#  A Maupertuis-type principle in relativistic mechanics and applications
††thanks: Work written under the auspices of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM). The third author has been also
supported by the Ministry of Science, Technology and Universities of Spain,
under Research Grant PGC2018-097104-B-100, and by contract CT42/18-CT43/18 of
Complutense University of Madrid.

Alberto Boscaggin
Università degli Studi di Torino
Dipartimento di Matematica “Giuseppe Peano”
Via Carlo Alberto 10, 10123 Torino, Italy.
E-mail<EMAIL_ADDRESS>
Walter Dambrosio
Università degli Studi di Torino
Dipartimento di Matematica “Giuseppe Peano”
Via Carlo Alberto 10, 10123 Torino, Italy.
E-mail<EMAIL_ADDRESS>
Eduardo Muñoz-Hernández
Universidad Complutense de Madrid
Instituto de Matemática Interdisciplinar (IMI)
Departamento de Análisis Matemático y Matemática Aplicada
Plaza de las Ciencias 3, 28040 Madrid, Spain
E-mail<EMAIL_ADDRESS>
###### Abstract

We provide a Maupertuis-type principle for the following system of ODE, of
interest in special relativity:

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=\nabla
V(x),\qquad x\in\Omega\subset\mathbb{R}^{n},$

where $m,c>0$ and $V:\Omega\to\mathbb{R}$ is a function of class $C^{1}$. As
an application, we prove the existence of multiple periodic solutions with
prescribed energy for a relativistic $N$-centre type problem in the plane.

2010 Mathematics Subject Classification: 37J45, 49S05, 70H40.

Keywords and Phrases. Maupertuis principle, energy functional, special
relativity, periodic solutions, singular potentials.

##  1 Introduction

The well-known Maupertuis principle of classical mechanics (see, for instance,
[4, page 247] or [12, equation (8.89)]) states that solutions of the system of
ODEs

$\ddot{x}=\nabla V(x),\qquad x\in\Omega\subset\mathbb{R}^{n},$ (1.1)

lying on a fixed energy level

$\frac{1}{2}|\dot{x}(t)|^{2}-V(x(t))\equiv h\in\mathbb{R},$

correspond, up to a time-reparameterization, to critical points $q$ of the
energy functional

$\int_{0}^{1}|\dot{q}(\sigma)|^{2}(V(q(\sigma))+h)\,d\sigma,$

that is, to geodesic curves on the Hill’s region
$\\{q\in\Omega\,:\,V(q)+h>0\\}$, when endowed with the Riemannian metric
$g_{ij}(q)=(V(q)+h)\delta_{ij}$ (the so-called Jacobi metrix; here
$\delta_{ij}$ is the usual Kronecker delta).

Taking advantage of this variational principle, in the last decades several
existence and multiplicity results for solutions with _prescribed energy_ of
equation (1.1) have been proved by the use of the methods of critical point
theory. In particular, due to its mechanics relevance, cases when the
potential $V$ is singular have been the object of extensive investigations: we
refer the reader to the book [1] for a recap of most of the results obtained
until the early nineties, and to [8, 10, 11] and the references therein for
some more recent contributions.

In this paper, we look for a corresponding variational principle for the
equation

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=\nabla
V(x),\qquad x\in\Omega\subset\mathbb{R}^{n},$ (1.2)

where $m,c$ are positive constants: here, according to the underlying Lagrange
structure (see the beginning of Section 2 for more details), preservation of
energy means that

$mc^{2}\left(\frac{1}{\sqrt{1-|\dot{x}(t)|^{2}/c^{2}}}-1\right)-V(x(t))\equiv
h.$ (1.3)

As well known (see, for instance, [12, Chapter 7] or [14, Chapter 33]),
equation (1.2) finds its motivation from special relativity, providing an
approximate model for the motion of a particle $x$ of mass $m$, moving at
speed close to that of light, denoted here111Even if in the mathematical
literature it is common to use the normalization assumption $m=c=1$, as in [6,
7]we find useful to keep track of both these constants, in order to have a
clearer physical interpretation of the results obtained, as well as to provide
some considerations about the non-relativistic limit $c\to+\infty$. by $c$.
Just to mention some few references in this direction, we refer to the papers
[2, 9, 18], dealing with equation (1.2) in the case when $V$ is a Kepler or
Coulomb potential.

From a genuinely mathematical point of view, equation (1.2) has been widely
investigated in the last decades, even in the more general case when the
right-hand side depends explicitly on time, and possibly does not have a
gradient-type structure. Nowadays, an enormous amount of existence and
multiplicity results, obtained with both variational and topological methods,
is available for the associated boundary value problems: we refer to [16] for
a ten-year-old survey on the topic, and to [3, 7] for some recent papers. In
spite of this extensive bibliography, however, to the best of our knowledge
there are no general results dealing with solutions of equation (1.2) having
prescribed energy $h\in\mathbb{R}$ and, in fact, the underlying Maupertuis
principle seems to be missing in the literature.

Our paper aims at filling this gap and is organized as follows. In Section 2
we provide the relativistic Maupertuis-type principle, cf. Proposition 2.1.
Roughly, it states the following: solutions to equation (1.2) having
prescribed energy (1.3) correspond, up to a time-reparameterization, to
critical points $q$ of the functional

$\mathcal{E}_{h}(q)=\int_{0}^{1}|\dot{q}(\sigma)|^{2}(Z_{h}(q(\sigma))+2hm)\,d\sigma,$
(1.4)

where

$Z_{h}(q)=2mV(q)+\dfrac{1}{c^{2}}\,(V(q)+h)^{2}.$ (1.5)

Thus, solutions of (1.2) are again geodesics on the Hill’s region
$\\{q\in\Omega\,:\,V(q)+h>0\\}$, but relativistic effects result into a
perturbation of the usual Jacobi metric of classical mechanics: indeed, notice
that in the non-relativistic limit $c\to+\infty$ the relativistic metric
$g_{ij}(q)=(Z_{h}(q)+2hm)\delta_{ij}$ reduces, up to a constant factor, to the
usual one. We emphasize that, in the functional (1.4), no bounds on the
modulus of $\dot{q}$ are needed. At first glance, this might seem quite
surprising, since equation (1.2) implicitly requires $|\dot{x}|<c$; however,
it is coherent with the fact that the fixed energy problem is of purely
geometrical nature: the kinematic property $|\dot{x}|<c$, indeed, will be
automatically obtained by the time-reparameterization.

