\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 120, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/120\hfil Jacobi-Maupertuis metric]
{Jacobi-Maupertuis metric of Li\'enard type equations and Jacobi last multiplier}

\author[S. Chanda, A. Ghose-Choudhury, P. Guha \hfil EJDE-2018/120\hfilneg]
{Sumanto Chanda, Anindya Ghose-Choudhury, Partha Guha}

\address{Sumanto Chanda \newline
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector-3, Salt Lake, Kolkata  700098, India}
\email{sumanto12@bose.res.in}

\address{Anindya Ghose-Choudhury \newline
Department of Physics, Surendranath  College,
24/2 Mahatma Gandhi Road,
 Kolkata  700009, India}
\email{aghosechoudhury@gmail.com}

\address{Partha Guha \newline
Instituto de F\'isica de S\~ao Carlos; IFSC/USP,
Universidade de S\~ao Paulo Caixa Postal 369,
CEP 13560-970,  S\~ao Carlos, SP, Brazil}
\email{partha@bose.res.in}

\dedicatory{Communicated by Anthony Bloch}

\thanks{Submitted January 15, 2018. Published June 15, 2018.}
\subjclass[2010]{34C14, 34C20}
\keywords{Jacobi-Maupertuis metric; position-dependent mass;
\hfill\break\indent  Jacobi's last multiplier}

\begin{abstract}
 We present a construction of the Jacobi-Maupertuis (JM) principle for an
 equation of the Li\'enard type,
 $$
 \ddot{x} + f(x) \dot{x}^2 + g(x) = 0,
 $$
 using Jacobi's last multiplier. The JM metric allows us to reformulate the
 Newtonian equation of motion  for a variable mass as a geodesic equation
 for a Riemannian metric. We illustrate the procedure with examples of
 Painlev\'e-Gambier XXI, the Jacobi equation and the Henon-Heiles system.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Nonlinear differential equations of the Li\'enard type occupy a special
place in the study of dynamical systems as they serve to model various physical,
chemical and biological processes. The standard Li\'enard equation involves
a dissipative term depending linearly on the velocity. However there are
practical problems in which higher order dependance on velocities are appropriate.
Such equations have the generic form $\ddot{x}+f(x)\dot{x}^2+g(x)=0$.
It is interesting to note that equations of this type naturally arise in
Newtonian dynamics when the mass, instead of being a constant, is allowed
to vary with the position coordinate --
the so called position dependent mass (PDM) scenario.
Recently it has been shown \cite{Morris,BGN} that  an inhomogeneous
compactification of the extra dimension of a
five-dimensional Kaluza-Klein metric has been shown to generate a PDM in
the corresponding four-dimensional system. But this method yields a very
special class of PDM Hamiltonian
which directly related to the solution of the Lioville equation.
In a related project Cari\~nena et al \cite{CRS} formulated a method that
starts with the study of the existence of Killing vector fields for the PDM
geodesic motion and the construction of the associated
Noether momenta.

There is also an alternative mechanism in which this dependance on a mass
function manifests itself  in the context of differential systems, namely
through Jacobi's last multiplier (JLM). The JLM originally arose in the
problem of reducing a system of first-order ordinary differential equations
to quadrature and has a long and chequered history. In recent years its role
in the context of the inverse problem of dynamical systems has led to a revival
 of interest in the JLM. In this brief note we examine the connection between
the JLM and the principle of least action within the framework of a Li\'enard
type differential equation with a quadratic dependance on the velocity.

