vector field. This geometric
approach allows us to implement new applications and to understand why such
transformations can be useful when additional geometric structures are
present. The case of systems of second-order differential equations is not so
easy because the concept of velocity also changes. However, those derivable
from a variational principle as those related to geodesic motions or Newtonian
mechanical systems have been studied in the framework of Riemannian geometry.
We can give a geometric answer to what is the relation between vector fields
if we can relate their integral curves or we impose some particular properties
to these integral curves. This is stated and solve in Sections 5 and 6.

As we relate the Sundman transformation with a conformal change, we use the
same idea to obtain the Jacobi metric as a kind of Sundman transformation, in
fact a conformal one of the metric of the manifold which is the configuration
space of the mechanical system. This is made in Section 7, Theorem 7.1, and
the relation with Hamilton-Jacobi equation for Newtonian systems is commented
in Section 8. This application is also related with the results obtained in
Section 6.

There are several future lines of development, most of them related with the
problems included in Section 3. On the other side, the geometric definition of
Sundman transformation for general systems of second-order differential
equation needs to be more carefully studied. As the concept of velocity after
a Sundman transformation is different, we have to consider alternative tangent
bundle structures. This shows the convenience of the study of the different
structures associated with the tangent and cotangent bundles of a manifold,
and this also affects to the idea of linear and Hamiltonian systems
respectively. We hope to present some results in these lines in a forthcoming
paper.

## Acknowledgements

The authors acknowledge the financial support from the Spanish Ministerio de
Ciencia, Innovación y Universidades projects PGC2018-098265-B-C31 and
PGC2018-098265-B-C33.

## Abbrevations

Not applicable

## Declarations

Ethics approval and consent to participate:

Not applicable.

Consent for publication:

Informed consent was obtained from all individual participants included in the
study.

Availability of data and material:

Not applicable.

Competing interests:

The authors declare that they have no Competing interests.

Funding:

Financial support from the Spanish Ministerio de Ciencia, Innovación y
Universidades, projects PGC2018-098265-B-C31 and PGC2018-098265-B-C33 (MINECO,
Madrid) and DGA project E48/20R Análisis y Física Matemática.

Authors’ contributions:

The authors, JFC, EMF and MCML, equally contributed to the paper and have
approved the submitted version