At the end of Section 2, an equivalent formulation of the variational
principle is given in Proposition 2.2, using instead of $\mathcal{E}_{h}$ the
so-called Maupertuis functional

$\mathcal{M}_{h}(q)=\int_{0}^{1}|\dot{q}(\sigma)|^{2}\,d\sigma\int_{0}^{1}(Z_{h}(q(\sigma))+2hm)\,d\sigma,$

which, as well known, is more manageable when using critical point theory, cf.
[1, Remark 4.2].

As an application of the Maupertuis principle, in Section 3 we investigate the
existence of periodic solutions, with prescribed energy, for the equation

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=-\sum_{i=1}^{N}\frac{\kappa_{i}(x-\sigma_{i})}{|x-\sigma_{i}|^{\alpha+2}}+\nabla
W(x),\qquad x\in\mathbb{R}^{2},$ (1.6)

where $\sigma_{1},\ldots,\sigma_{N}$ are distinct points of $\mathbb{R}^{2}$,
$\kappa_{i}>0$, $\alpha>1$ and $W$ is meant as a perturbation term as
$|x|\to+\infty$. Such an equation can be interpreted as a (perturbed)
relativistic $N$-centre problem, providing the motion of a charge $x$ in the
electric field generated by $N$-fixed charges $\sigma_{1},\ldots,\sigma_{N}$,
cf. [3]. By minimization arguments for the associated energy functional, and
taking advantage of the topology of the punctured plane (see [11] for a
somehow related result in the non-relativistic setting), we prove the
existence of multiple periodic solutions of equation (1.6), with prescribed
homotopy class and prescribed energy (1.3), for any $h>0$ arbitrarily fixed,
cf. Theorem 3.1 for the precise statement. We point out that, for the whole
argument, the restriction $\alpha>1$ is needed: this has to be interpreted as
a relativistic version of the usual strong-force condition of classical
mechanics, see [13]. Notice however that, while in the classical case one is
led to assume $\alpha\geq 2$, here a weaker assumption is enough: this comes
from the fact that the potential $Z_{h}$ given in (1.5) behaves, near the
singularity, as the square of the potential $V$. There is however a subtle
point, since the physically relevant case $\alpha=1$, which would give rise to
a potential $Z_{h}$ of order $2$ near the singularity, cannot be treated
within our approach: indeed, as explained in Remark 3.3, a further condition
for $Z_{h}$ is needed, requiring eventually $\alpha>1$. At first sight,
Theorem 3.1 thus seems to be in sharp contrast with the results obtained in
[7] for the fixed period problem: as a matter of fact, however, the difficulty
comes from the specific structure of the Maupertuis functional, which,
compared with the action functional, requires an independent control of the
kinetic and potential parts, cf. [1, Remark 5.4]. We stress that this
difficulty is not only of technical nature: indeed, as pointed out in Remark
4.2, in the case $\alpha=1$ a different range for the admissible energies is
expected, giving rise to a different geometry for the associated functional.
Of course, due do its mechanical significance, the case $\alpha=1$ deserves
further investigation in the future.

Finally, in Section 4 we use phase-plane analysis (for the model problem (1.6)
for $N=1$ and $W\equiv 0$) to investigate the optimality of the condition
$h>0$ appearing in Theorem 3.1, finally proving that it is sharp when
$\alpha\geq 2$.

##  2 A variational principle for prescribed energy relativistic problems

Let us consider the system of ODEs

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=\nabla
V(x),\qquad x\in\Omega,$ (2.1)

where $m,c>0$, $\Omega\subset\mathbb{R}^{n}$ is an open set and
$V:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ is a function of class $C^{1}$.
This equation can be meant as the Euler-Lagrange equation

$\frac{\rm d}{{\rm d}t}\left(\frac{\partial
L}{\partial\dot{x}}\right)=\frac{\partial L}{\partial x}$

for the Lagrangian

$L(x,\dot{x})=-mc^{2}\sqrt{1-\dfrac{|\dot{x}|^{2}}{c^{2}}}+V(x).$

We can then pass to an equivalent Hamiltonian formulation defining the dual
variable

$p=\frac{m\dot{x}}{\sqrt{1-\dfrac{|\dot{x}|^{2}}{c^{2}}}};$ (2.2)

accordingly, by Legendre transformation, the associated Hamiltonian function
turns out to be

$H(x,p)=mc^{2}\sqrt{1+\frac{|p|^{2}}{m^{2}c^{2}}}-V(x).$ (2.3)

As well known, the Hamiltonian $H$ is a constant of motion. The _relativistic
energy_ is nothing but the Hamiltonian $H$ written in terms of $x$ and
$\dot{x}$; using (2.2), we thus find

$E(x,\dot{x})=\frac{mc^{2}}{\sqrt{1-\dfrac{|\dot{x}|^{2}}{c^{2}}}}-V(x)$

and, so, for any solution $x$ of (2.1) there exists
$\mathfrak{E}\in\mathbb{R}$ such that

$\frac{mc^{2}}{\sqrt{1-\dfrac{|\dot{x}(t)|^{2}}{c^{2}}}}-V(x(t))\equiv\mathfrak{E}.$
(2.4)

Let us notice that (2.4) implies that, in the case of potential $V$ such that
$V(x)\to 0$ as $|x|\to+\infty$, the particle rest energy is $mc^{2}$. Thus, it
is convenient to write

$\mathfrak{E}=h+mc^{2},$

in such a way that $h$ is the difference from the relativistic energy and
particle rest energy, cf. with (1.2) of the Introduction. Incidentally, let us
notice that (see Remark 3.2 for more details) that the non-relativistic limit
of $h=\mathfrak{E}-mc^{2}$ as $c\to+\infty$ is the usual classical mechanical
energy.

Let us also observe that solutions $x$ with energy $\mathfrak{E}=h+mc^{2}$
satisfy

$V(x(t))+h>0,\quad\forall\ t,$

and so are confined in the

$\Omega_{h}=\\{x\in\Omega:\ V(x)+h>0\\},$ (2.5)

which is then the analogous of the Hill’s region in classical mechanics.