It is known that the Li\'enard type equation is connected to the Painlev\'e-Gambier
equations \cite{CGK,GGCG}. So it is natural for us to ask whether we can
reformulate the subclass of the Painlev\'e-Gambier family as geodesic equations
for a Riemannian metric using the Jacobi-Maupertuis principle.
There are several choices for a Riemannian manifold and metric tensor: a
space-time configuration manifold and the Eisenhart metric
(for example, \cite{cgg,cgg1,Gibbons,SHS},
a configuration manifold and the Jacobi-Maupertuis metric \cite{nom,Pettini}.
 In this paper we choose a configuration space of an analyzed system for a
Riemannian manifold. The crux of the matter is that the Hamiltonian or energy
function provided by the JLM should remain constant for these equations.
In \cite{Casetta},  geometrical theory for the mechanics of a
position-dependent mass particle is developed using proper generalization of
Euler-Maupertuis' theory
and generalized Jacobi's principle. In this paper we generalize this to
Li\'enard type equation and show that these are equivalent
to the geodesic equations for the JM metric. We illustrate our construction
 with some interesting examples.

\subsection*{Main Result}
Let $\mathcal{V}$ be a Hamiltonian vector field of
the Li\'enard type equation $\ddot{x} + f(x)\dot{x}^{2} + g(x) = 0$ in
$\mathbb{R}^2$ with Hamiltonian
$H = \frac{1}{2}M(x)\dot{x}^{2} + U(x)$, where $M(x) = exp(2\int^x f(s)ds)$ and
$U(x) = \int^x M(s)g(s)ds$. Then by Maupertuis principle,  $\mathcal{V}$
coincides with the trajectories of the modified vector field
$\mathcal{V}^{\prime}$ on the fixed isoenergy level
$H(x,\dot{x}) = E$ for the Hamiltonian
${\tilde H} = \frac{1}{2(E - U(x))}M(x)\dot{x}^{2}$.
This defines a geodesic flow of some Riemannian metric given by Jacobi.
In other words, solutions to the Li\'enard type equation with energy $E$ are,
after reparametrization, geodesics for the Jacobi-Maupertuis metric.


A corollary of the main result shows that we can reformulate the Newtonian equation
of motion for a variable mass, Painlev\'e-Gambier XXI equation,
the Jacobi equation and Henon-Heiles system
in terms of geodesic flows of the Jacobi-Maupertuis metric.

The outline of this article is as follows: in section 2 we introduce
 the Jacobi Last Multiplier and point out its connection to the Lagrangian
of a second-order ODE. Thereafter we explicitly derive the Lagrangian and
the Hamiltonian functions for a Li\'enard equation of the second kind,
i.e., with a quadratic dependance on the velocity and highlight the role of
the position dependant mass term. In section 3 we express the equation in
terms of geodesic flows of the Jacobi-Maupertuis metric and some observations
regarding the geometric  consequences of the PDM are outlined.
Explicit examples from the Painlev\'e-Gambier family of equations are
considered along with the two-dimensional Henon-Heiles system.

\section{Lagrangians and the Jacobi last multiplier}

 Let $M=M(x^1,\dots ,x^n)$ be a non-negative $C^1$ function
 non-identically vanishing on any open subset of $\mathbb{R}^n$, then $M$
 is a Jacobi multiplier of the vector field
 $\mathbb{X}=W^i\frac{\partial}{\partial
 x^i}$ if
\begin{equation}
\int_DM(x^1,\dots ,x^n)dx^1\dots dx^n=\int_{\phi_t(D)}
 M(x^1,\dots ,x^n)dx^1\dots dx^n
\end{equation}
where $D$ is any open subset of $\mathbb{R}^n$ and $\phi_t(\cdot)$ is the flow
generated by the solution $\mathbf{x}= \mathbf{x}(t)$ of the system
\begin{equation}
\frac{dx^i}{dt} = W^i(x^1,\dots ,x^n) \quad i=1,\dots ,n.
\end{equation}
 Thus the Jacobi multiplier can be viewed as the density
 associated with the invariant measure $\int_D M dx$.
The divergence free condition is
\begin{equation}
\frac{dM}{dt}+\frac{\partial W^i}{\partial x^i} M=0.
\end{equation}
The appellation `last' is a  historical legacy.
 If a Jacobi multiplier is known together with $(n-2)$ first
 integrals, we can reduce locally the $n$ dimensional system to a
 two-dimensional vector field on the intersection of the $(n-2)$
 level sets formed by the first integrals. The existence of a Jacobi
 Last Multiplier \cite{CG} then implies  the existence of an extra first
 integral and the system may therefore be reduced to quadrature.