## Conflicts of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Sundman, K.F.: Mémoire sur le problème des trois corps, Acta Mathematica 36, 105–179 (1913).
  * [2] Levi-Civita, T.: Sur la régularisation du problème des trois corps, Acta Math. 42, 99–144 (1920).
  * [3] Levi-Civita, T.: Traiettorie singolari ed urti nel problema ristretto dei tre corpi, Annali di Matematica Pura ed Applicata 9, 1–32 (1904).
  * [4] Levi-Civita, T.: Sur la résolution qualitative du problème restreint des trois corps, Acta Math. 30, 305–327 (1906).
  * [5] Bohlin, K.: Note sur le problème des deux corps et sur une intégration nouvelle dans le problème des trois corps, Bulletin Astronomique 28, 113–119 (1911).
  * [6] Duarte, L.G.S., Moreira, I.C., and Santos F.C.: Linearization under non-point transformations, J. Phys. A: Math. Gen. 27, L739–L743 (1994).
  * [7] Blanes, S., and A. Iserles, A.: Explicit adaptive symplectic integrators for solving Hamiltonian systems, Celest. Mech. Dyn. Astr. 114, 297–317 (2012).
  * [8] Cirilli, S., Hairer, E., and Leimkuhler, B.: Asymptotic Error Analysis of the Adaptive Verlet Method, BIT Numerical Mathematics 39, 25–33 (1999).
  * [9] Nacozy, P.: A discussion of time transformations and local truncation errors, Celest. Mechanics 13, 495–501 (1976).
  * [10] Euler, N.: Transformation Properties of $\ddot{x}+f_{1}(t)\,\dot{x}+f_{2}(t)\,x+f_{3}(t)\,x^{n}=0$, J. Nonl. Math. Phys. 4, 310–337 (1997).
  * [11] Euler, N., and Euler, M.: Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations, J. Nonl. Math. Phys. 11, 399–423 (2004).
  * [12] Bond, V.R.: A transformation of the two-body problem, Celest. Mechanics 35, 1–7 (1985).
  * [13] Ferrer, S., and Sein-Echaluce, M.L.: On the Szebehely-Bond equation generalized Sundman’s transformation for the perturbed two-body problem, Celest. Mechanics 32, 333–347 (1984).
  * [14] Szebehely, V. and Bond, V.: Transformations of the perturbed two-body problem to unperturbed harmonic oscillators, Celest. Mechanics 30, 59–69 (1983).
  * [15] Kavle, H., Offin, D., and Portaluri, A.: Keplerian orbits through the Conley–Zehnder index, Qual. Theory Dyn. Syst. 20, 10 (2021).
  * [16] Pellegrini, E., Russell, R.P., and Vittaldev, V.: $F$ and $G$ Taylor series solutions to the Stark and Kepler problems with Sundman transformations, Celest. Mech. Dyn. Astr. 118, 355–378 (2014).
  * [17] Cariñena, J.F., Clemente-Gallardo, J., Jover-Galtier, J.A., and Marmo, G.: Tangent bundle geometry from dynamics: application to the Kepler problem, Int. J. Geom. Methods Mod. Phys. 14, 1750047 (2017).
  * [18] Moyo, S., and Meleshko, S.V.: Application of the Generalised Sundman Transformation to the Linearisation of Two Second-Order Ordinary Differential Equations, J. Nonl. Math. Phys. 18 S 1, 213–236 (2011).
  * [19] Cariñena, J.F., Ibort, L.A., and Lacomba, E.: Time scaling as an infinitesimal canonical transformation in Celestial Mechanics, Celest. Mechanics 42, 201–213 (1988).
  * [20] Bond, V.R., and Janin, G.: Canonical orbital elements in terms of an arbitrary independent variable, Celest. Mechanics 23, 159–172 (1981).
  * [21] Cariñena, J.F., Ibort, A., Marmo, G., and Morandi, G.: Geometry from Dynamics: Classical and Quantum. Springer, Dordrecht, 2015.
  * [22] Jacobi, C., Clebsch, A, and Brockhardt, C.: Jacobi’s Lectures on Dynamics, Texts and Readings in Mathematics, Hindustan Book Agency, 2009.
  * [23] Cariñena, J.F., and Fernández–Núñez, J.: Jacobi multipliers in integrability and the inverse problem of mechanics, Symmetry 13, 1413 (2021).
  * [24] Cariñena, J.F., and Ibort, L.A.: Non-Noether constants of motion, J. Phys. A: Math. Gen. 16, 1–7 (1983).
  * [25] Koszul, J.L.: Crochet de Schouten-Nijenhuis et cohomologie, in E. Cartan et les Mathématiciens d’aujourd’hui, Lyon, 25-29 Juin, 1984, Astérisque, hors-série, 251-271 (1985).
  * [26] Lichnerowicz, A.: Les variétés de Poisson et leur algèbres de Lie associées, J. Diff. Geom. 12, 253–300 (1977).
  * [27] Bizyaev, I.A., , A.V., and Mamaev, I.S.: Hamiltonization of Elementary Nonholonomic Systems, Russian J. Math. Phys. 22, 444–453 (2015).
  * [28] Bizyaev, I.A., Borisov, A.V., and I.S. Mamaev, I.S.: The Hojman construction and Hamiltonization of Nonholonomic Systems, SIGMA 12, 012 (2016).
  * [29] Lee, J.M.: Introduction to Riemannian manifolds, Graduate Texts in Mathematics 176, Springer, 2018.
  * [30] Conlon, L.: Differentiable Manifolds, Modern Birkhäuser Classics, Birkhäuser, Boston, 2008.
  * [31] Matveev, V.S., and Trautman, A.: A Criterion for Compatibility of Conformal and Projective Structures, Commun. Math. Phys. 329, 821–825 (2014).
  * [32] Crampin, M.: On the differential geometry of the Euler–Lagrange equations and the inverse problem in Lagrangian dynamics, J. Phys. A: Math. Gen. 14, 2567–2575 (1981).
  * [33] Crampin, M.: Tangent bundle geometry for Lagrangian dynamics, J. Phys. A: Math. Gen. 16, 3755–3772 (1983).
  * [34] Cariñena, J.F., and López, C.: Symplectic Structure on the set of Geodesics of a Riemannian manifold, Int. J. Mod. Phys. A, 6, 431–444 (1991).
  * [35] Deshmukh S., Peska, P., and Bin Turki, N.: Geodesic Vector Fields on a Riemannian Manifold, Mathematics 8, 137 (2020).
  * [36] V Berestovskii, V.N., and Nikonorov, Y.G.: Clifford-Wolf homogeneous Riemannian manifolds, J. Diff. Geom. 82, 467–500 (2009).
  * [37] Nikonorov, Y.G.: Spectral properties of Killing vector fields of constant length, J. Geom. Phys. 145, 103485 (2019).
  * [38] Deshmukh, S., and Chen, B.Y.: A note on Yamabe solitons, Balkan J. Geom. Appl. 23, 37–42 (2018).
  * [39] Cariñena, J.F., Rañada, M.F., and Santander, M.: Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems, J. Phys. A: Math. Theor. 54, 105201 (2021).
  * [40] Liouville, J.: L’integration des équations différentielles du mouvement d’un nombre quelconque de points matériels, Journal des Mathématiques Pures et appliqúees 14, 257–299 (1849).
  * [41] A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras (Basel; Birkhauser, 1990).
  * [42] M.A. González-León, J. Mateos-Guilarte, and M. de la Torre Mayado, “On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere”, Integrability, supersymmetry and coherent states, 359–373, CRM Ser. Math. Phys. (Springer, 2019).
  * [43] Boyer, C.P., Kalnins, E.G. and Miller W.: Stäckel-Equivalent Integrable Hamiltonian Systems, SIAM J. Math. Anal. 17, 778–797 (1986).
  * [44] Hietarinta, J., Grammaticos, B., Dorizzi, B., and Ramani, A.: Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems, Phys. Rev. Lett. 53, 1707–1710 (1984).
  * [45] A. Sergyeyev, A., and Blaszak, M.: Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems, J. Phys. A: Math. Theor. 41, 105205 (2008).
  * [46] López, C., and Martínez, E.: SubFinslerian metric associated to an Optimal Control system, SIAM J. Cont. Opt. 39, 798–811 (2000).
  * [47] Cariñena, J.F., Gràcia, X., Marmo, G., Martínez, E., Muñoz–Lecanda, M.C., and Román–Roy, N.: Geometric Hamilton-Jacobi theory, Int. J. Geometric Methods Mod. Phys. 3, 1417–1458 (2006).