The aim of this section is to introduce a Maupertuis-type functional whose
critical points are solutions of (2.1) with prescribed energy: in order to do
this, we follow the same ideas of the non-relativistic case, relating the
action, length, energy and Maupertuis functionals.

Motivated by the application of Section 3, from now on we focus on the case of
periodic boundary conditions: it will be clear, however, that the whole
discussion could be performed with very minor changes if other boundary value
problems (like, for instance, the two-point one) are taken into account.

###  2.1 The heuristic derivation: from action to length

Let us begin our discussion with an heuristic approach leading to a length
functional starting from the action functional, without focusing on precise
statements and considerations.

First of all, let us fix $h\in\mathbb{R}$ such that

$\Omega_{h}\neq\emptyset,$ (2.6)

where $\Omega_{h}$ is defined in (2.5). For a given $T>0$, we consider the
space $W^{1,\infty}_{T}=\\{x\in W^{1,\infty}([0,T])\,:\,x(0)=x(T)\\}$ and the
set

$\Lambda_{h}=\\{x\in W^{1,\infty}_{T}:\ x(t)\in\Omega_{h},\ |\dot{x}(t)|\leq
c,\ \forall\ t\in[0,T]\\}.$

We then define the _action functional_ $\mathcal{A}_{T}$ by

$\mathcal{A}_{T}(x)=\int_{0}^{T}\left(-mc^{2}\sqrt{1-\frac{|\dot{x}(t)|^{2}}{c^{2}}}+V(x(t))+h+mc^{2}\right)\;dt,$

for every $x\in\Lambda_{h}$. As well known, its critical points (in an
appropriate sense, cf. [3, 7]) are $T$-periodic solutions of (2.1). Now, for
every increasing diffeomorphism $t:[0,1]\rightarrow[0,T]$, let us define

$q(\sigma)=x(t(\sigma)),$

for every $\sigma\in[0,1]$. Hence, we plainly deduce that the action
functional reads as

$\begin{array}[]{l}\displaystyle\int_{0}^{1}t^{\prime}(\sigma)\left(-mc^{2}\sqrt{1-\frac{|q^{\prime}(\sigma)|^{2}}{(t^{\prime}(\sigma))^{2}c^{2}}}+V(q(\sigma))+h+mc^{2}\right)\;d\sigma\,.\end{array}$

For every $\sigma\in[0,1]$, let

$b=V(q(\sigma))+h+mc^{2},\quad z=\dfrac{|q^{\prime}(\sigma)|^{2}}{c^{2}}$
(2.7)

and

$f(\zeta)=\zeta\left(-mc^{2}\sqrt{1-\frac{z}{\zeta^{2}}}+b\right),$

for every $\zeta$ with $\zeta\geq\sqrt{z}$. Let us observe that assumption
(2.6) implies that $b>mc^{2}$; hence, a simple computation shows that $f$ has
the unique minimum point

$\zeta_{0}=b\,\sqrt{\frac{z}{b^{2}-m^{2}c^{4}}}=\dfrac{V(q(\sigma))+h+mc^{2}}{\sqrt{2mV(q(\sigma))+\dfrac{1}{c^{2}}\,(V(q(\sigma))+h)^{2}+2hm}}\,\dfrac{|q^{\prime}(\sigma)|}{c^{2}},$

with

$\begin{array}[]{ll}\displaystyle
f(\zeta_{0})&\displaystyle=b\,\sqrt{\frac{z}{b^{2}-m^{2}c^{4}}}\left(-mc^{2}\sqrt{1-\frac{z}{\zeta_{0}^{2}}}+b\right)=\dfrac{{|q^{\prime}(\sigma)|}}{c}\,\sqrt{b^{2}-m^{2}c^{4}}\\\
&\\\
&\displaystyle=|q^{\prime}(\sigma)|\sqrt{2mV(q(\sigma))+\dfrac{1}{c^{2}}\,(V(q(\sigma))+h)^{2}+2hm},\end{array}$
(2.8)

by (2.7). Now, let us define

$Z_{h}(x)=2mV(x)+\dfrac{1}{c^{2}}\,(V(x)+h)^{2},\quad\forall\ x\in\Omega_{h},$
(2.9)

and observe that

$Z_{h}(x)+2hm>0,\quad\forall\ x\in\Omega_{h}.$

Accordingly, we can define the _length functional_

$\mathcal{L}_{h}(q)=\int_{0}^{1}|q^{\prime}(\sigma)|\sqrt{Z_{h}(q(\sigma))+2hm}\,d\sigma.$
(2.10)

From (2.8) we conclude that

$\begin{split}\int_{0}^{1}\left(\min_{t^{\prime}(\sigma)}f(t^{\prime}(\sigma))\right)\,d\sigma&=\mathcal{L}_{h}(q).\end{split}$
(2.11)

Relation (2.11) is reminiscent of the well-known connection between action and
length in the non-relativistic case, suggesting that solutions of equation
(2.1) can be obtained as geodesics with respect to a suitable Riemannian
metric.

###  2.2 The formal statement: energy and Maupertuis functionals

In this section we provide formal statements and proofs of the variational
principle. Since, as well known, the length functional (2.10) has the drawback
of being invariant under time-reparameterizations and is thus not suited for
the calculus of variations and min-max critical point theory, we first follow
the usual strategy of replacing the length functional with the energy
functional: the resulting variational principle is contained in Proposition
2.1. As a second step, we give an equivalent formulation of this principle,
Proposition 2.2: in this second version, the energy functional is replaced by
the so-called Maupertuis functional, an even more convenient choice for the
use of critical point theory.