For the second-order ODE
\begin{equation}\label{eom}
\ddot{x} = F(x,\dot{x},t) \quad \Rightarrow \quad
 \dot{x} = y, \; \dot{y} = F(x,y,t)
\end{equation}
we have
\begin{equation}\label{dfree1}
\frac{dM}{dt} + \frac{\partial F}{\partial y} M = 0.
\end{equation}
On the other hand by expanding the Euler-Lagrange equation of motion
\begin{equation}\label{EL1}
\frac{\partial L}{\partial x} - \frac{d}{dt}
\Big(\frac{\partial L}{\partial \dot x}\Big) = 0,
\end{equation}
we have
$$
\frac{\partial L}{\partial x}
= {\dot y} \Big(\frac{\partial^2 L}{\partial \dot x^2}\Big)
 + {\dot x} \frac{\partial }{\partial x} \Big(\frac{\partial L}{\partial \dot x}\Big)
 = {\dot y} \Big(\frac{\partial^2 L}{\partial \dot x^2}\Big)
 + y \frac{\partial }{\partial \dot x} \Big(\frac{\partial L}{\partial x}\Big).
$$
 Differentiating it with respect to, $\dot{x}=y$, gives
\[
\frac{\partial \ }{\partial \dot x}
\Big( \frac{\partial L}{\partial x} \Big)
= \frac{\partial \dot y}{\partial y}
 \Big(\frac{\partial^2 L}{\partial \dot x^2}\Big)
 + \dot y \Big(\frac{\partial^3 L}{\partial \dot x^3}\Big)
 + \frac{\partial }{\partial \dot x} \Big(\frac{\partial L}{\partial x}\Big)
 + y \frac{\partial^2 \ }{\partial \dot x^2}
  \Big(\frac{\partial L}{\partial x}\Big),
\]
implies
\[
 \frac{\partial F}{\partial y} \Big(\frac{\partial^2 L}{\partial \dot x^2}\Big)
+ \Big[ \dot y \frac{\partial \ }{\partial \dot x}
\Big( \frac{\partial^2 L}{\partial \dot x^2} \Big)
 + y \frac{\partial \ }{\partial x}
 \Big(  \frac{\partial^2 L}{\partial \dot x^2} \Big) \Big] = 0.
\]
Therefore
\begin{equation}\label{dfree2}
\frac{d}{dt}\Big(\frac{\partial ^2 L}{\partial \dot{x}^2}\Big)
+ \Big(\frac{\partial F}{\partial y}\Big)
\Big(\frac{\partial ^2 L}{\partial \dot{x}^2}\Big) = 0.
\end{equation}
Thus, by comparing \eqref{dfree2} to \eqref{dfree1}, we may identify the
JLM as  follows:
\begin{equation}\label{density}
M=\frac{\partial ^2 L}{\partial \dot{x}^2}.
\end{equation}
Given a JLM we can easily integrate \eqref{density} twice to obtain
\begin{equation}\label{lagsol}
L(x,\dot{x},t) = \int^{\dot{x}} \Big(\int^y M dz\Big) dy +R(x,t) \dot{x}+S(x,t).
\end{equation}
where $R$ and $S$ are functions arising from integration.
 To determine these functions we substitute the
 Lagrangian  of \eqref{lagsol} into the Euler-Lagrange equation of motion
\eqref{EL1}  and
 compare the resulting equation with the given ODE  \eqref{eom}.