Let us first fix $h\in\mathbb{R}$ such that $\Omega_{h}\neq\emptyset$ and
consider the set

$\Lambda^{\prime}_{h}=\\{q\in H^{1}([0,1]):\ q(0)=q(1),\
q(\sigma)\in\Omega_{h},\ \forall\ \sigma\in[0,1]\\}.$

The _energy functional_ for problem (2.1) is defined by

$\mathcal{E}_{h}(q)=\int_{0}^{1}|q^{\prime}(\sigma)|^{2}\left(Z_{h}(q(\sigma))+2hm\right)\,d\sigma$

for every $q\in\Lambda^{\prime}_{h}$. It is well known (see the proof of Lemma
2.1 for more details) that critical points $q\in\Lambda^{\prime}$ of
$\mathcal{E}_{h}$ fulfill the geodesic equation

$\frac{\rm d}{{\rm
d}\sigma}(q^{\prime}\,(Z_{h}(q)+2hm))=\frac{1}{2}|q^{\prime}|^{2}\nabla
Z_{h}(q)$ (2.12)

and, up to a reparameterization, are solutions of the second order equation

$z^{\prime\prime}=\nabla Z_{h}(z)$

satisfying the conservation law

$\dfrac{1}{2}|z^{\prime}|^{2}-Z_{h}(z)=2hm.$

Nevertheless, up to a different reparameterization, they also give rise to
solutions of fixed relativistic energy of (2.1). This is the main result of
this section and is the content of the next Proposition.

######  Proposition 2.1.

The following hold true.

Part A. Let $q\in\Lambda^{\prime}_{h}$ be a non-constant critical point of
$\mathcal{E}_{h}$ and define

$t(\sigma)=\int_{0}^{\sigma}\dfrac{V(q(\xi))+h+mc^{2}}{\sqrt{Z_{h}(q(\xi))+2hm}}\,\dfrac{|q^{\prime}(\xi)|}{c^{2}}\,d\xi,$
(2.13)

for every $\sigma\in[0,1]$. Let $T=t(1)$ and $\varsigma:[0,T]\to[0,1]$ be the
inverse of $t$; define

$x(t)=q(\varsigma(t)),$

for every $t\in[0,T]$. Then:

  1. 1.

$x(t)\in\Omega_{h}$ and $|\dot{x}(t)|<c$, for every $t\in[0,T]$,

  2. 2.

$x$ is a periodic solution of (2.1),

  3. 3.

$x$ satisfies

$\begin{array}[]{ll}&\displaystyle\frac{mc^{2}}{\sqrt{1-|\dot{x}(t)|^{2}/c^{2}}}-V(x(t))=h+mc^{2},\quad\forall\
t\in[0,T].\end{array}$ (2.14)

Part B. Let $x:[0,T]\to\Omega_{h}$ be a non-constant periodic solution of
(2.1), satisfying the energy law (2.14). Define

$\varpi(t)=\frac{c^{2}}{\sqrt{2\lambda}}\int_{0}^{t}\frac{Z_{h}(x(\xi))+2hm}{V(x(\xi))+h+mc^{2}}\,d\xi,$

for every $t\in[0,T]$, where $\lambda$ is such that $\varpi(T)=1$. Let
$\tau:[0,1]\to[0,T]$ be the inverse of $\varpi$ and define

$q(\sigma)=x(\tau(\sigma)),$

for every $\sigma\in[0,1]$. Then $q\in\Lambda_{h}^{\prime}$ is a non-constant
critical point of $\mathcal{E}_{h}$.

###### Proof.

Part A.

1\. From the definition of $x$ and the fact that $q\in\Lambda^{\prime}_{h}$ we
immediately deduce that $x(t)\in\Omega_{h}$, for every $t\in[0,T]$. Moreover,
from (2.13) we deduce that

$\dot{x}(t)=q^{\prime}(\varsigma(t))\,\dot{\varsigma}(t)=c^{2}\,\dfrac{q^{\prime}(\varsigma(t))}{|q^{\prime}(\varsigma(t))|}\,\dfrac{\sqrt{Z_{h}(q(\varsigma(t)))+2hm}}{V(q(\varsigma(t)))+h+mc^{2}},$
(2.15)

for every $t\in[0,T]$. Recalling (2.9), we have

$\begin{split}\dfrac{\sqrt{Z_{h}(q(\varsigma(t)))+2hm}}{V(q(\varsigma(t)))+h+mc^{2}}&=\dfrac{\sqrt{2mV(q(\varsigma(t)))+\dfrac{1}{c^{2}}\,(V(q(\varsigma(t)))+h)^{2}+2hm}}{V(q(\varsigma(t)))+h+mc^{2}}\\\
&=\dfrac{1}{c}\dfrac{\sqrt{(V(q(\varsigma(t)))+h)\,(V(q(\varsigma(t)))+h+2mc^{2})}}{V(q(\varsigma(t)))+h+mc^{2}}<\dfrac{1}{c},\end{split}$

for every $t\in[0,T]$. As a consequence, from (2.15) we infer

$|\dot{x}(t)|<c,\quad\forall\ t\in[0,T].$

2\. Let us now prove that $x$ is a periodic solution of (2.1). To this aim,
let us first observe that the periodicity of $x$ immediately follows from the
periodicity of $q$.

As already observed, if $q\in\Lambda^{\prime}_{h}$ is a critical point of
$\mathcal{E}_{h}$, then $q$ is a solution of the geodesic equation (2.12).
Moreover, a computation shows that $\frac{1}{2}|q^{\prime}|^{2}(Z_{h}(q)+2hm)$
is a first integral of (2.12). Then, there exists $\lambda>0$ such that

$\frac{1}{2}|q^{\prime}(\sigma)|^{2}(Z_{h}(q(\sigma))+2hm)=\lambda,\quad\forall\
\sigma\in[0,1].$ (2.16)

As a consequence, from (2.13) we obtain

$\begin{split}t^{\prime}(\sigma)&=\dfrac{\sqrt{2\lambda}}{c^{2}}\dfrac{V(q(\sigma))+h+mc^{2}}{Z_{h}(q(\sigma))+2hm},\end{split}$
(2.17)

for every $\sigma\in[0,1]$. On the other hand, from (2.9) and (2.15) we deduce
that

$\begin{array}[]{l}\displaystyle\sqrt{1-\dfrac{|\dot{x}(t)|^{2}}{c^{2}}}=\sqrt{1-c^{2}\dfrac{Z_{h}(q(\varsigma(t)))+2hm}{(V(q(\varsigma(t)))+h+mc^{2})^{2}}}=\frac{mc^{2}}{V(x(t))+h+mc^{2}},\end{array}$
(2.18)

for every $t\in[0,T]$. Hence, we obtain

$\begin{split}\frac{m\dot{x}(t)}{\sqrt{1-|\dot{x}(t)|^{2}/c^{2}}}&=\dfrac{V(q(\varsigma(t)))+h+mc^{2}}{c^{2}}\,q^{\prime}(\varsigma(t))\,\dot{\varsigma}(t)\\\
&=\dfrac{V(q(\varsigma(t)))+h+mc^{2}}{c^{2}}\,\dfrac{c^{2}}{\sqrt{2\lambda}}\,\dfrac{Z_{h}(q(\varsigma(t)))+2hm}{V(q(\varsigma(t)))+h+mc^{2}}\,q^{\prime}(\varsigma(t))\\\
&=\dfrac{1}{\sqrt{2\lambda}}\,q^{\prime}(\varsigma(t))\,(Z_{h}(q(\varsigma(t)))+2hm),\end{split}$
(2.19)

for every $t\in[0,T]$. Hence, from (2.12), (2.17) and (2.19) it follows that

$\begin{split}\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}(t)}{\sqrt{1-|\dot{x}(t)|^{2}/c^{2}}}\right)&=\frac{1}{\sqrt{2\lambda}}\left(\frac{{\rm
d}}{{\rm
d}\sigma}\left(q^{\prime}(\sigma)\,(Z_{h}(q(\sigma)+2hm)\right)\right)|_{\sigma=\varsigma(t)}\,\dot{\varsigma}(t)\\\
&=\dfrac{1}{\sqrt{2\lambda}}\frac{1}{2}|q^{\prime}(\varsigma(t))|^{2}\nabla
Z_{h}(q(\varsigma(t)))\,\dot{\varsigma}(t)\\\
&=\dfrac{c^{2}}{4\lambda}\,\dfrac{Z_{h}(q(\varsigma(t)))+2hm}{V(q(\varsigma(t)))+h+mc^{2}}\,|q^{\prime}(\varsigma(t))|^{2}\,\nabla
Z_{h}(q(\varsigma(t)),\end{split}$ (2.20)

for every $t\in[0,T]$. Now, let us observe that (2.9) implies that

$\nabla Z_{h}(w)=\dfrac{2}{c^{2}}\,\left(V(w)+h+mc^{2}\right)\,\nabla
V(w),\quad\forall\ w\in\Omega_{h}.$

Taking into account (2.16) and this relation, from (2.20) we deduce that

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}(t)}{\sqrt{1-|\dot{x}(t)|^{2}/c^{2}}}\right)=\frac{c^{2}\nabla
Z_{h}(q(\varsigma(t)))}{2(V(q(\varsigma(t)))+h+mc^{2})}\\\ =\nabla
V(q(\varsigma(t)))=\nabla V(x(t)),$

for every $t\in[0,T]$.

3\. The energy law (2.14) immediately follows from (2.18).

Part B. Let us first observe that the fact that $q\in\Lambda^{\prime}_{h}$
plainly follows from the definition of $q$. We now check that $q$ satisfies
the geodesic equation (2.12): for every $\sigma\in[0,1]$ we have

$q^{\prime}(\sigma)=\dot{x}(t(\sigma))\,t^{\prime}(\sigma)=\dot{x}(t(\sigma))\,\dfrac{\sqrt{2\lambda}}{c^{2}}\,\dfrac{V(x(t(\sigma)))+h+mc^{2}}{Z_{h}(x(t(\sigma)))+2hm}$
(2.21)

Moreover, from the energy law (2.14) and the definition of $Z_{h}$ given in
(2.9), we deduce that

$|\dot{x}(t)|^{2}=c^{4}\,\dfrac{Z_{h}(x(t))+2hm}{(V(x(t))+h+mc^{2})^{2}},$

for every $t\in[0,T]$. As a consequence, we obtain

$\dfrac{1}{2}\,|q^{\prime}(\sigma)|^{2}\,(Z_{h}(q(\sigma))+2hm)=\lambda,$
(2.22)

for every $\sigma\in[0,1]$. On the other hand, recalling again the energy law
(2.14), from (2.21) we deduce that

$q^{\prime}(\sigma)=\dot{x}(t(\sigma))\,t^{\prime}(\sigma)=\dfrac{\sqrt{2\lambda}}{c^{2}}\,\dfrac{1}{Z_{h}(x(t(\sigma)))+2hm}\,\dfrac{mc^{2}\dot{x}(t(\sigma))}{\sqrt{1-|\dot{x}(t(\sigma))|^{2}/c^{2}}}$

and

$q^{\prime}(\sigma)\,\left(Z_{h}(q(\sigma))+2hm\right)=\sqrt{2\lambda}\,\dfrac{m\dot{x}(t(\sigma))}{\sqrt{1-|\dot{x}(t(\sigma))|^{2}/c^{2}}},$

for every $\sigma\in[0,1]$. Now, from the fact that $x$ is a solution of (2.1)
we infer that

$\begin{array}[]{ll}\displaystyle\frac{\rm d}{{\rm
d}\sigma}(q^{\prime}(\sigma)\,(Z_{h}(q(\sigma))+2hm))&=\displaystyle\sqrt{2\lambda}\,\nabla
V(x(t(\sigma)))\,t^{\prime}(\sigma)\\\ &\\\
&\displaystyle=\dfrac{2\lambda}{c^{2}}\,\dfrac{V(q(\sigma))+h+mc^{2}}{Z_{h}(q(\sigma))+2hm}\,\nabla
V(q(\sigma)),\end{array}$

for every $\sigma\in[0,1]$. Recalling (2.22) and the fact that

$\nabla Z_{h}(x)=\dfrac{2}{c^{2}}\,(V(x)+h+mc^{2})\,\nabla V(x),\quad\forall\
x\in\Omega,$

we conclude that $q$ satisfies (2.12). ∎

Let us now introduce the _Maupertuis functional_ defined by

$\mathcal{M}_{h}(u)=\int_{0}^{1}|u^{\prime}(s)|^{2}\,ds\,\int_{0}^{1}\left(Z_{h}(u(s))+2hm\right)\,ds,$
(2.23)

for every $u\in\Lambda^{\prime}_{h}$; we point out that this is the functional
that will be used in our application of Section 3. The relation between
critical points of $\mathcal{E}_{h}$ and $\mathcal{M}_{h}$ is well known: we
recall it in the next lemma, whose proof is given for the reader’s convenience
at the end of this section.