Consider  now a Li\'enard equation of the second kind,
\begin{equation} \label{lien}
\ddot{x}+f(x)\dot{x}^2 +g(x)=0,
\end{equation}
where $f$ and $g$ are defined in a neighbourhood of $0 \in \mathbb{R}$.
 We assume that $g(0) = 0$, which says that
$O$ is a critical point, and $x g(x) > 0$ in a punctured neighbourhood
of $0 \in \mathbb{R}$, which ensures that the origin
is a centre.

\begin{proposition}\label{P1}
A Li\'enard equation of the second kind, $\ddot{x}+f(x)\dot{x}^2 +g(x)=0$,
admits a Hamiltonian of the form $H=1/2 M(x)\dot{x}^2 +U(x)$ which is a
constant of motion where $M(x)$ is the Jacobi last multiplier and $U(x)$
is a potential function.
\end{proposition}

\begin{proof}
 From the definition  \eqref{dfree1} of the last multiplier
it follows that for the equation under consideration
\begin{equation}
M(x)=\exp(2F(x)) \quad \text{where }  F(x)=\int^x f(s) ds.
\end{equation}
Consequently by \eqref{lagsol}, we have
\begin{equation}
L = \frac12 M(x) \dot{x}^2 + R(x,t) \dot{x} + S(x,t).
\end{equation}
From the Euler-Lagrange equation one finds that the functions
   $R$ and $S$ must satisfy
   $$
S_x - R_t = - M(x)g(x)
$$
This gives us the freedom to set
   $S=G_t - U(x)$ and $R=G_x$ for some gauge function $G$,
so that there exists a potential function $U(x)$ given by
\begin{equation}
U(x) = \int^x M(s) g(s) ds.
\end{equation}
The Lagrangian then has the form
\begin{equation}\label{lienlag}
L=\frac12 M(x) \dot{x}^2 - U(x) + \frac{dG}{dt}.
\end{equation}
Clearly the total derivative term may be ignored and by means of the
standard Legendre transformation the Hamiltonian is given by
\begin{equation}\label{ham}
H = \frac12 M(x) \dot{x}^2 + U(x).
\end{equation}
It is now straight forward to verify that $dH/dt=0$ so that $H=E$(say)
is a constant of motion. This completes the proof.
\end{proof}

From \eqref{ham} it is evident  that the JLM, $M(x)$, plays the role of a variable
mass term. We can reduce the differential system to a unit mass problem
by defining a transformation $x\longrightarrow X=\int_0^x \sqrt{M(s)}ds=\psi(x)$
whence
\begin{equation}
\frac12 \dot{X}^2 + \int_0^{\psi^{-1}(X)}M(s)g(s) ds = E.
\end{equation}
In terms of $X$ the equation of motion is given by
\begin{equation}
\ddot{X} + e^{F(\psi^{-1}(X))} g(\psi^{-1}(X)) = 0.
\end{equation}
We  now proceed to cover some fundamentals regarding the Jacobi metric,
and deduce it for the Li\'enard equation. We mainly follow the Nair et al
  formalism of Jacobi-Maupertuis principle \cite{nom} and
elaborate on it in the next section.