######  Lemma 2.1.

Let $q\in\Lambda^{\prime}_{h}$ be a non-constant critical point of
$\mathcal{E}_{h}$. Then, there exists a change of variable
$\chi:[0,1]\to[0,1]$ such that the function

$u(s)=q(\chi(s)),\qquad s\in[0,1],$

is a non-constant critical point of $\mathcal{M}_{h}$. Conversely, if
$u\in\Lambda^{\prime}_{h}$ is a non-constant critical point of
$\mathcal{M}_{h}$, then there exists a change of variable
$\varrho:[0,1]\to[0,1]$ such that the function

$q(\sigma)=u(\varrho(\sigma)),\qquad\sigma\in[0,1],$

is a non-constant critical point of $\mathcal{E}_{h}$.

From Proposition 2.1 and Lemma 2.1 we can thus obtain the following equivalent
formulation of the variational principle, relating critical points of (2.23)
and solutions with prescribed relativistic energy of (2.1).

######  Proposition 2.2.

Let $u\in\Lambda^{\prime}_{h}$ be a non-constant critical point of
$\mathcal{M}_{h}$. Then, there exist $T>0$ and a change of variable
$\upsilon:[0,T]\to[0,1]$ such that the function $x:[0,T]\to\mathbb{R}^{n}$
defined by

$x(t)=u(\upsilon(t)),\qquad t\in[0,T],$

is a non-constant periodic solution of (2.1), satisfying the energy law
(2.14). Conversely, let $x:[0,T]\to\Omega_{h}$ be a non-constant periodic
solution of (2.1), satisfying the energy law (2.14). Then, there exists a
change of variable $\varphi:[0,1]\to[0,T]$ such that the function
$u:[0,1]\to\mathbb{R}^{n}$ defined by

$u(s)=x(\varphi(s)),\qquad s\in[0,1],$

is a non-constant critical point of $\mathcal{M}_{h}$ in
$\Lambda^{\prime}_{h}$.

###### Proof of Lemma 2.1.

In order to prove the result, it is sufficient to show that non-constant
critical points of the two functionals $\mathcal{E}_{h}$ and $\mathcal{M}_{h}$
correspond, up to suitable reparameterizations, to solutions $z$ of the
differential equation

$z^{\prime\prime}=\nabla Z_{h}(z)$ (2.24)

satisfying

$\dfrac{1}{2}\,|z^{\prime}|^{2}-Z_{h}(z)=2hm.$ (2.25)

In the case of the Maupertuis functional, this follows from a well-known
elementary computation: indeed, non-constant critical points
$u\in\Lambda^{\prime}_{h}$ of $\mathcal{M}_{h}$ and solutions $z$ of
(2.24)-(2.25) are related by

$z(\rho)=u\left(\dfrac{\rho}{\omega}\right),\quad\forall\ \rho\in[0,\omega],$

where $\omega>0$ is given by

$\displaystyle\omega=\left(\dfrac{\bigintss_{0}^{1}|u^{\prime}(\sigma)|^{2}\,d\sigma}{2\bigintss_{0}^{1}\left(Z_{h}(u(\sigma))+2hm\right)\,d\sigma}\right)^{\frac{1}{2}}.$

The relation bewteen non-constant critical points $q\in\Lambda^{\prime}_{h}$
of $\mathcal{E}_{h}$ and solutions $z$ of (2.24)-(2.25) is less obvious. We
write here the expressions of the change of variables relating $q$ and $z$ and
we refer to [5] for the details of the proof.

Given $q\in\Lambda^{\prime}_{h}$, define

$\varrho(s)=\dfrac{1}{\sqrt{2}}\,\int_{0}^{s}\dfrac{|q^{\prime}(\xi)|}{\sqrt{Z_{h}(q(\xi))+2hm}},\quad\forall\
s\in[0,1],$

and let $\varsigma$ be the inverse of $\varrho$. Then, the function
$z:[0,\varrho(1)]\to\mathbb{R}^{n}$ defined by

$z(\rho)=q(\varsigma(\rho)),\quad\forall\ \rho\in[0,\varrho(1)],$

satisfies (2.24) and (2.25).

Conversely, for every function $z$ satisfying (2.24) and (2.25) in some
interval $[0,\textrm{T}]$, let us define

$s(t)=\dfrac{1}{\lambda}\,\int_{0}^{t}(Z_{h}(z(\xi))+2hm)\,d\xi,\quad\forall\
t\in[0,\textrm{T}],$

where $\lambda>0$ is taken in such a way that $s(\textrm{T})=1$, and let
$\tau$ be the inverse of $s$. Then, the function $q:[0,1]\to\mathbb{R}^{n}$
defined by

$q(\sigma)=z(\tau(s)),\quad\forall\ \sigma\in[0,1],$

is a non-constant critical point of $\mathcal{E}_{h}$. ∎

##  3 Prescribed energy periodic solutions for a relativistic problem with
singular potential

In this section, we present an application of the Maupertuis principle
developed in Section 2 to the existence of fixed energy periodic solutions for
a relativistic problem with singular potential (for the case of smooth
potentials, see Remark 3.5).

More precisely, we deal with the equation

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=-\sum_{i=1}^{N}\frac{\kappa_{i}(x-\sigma_{i})}{|x-\sigma_{i}|^{\alpha+2}}+\nabla
W(x),\qquad x\in\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\},$
(3.1)

where, as usual, $m,c>0$, $\sigma_{1},\ldots,\sigma_{N}$ are distinct points
of $\mathbb{R}^{2}$, $\kappa_{i}>0$ for $i=1,\ldots,N$, and
$W:\mathbb{R}^{2}\to\mathbb{R}$ is a function of class $C^{1}$ satisfying the
following assumptions:

  * (W1)

there exists $M>0$ such that $0<W(x)\leq M$ for every $x\in\mathbb{R}^{2}$

  * (W2)

$\nabla W(x)\to 0$ for $|x|\to+\infty$.

Finally, we assume the “relativistic strong force condition”

$\alpha>1,$

see Remark 3.3 for some comments. Of course, equation (3.1) is of the type
(2.1) with

$V(x)=\sum_{i=1}^{N}\frac{\kappa_{i}}{\alpha|x-\sigma_{i}|^{\alpha}}+W(x)$
(3.2)

and $\Omega=\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$, so
that the theory developed in Section 2 can be applied.