\section{Jacobi-Maupertuis metric and Li\'enard type equation}

When the Hamiltonian is not explicitly time dependent, i.e.,
 $H=E_0$, a constant, then the solutions may be restricted to the energy surface
$E=E_0$. Suppose $Q$ is a manifold with local coordinates $x=\{x^i\}, i=1,\dots ,n$
and $x(\tau) \in Q\subseteq \mathbb{R}^n$ be a curve with $\tau\in[0, T]$.
Let $T_xQ$ and $T_x^*Q$ be the tangent and cotangent spaces with velocity
$\dot{x}(\tau)\in T_xQ\subseteq\mathbb{R}^n$ and momenta $p(\tau)\in T_x^*Q\subseteq\mathbb{R}^n$.
Denote by $\gamma$ a curve in the manifold $Q$ parametrized by $t\in[a, b]$ with
$\gamma(a)=x_0$ and $\gamma(b)=x_N$. The according to the Maupertuis principle
among all the curves $x(t)$ connecting the two points $x_0$ and $x_n$ parametrized
such that $H(x, p)=E_0$ the trajectory of the Hamiltons equation of motion is an
extremal of the integral of action
\begin{equation}\label{x0}
\int_\gamma p dx=\int_\gamma p \dot{x} dt
=\int_\gamma \frac{\partial L(t)}{\partial\dot{x}} \dot{x}(t) dt.
\end{equation}
Here $L$ is assumed to be a regular Lagrangian $L:TQ\rightarrow \mathbb{R}$ where
$L=K-U$ and the kinetic energy $K:TQ\rightarrow \mathbb{R}$.

\begin{proposition} \label{prop3.1}
Let the Hamiltonian $H=K+U$ be a constant of motion i.e., $H=E$ (say) with
the kinetic energy $K$ being a homogeneous quadratic function of the
velocities and $U(x)$ is some potential function such that $U(x)<E$: then
 there exists a Riemannian metric defined by $d\widetilde{s}=\sqrt{2(E-U(x))}ds$
with $K=1/2(ds/dt)^2$ such that the trajectories are the geodesic equations
corresponding to the Jacobi-Maupertuis principle of least action.
\end{proposition}

\begin{proof}
 Let $ds^2$ be a Riemannian metric on the configuration space with kinetic energy
\begin{equation} \label{x1}
K=\frac 12 g_{ij}(x) \dot{x}^i \dot{x}^j=\frac 12\left(\frac{ds}{dt}\right)^2.
\end{equation}
As the total energy is a constant $E$ with potential $U(x)<E$ the Hamiltonian
satisfies $H=K+U=E$. Because $K$ is a homogeneous quadratic function hence
Euler theorem implies $ 2K = \dot{x}^i {\partial L}/{\partial \dot{x}^i}=(ds/dt)^2$.
Therefore from \eqref{x0} we have
\begin{align*}
\int_\gamma \frac{\partial L(t)}{\partial\dot{x}} \dot{x}(t) dt
&=\int_\gamma 2K dt=\int_\gamma 2K\frac{ds}{\sqrt{2K}}
=\int_\gamma \sqrt{2K} ds \\
&=\int_\gamma \sqrt{2(E-U(x))}ds
=\int_\gamma d\widetilde s,
\end{align*}
where the Riemannian metric $\widetilde s$ is defined by
$d\widetilde s=\sqrt{2(E-U(x))}ds$.
This shows that it is possible to derive a metric which is given by the
kinetic energy itself \cite{cgg} and the trajectories are geodesics in
the metric $d\widetilde s$. From \eqref{x1} one finds
$ds=\sqrt{q_{ij} dx^idx^j}$ and the Maupertuis principle involves solving
for the stationary points of the action $\int \sqrt{2K} ds$, i.e.,
\begin{equation} \label{x3}
\delta\int \sqrt{2K} ds=0\quad \text{or}\quad
\delta\int \sqrt{2(E-U(x))g_{ij} dx^idx^j}=0,
\end{equation}
with the integral being over the generalized coordinates $\{x^i\}$ along all
paths connecting $\gamma(a)$ and $\gamma(b)$.

It is evident from $d\widetilde s=\sqrt{2(E-U(x))g_{ij} dx^idx^j}$ that
\begin{equation}\label{x4}
d{\widetilde s}^2=\widetilde{g}_{ij} dx^idx^j\quad\text{ where }
 \widetilde{g}_{ij}(x)=2(E-U(x))g_{ij}(x).
\end{equation}
The geodesic equation corresponding to the least action
$\delta \int_{s_1}^{s_2} dt \sqrt{\widetilde{g}_{ij} \dot x^i \dot x^j} = 0$
is given by
\begin{equation} \label{x5}
\frac{d^2x^i}{d \widetilde s^2}
 + \Gamma^i_{jk}\frac{dx^j}{d \widetilde s}\frac{dx^j}{d \widetilde s} = 0,
\quad \text{where }
 \Gamma^i_{jk} = \frac12 \widetilde{g}^{il}
\Big(\frac{\partial \widetilde{g}_{jl}}{\partial x^k}
 + \frac{\partial \widetilde{g}_{kl}}{\partial x^j}
- \frac{\partial \widetilde{g}_{jk}}{\partial x^l}\Big).
\end{equation}
This complete the proof.
\end{proof}

 For an equation of the Li\'enard type \eqref{lien},
from Proposition \eqref{P1} we have
$$
K=\frac{1}{2}M(x) \dot{x}^2\quad \text{where }
M(x)=\exp(2F(x))
$$
so that $g_{11}(x)=M(x)$ while from the
Jacobi-Maupertuis (JM) metric  \eqref{x4} we observe that
$\widetilde{g}_{11}=2(E-U(x))M(x)$. The geodesic equation \eqref{x5}
therefore reduces to
$$
\frac{d^2x}{d \widetilde s^2} + \Gamma^1_{11}\Big(\frac{dx}{d\widetilde s}\Big)^2
=0\quad \text{with }\Gamma^1_{11}=\frac{M'(x)}{2M(x)}
-\frac{U'(x)}{2(E-U(x))},
$$
or in explicit terms
\begin{equation} \label{x6}
\frac{d^2x}{d \widetilde s^2}+\Big(\frac{M'(x)}{2M(x)}
-\frac{U'(x)}{2(E-U(x))}\Big)
\Big(\frac{dx}{d\widetilde s}\Big)^2=0.
\end{equation}
Equation \eqref{x6} gives the geodesic for the JM-metric of a Li\'enard
equation of the type \eqref{lien}.

\begin{proposition} \label{prop3.2}
 The geodesic equation \eqref{x6} and \eqref{lien} are equivalent.
\end{proposition}


\begin{proof}
 From $K=E-U(x)=1/2 M(x)\dot{x}^2$  we have
\begin{equation} \label{x6a}
\dot{x}^2=2(E-U(x))/M(x)
\end{equation}
 and as $d\widetilde{s}^2=\widetilde{g}_{11} dx^2=2((E-U(x))M(x) dx^2$,
 it follows that
\begin{equation} \label{x7}
\frac{d\widetilde{s}}{dt}=2(E-U(x))\;\Rightarrow\;
\frac{dx}{dt}=2(E-U(x))\frac{dx}{d\widetilde{s}}.
\end{equation}
This enables us to obtain
\begin{equation}
\begin{aligned}
\frac{d^2 x}{d\widetilde{s}^2}
&=\frac{1}{2(E-U(x))}\frac{d}{dt}\Big\{\frac{1}{2(E-U(x))}
\frac{dx}{dt}\Big\} \\
&=\frac{1}{4(E-U(x))^2}\Big[\frac{d^2 x}{dt^2}+\frac{U'(x)}{(E-U(x))}
 \dot{x}^2\Big]
\end{aligned}
\end{equation}
Consequently \eqref{x6}, taking \eqref{x6a} into account, assumes the form
$$
\frac{1}{4(E-U(x))^2}\Big[\frac{d^2 x}{dt^2}+\frac{U'(x)}
{(E-U(x))}\dot{x}^2\Big]
= \Big[\frac{U'(x)}{2(E-U(x))}-\frac{M'(x)}{2M(x)}\Big]
\frac{1}{4(E-U(x))^2}\dot{x}^2;
$$
 in other words we have
 \begin{equation} \label{x8}
\frac{d^2 x}{dt^2}+\frac{M'(x)}{2M(x)}\dot{x}^2
+\frac{U'(x)}{2(E-U(x))}\dot{x}^2=0.
\end{equation}
However as $\dot{x}^2=2(E-U(x))/M(x)$ the last term of the above equation
can be expressed as $U'(x)/M(x)$ and as a result the equation has the appearance
\begin{equation} \label{x9}
\frac{d^2 x}{dt^2}+\frac{M'(x)}{2M(x)}\dot{x}^2 +\frac{U'(x)}{M(x)}=0.
\end{equation}
This equation reduces to \eqref{lien} upon making the identifications
$M(x)=\exp(2F(x))$ which implies $M'(x)/2M(x)=f(x)$ and
$U(x)=\int^x M(y) g(y) dy$ which implies $U'(x)/M(x)=g(x)$ where $g(x)$
refers to the forcing term of the Li\'enard equation \eqref{lien}.
\end{proof}