In the above setting, we are going to prove the existence of (non-collision)
periodic solutions of equation (3.1) with prescribed energy (in the sense of
(2.4)) and prescribed homotopy class in the punctured plane
$\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$ (see Remark 3.1).
More precisely, we establish the following result.

######  Theorem 3.1.

In the above setting, let us fix $h>0$ and a non-trivial homotopy class
$[\gamma]\in\pi_{1}(\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\})$.
Then, there exists a non-collision periodic solution of equation (3.1), having
energy $\mathfrak{E}=h+mc^{2}$ and belonging to the homotopy class $[\gamma]$.

######  Remark 3.1.

Since we deal with solutions with prescribed energy (and unprescribed period),
the expression “belonging to the homotopy class $[\gamma]$” used in the
statement of Theorem 3.1 requires some clarification. In fact, the periodic
solution $x$ provided by Theorem 3.1 arises as a reparameterization of a
critical point
$u:[0,1]\to\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$ of the
Maupertuis functional. The condition that we impose is that the path $u$
belongs to the homotopy class $[\gamma]$; in terms of the solution $x$, this
just means that, on some period $T>0$ (unknown a-priori, and not necessarily
the minimal one ), the path $x|_{[0,T]}$ belong (after a linear
reparameterization of time) to the class $[\gamma]$. As a consequence, it is
not possible to ensure that all the solutions provided by Theorem 3.1, in
spite of coming from different homotopy classes, are distinct the one from the
other.

To understand this delicate issue, it can be useful to first focus on the case
$N=1$. In this situation, as well known, the fundamental group
$\pi_{1}(\mathbb{R}^{2}\setminus\\{\sigma_{1}\\})$ is isomorphic to
$\mathbb{Z}$ and the homotopy class of a closed path is completely determined
by its winding number around the point $\sigma_{1}$. Thus, in principle we
could apply Theorem 3.1 to get the existence of a periodic solution $x_{k}$
with winding number $k$, for any $k\in\mathbb{Z}$, with $k\neq 0$. If $k=\pm
1$, of course we can be sure that the winding number is evaluated on the
minimal period, say $T_{k}$, of the solution $x_{k}$. On the other hand, a
solution $x_{k}$ having winding number $k$ with $|k|>1$ could, in principle,
just correspond to the solution $x_{\textnormal{sgn}(k)}$, when regarded on
the period $|k|T_{k}$. Thus, summing up, in the case $N=1$ Theorem 3.1 just
ensures the existence of two solutions, a first one winding around the origin
in the clockwise sense and the second one in the counter-clockwise sense.
Since, moreover, the problem is time-reversible, these two solutions could
correspond to the same orbit, just covered in different directions: summing,
just _one_ periodic solution can be provided by Theorem 3.1 in this case.

In the case $N\geq 2$, the situation is similar, but more involved. Indeed,
now the fundamental group
$\pi_{1}(\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\})$ is
homeomorphic to the free group with $n$ generators $\\{a_{1},\ldots,a_{n}\\}$,
that is, an homotopy class is described by a finite word formed with the
symbols $a_{i}$ and their inverses $a_{i}^{-1}$ (cf. [10, 15]). Taking into
account that the group product corresponds to the concatenation of paths, we
deduce that homotopy classes described by words which are formed by a
repetition of a same sub-word (like, for instance,
$a_{1}a_{2}^{-1}a_{1}a_{2}^{-1}$, which consists of two repetitions of the
sub-word $a_{1}a_{2}^{-1}$) must not be considered, since they could give rise
to the same periodic solutions provided by the homotopy class described by the
sub-word. Moreover, again by the time-reversibility of the problem, a word and
its inverse must be identified, since they could provide the same periodic
solution, up to time-inversion. Clearly enough, however, still an infinite
number of distinct periodic solutions (with the same energy) can be provided
by Theorem 3.1 for any $N\geq 2$.

We now give the proof of Theorem 3.1. Before going into the details, however,
let us state and prove the following preliminary lemma.

######  Lemma 3.1.

Let $u\in H^{1}([0,1])$ be such that $u(0)=u(1)$ and $u(s)\neq\sigma_{i}$ for
every $s\in[0,1]$ and $i=1,\ldots,N$. If, moreover, $u$ is not null-homotopic
as a loop in $\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$, then

$\|u\|_{L^{\infty}}\leq R+\|u^{\prime}\|_{L^{2}},$ (3.3)

where $R=\max_{i}|\sigma_{i}|$.

###### Proof.

Let us assume that $\|u\|_{L^{\infty}}>R$ (otherwise there is nothing to
prove) and take $s^{*}\in[0,1]$ such that $|u(s^{*})|=\|u\|_{L^{\infty}}$.
Now, let us consider the open ball $B\subset\mathbb{R}^{2}$ centered at
$u(s^{*})$ and with radius $r=|u(s^{*})|-R$: clearly, such a ball does not
contain any of the centers $\\{\sigma_{1},\ldots,\sigma_{N}\\}$. Accordingly,
the image of the path $u$ cannot be entirely contained in $B$, since otherwise
$u$ would be a null-homotopic loop in
$\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$. We thus infer
that there exists $\bar{s}\in[0,1]$ such that

$|u(\bar{s})-u(s^{*})|\geq|u(s^{*})|-R.$

Then

$\displaystyle\|u\|_{L^{\infty}}$ $\displaystyle=|u(s^{*})|\leq
R+|u(\bar{s})-u(s^{*})|\leq
R+\left|\int_{s^{*}}^{\bar{s}}|u^{\prime}|\,ds\right|$ $\displaystyle\leq
R+|s^{*}-\bar{s}|^{1/2}\left|\int_{s^{*}}^{\bar{s}}|u^{\prime}|^{2}\,ds\right|^{1/2}\leq
R+\|u^{\prime}\|_{L^{2}},$

thus proving (3.3). ∎

###### Proof of Theorem 3.1.

Let us first observe that, since $V(u)>0$ for every
$u\in\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$ and $h>0$,
with the notation of Section 2 we have
$\Omega_{h}=\Omega=\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\}$
and