\begin{remark} \rm
Finally it is interesting to note how \eqref{lien} or equivalently \eqref{x9}
may be viewed geometrically. To this end we write \eqref{x9}  as
\begin{equation} \label{x10}
\frac{d^2 x}{dt^2}+\frac{M'(x)}{2M(x)}\dot{x}^2 =-\frac{U'(x)}{M(x)}
\end{equation}
and look upon the right hand side as an external force function.
Restricting ourselves to the left hand side we consider a 1+1 dimensional
line element of the form $ds^2=c^2dt^2-M(x)dx^2=c^2d\tau^2$ which
yields the following geodesic equations for a free particle moving in this
spacetime, namely
$$
\frac{d^2x}{d\tau^2}+\frac{M'(x)}{2M(x)}
\Big(\frac{dx}{d\tau}\Big)^2=0, \quad \frac{d^2t}{d\tau^2}=0.
$$
These equations imply upon elimination of the proper time $\tau$ the
left hand side of \eqref{x10}and the latter may be recast as
$$
\frac{d}{dt}\left(M(x)\dot{x}\right)=\frac{M'(x)}{2}\dot{x}^2.
$$
Thus from a Newtonian perspective we see that the position dependent mass
function $M(x)$ changes the geometry of spacetime in a manner such that the
particle experiences an additional geometric force $f_G=M'(x)\dot{x}^2/2$. However unlike the case when the PDM is also a function of time \cite{MH} the curvature of spacetime is flat because as a result of the  transformation $dX=\sqrt{M(x)}dx$ one has $ds^2=c^2dt^2-dX^2$  and the resulting geodesic equation of a free particle in this transformed spacetime is just $\frac{d^2X}{dt^2}=0$ or
$$
\frac{d}{dt}\Big(\sqrt{M(x)}\frac{dx}{dt}\Big)=0, \quad \text{or}\quad
\frac{1}{2}M(x)\dot{x}^2=const.
$$
which implies the conservation of the kinetic energy.
\end{remark}

To complete this article we illustrate our results with a few examples.

\begin{example} \label{examp1} \rm
We consider the Painl\'eve-Gambier XXI
$$
\ddot{x}-\frac{3}{4x}\dot{x}^2-3x^2=0,
$$
for which we have $F(x)=-3/4\int dx/x=-3/4\log|x|$ so that
$M(x)=|x|^{-3/2}$; and as $2K=M(x)\dot{x}^2=g_{11}(x)\dot{x}^2$ we have
 $g_{11}(x)=M(x)=|x|^{-3/2}$ while $U(x)=\int^x M(z)g(z) dz=\mp 2x^{3/2}$
depending on whether $x>0$ or $x<0$. As a result we find
$\widetilde{g}_{11}=2(E\pm 2x^{3/2})|x|^{-3/2}$.
\end{example}

\begin{example} \label{examp2}\rm
We consider the Jacobi equation
 $$
\ddot{x}+\frac{1}{2}\phi_x \dot{x}^2+\phi_t\dot{x}+B(t, x)=0,
$$
for which we have  $M(x,t)=\exp(\phi(x,t))=g_{11}$ and the Lagrangian
$$
L=\frac{1}{2}e^\phi \dot{x}^2-U(x,t), \quad \text{where}\quad
U(x,t)=\int^x e^{\phi(y,t)}B(y,t) dy\,.
$$
It canb be verified that the Hamiltonian is a constant of motion and
$\widetilde{g}_{11}=2(E-U(x,t))\exp(\phi(x,t))$.
The geodesic equation is
$$
\frac{d^2x}{d\widetilde{s}^2}+\Gamma^1_{11}\Big(\frac{dx}{d\widetilde{s}}\Big)^2=0,
\quad\text{with }
\Gamma^1_{11}=\frac{\phi_x}{2}-\frac{U_x}{2(E-U(x,t))}.
$$
\end{example}

\begin{example} \label{examp3} \rm
We consider the Henon-Heiles system
\begin{gather*}
\ddot{x}=-(Ax+2\alpha xy), \\
\ddot{y}=-(By+\alpha x^2-\beta y^2)
\end{gather*}
which has has the Lagrangian
$$
L(x,y, \dot{x}, \dot{y})=\frac{1}{2}(\dot{x}^2+\dot{y}^2)
-\Big(\frac{A}{2}x^2+\frac{B}{2}y^2+\alpha x^2 y-\frac{\beta}{3}y^3\Big)\,.
$$
It is therefore easily seen that $M_{xx}=M_{yy}=1$ and it admits the
first integral
$$
I=\frac{1}{2}(\dot{x}^2+\dot{y}^2)+\Big(\frac{A}{2}x^2
+\frac{B}{2}y^2+\alpha x^2 y-\frac{\beta}{3}y^3\Big),
$$
which is just the Hamiltonian of the system. Consequently we have
$g_{11}=M_{xx}=1$ and $g_{22}=M_{yy}=1$ while
\[
\widetilde{g}_{11}=2(E-U(x,y))
=\widetilde{g}_{22},
\]
where
\[
U(x,y)=\frac{1}{2}(\dot{x}^2+\dot{y}^2)
-\Big(\frac{A}{2}x^2+\frac{B}{2}y^2+\alpha x^2 y-\frac{\beta}{3}y^3\Big)
\]
The geodesic equations have the form
\begin{gather*}
\frac{d^2x}{d\widetilde{s}^2}-\frac{1}{2(E-U(x,y))}
\Big(U_x\Big(\frac{dx}{d\widetilde{s}}\Big)^2
+2U_y\Big(\frac{dx}{d\widetilde{s}}\Big)\Big(\frac{dy}{d\widetilde{s}}\Big)
+ U_x\Big(\frac{dy}{d\widetilde{s}}\Big)^2\Big)=0, \\
\frac{d^2y}{d\widetilde{s}^2}-\frac{1}{2(E-U(x,y))}
 \Big(U_y\Big(\frac{dx}{d\widetilde{s}}\Big)^2
+2U_x\Big(\frac{dx}{d\widetilde{s}}\Big)\Big(\frac{dy}{d\widetilde{s}}\Big)
+ U_y\Big(\frac{dy}{d\widetilde{s}}\Big)^2\Big)=0
\end{gather*}
\end{example}

\subsection*{Conclusion}

In this article, we studied the so called Li\'enard type equations, such equations
naturally appear in physical system such as position dependent mass particles,
 and show these are equivalent to the
geodesic equations for the Jacobi-Maupertuis (JM) metric.
We have illustrated our construction with some explicit examples including The
Painlev\'e-Gambier XXI, The Jacobi and the Henon-Heiles system of equations.

\subsection*{Acknowledgement}
We are grateful to Professor Jaume Llibre for his valuable comments.
The research of PG is supported by FAPESP through
Instituto de Fisica de S\~ao Carlos, Universidade de Sao Paulo with
grant number 2016/06560-6.



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\end{document}