$\Lambda_{h}^{\prime}=\\{u\in
H^{1}([0,1])\,:\,u(0)=u(1),\;u(s)\neq\sigma_{i}\;\forall\
s\in[0,1],\,i=1,\ldots,N\\}.$

Accordingly, we can consider the Maupertuis functional $\mathcal{M}_{h}$ as
defined in (2.23), that is,

$\mathcal{M}_{h}(u)=\int_{0}^{1}|u^{\prime}|^{2}\,ds\,\int_{0}^{1}\left(Z_{h}(u)+2hm\right)\,ds,\qquad
u\in\Lambda_{h}^{\prime},$ (3.4)

where, as in (2.9),

$Z_{h}(u)=2mV(u)+\dfrac{1}{c^{2}}\,(V(u)+h)^{2},$ (3.5)

for $u\in\Omega$. By standard arguments (cf. [17, Theorem 1.4]), the
functional $\mathcal{M}_{h}$ is of class $C^{1}$, with

$d\mathcal{M}_{h}(u)[v]=2\int_{0}^{1}\langle
u^{\prime},v^{\prime}\rangle\,ds\int_{0}^{1}\left(Z_{h}(u)+2hm\right)\,ds+\int_{0}^{1}|u^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(u),v\rangle\,ds.$ (3.6)

For further convenience, we now claim that $Z_{h}$ satisfies the following
crucial properties: for every $i=1,\ldots,N$, there exist $\beta_{i}>0$ and
$r_{i}>0$ such that

$Z_{h}(u)\geq\frac{\beta_{i}}{|u-\sigma_{i}|^{2}},\quad\mbox{ if
}0<|u-\sigma_{i}|<r_{i},$ (3.7)

and, moreover,

$\lim_{u\to\sigma_{i}}\left(Z_{h}(u)+\frac{1}{2}\langle\nabla
Z_{h}(u),u-\sigma_{i}\rangle\right)=-\infty.$ (3.8)

Indeed, (3.7) is an easy consequence of the fact that $\alpha>1$. As far as
(3.8) is concerned, recalling (3.5), we immediately deduce that

$\begin{array}[]{ll}\displaystyle Z_{h}(u)+\frac{1}{2}\langle\nabla
Z_{h}(u),u-\sigma_{i}\rangle=&\displaystyle\dfrac{1}{c^{2}}\,(V(u))^{2}+\left(2m+\dfrac{2h}{c^{2}}\right)\,V(u)+\dfrac{h^{2}}{c^{2}}\\\
&\\\
&\displaystyle+\left(\dfrac{V(u)}{c^{2}}+m+\dfrac{h}{c^{2}}\right)\,\langle\nabla
V(u),u-\sigma_{i}\rangle,\end{array}$ (3.9)

for every $u\in\Omega$ and $i=1,\ldots,N$. Now, let us observe that (3.2)
implies that for every $i=1,\ldots,N$ there exists a neighbourhood
$\Sigma_{i}$ of $\sigma_{i}$ such that

$V(u)=\dfrac{\kappa_{i}}{\alpha\
|u-\sigma_{i}|^{\alpha}}+\widetilde{W}(u),\quad\forall\ u\in\Sigma_{i},\
u\neq\sigma_{i},$

with $\widetilde{W}$ and $\nabla\widetilde{W}$ bounded in $\Sigma_{i}$. Hence,
from (3.9) we infer that there exist a constant $K\in\mathbb{R}$ and a
function $W^{*}$ bounded in $\Sigma_{i}$ such that

$Z_{h}(u)+\frac{1}{2}\langle\nabla
Z_{h}(u),u-\sigma_{i}\rangle=\dfrac{\kappa_{i}^{2}}{\alpha^{2}c^{2}}\,(1-\alpha)\,\dfrac{1}{|u-\sigma_{i}|^{2\alpha}}+\dfrac{K}{|u-\sigma_{i}|^{\alpha}}+W^{*}(u),$

for every $u\in\Sigma_{i}$, $u\neq\sigma_{i}$. From this relation, recalling
that $\alpha>1$ we plainly obtain (3.8).

Let us now notice that from (3.8) we immediately infer that there exists
$\varepsilon\in(0,1)$ such that, for every $i=1,\ldots,N$,

$d\mathcal{M}_{h}(u)[u-\sigma_{i}]<0,\quad\mbox{ if }\|u-\sigma_{i}\|\leq
2\varepsilon,\quad\mbox{$u$ non-constant}.$ (3.10)

Indeed, by (3.6) we have

$d\mathcal{M}_{h}(u)[u-\sigma_{i}]=2\int_{0}^{1}|u^{\prime}|^{2}\,ds\int_{0}^{1}\left(Z_{h}(u)+2hm+\frac{1}{2}\langle\nabla
Z_{h}(u),u-\sigma_{i}\rangle\right)\,ds,$

so that the conclusion follows from (3.8), in view of the continuous embedding
$H^{1}([0,1])\subset L^{\infty}([0,1])$. Of course, such a $\varepsilon$ can
be chosen so small that

$\|u-\sigma_{i}\|\leq
2\varepsilon\;\Longrightarrow\;\|u-\sigma_{j}\|>2\varepsilon,\,\forall\ j\neq
i,$ (3.11)

that is, the balls $\\{B_{2\varepsilon}(\sigma_{i})\\}_{i=1}^{N}$ are
disjoint. From now on, we assume that this condition is satisfied.

We are now in a position to start with the proof. Fixed a non-trivial homotopy
class
$[\gamma]\in\pi_{1}(\mathbb{R}^{2}\setminus\\{\sigma_{1},\ldots,\sigma_{N}\\})$,
and with $\varepsilon>0$ as above, let us consider the set

$\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}=\left\\{u\in\Lambda^{\prime}_{h}\,:\,u\mbox{
lies in the homotopy class $[\gamma]$ and
}\|u-\sigma_{i}\|\geq\varepsilon,\,\forall\ i=1,\ldots,N\right\\}.$

Observe that, by the non-triviality of $\gamma$, any
$u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$ is non-constant Moreover,
recalling that $Z_{h}$ is always positive, we plainly deduce the estimate

$\mathcal{M}_{h}(u)\geq 2hm\int_{0}^{1}|u^{\prime}|^{2}\,ds>0,\quad\forall\
u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}.$ (3.12)

With this in mind, let us consider the minimization problem

$\inf_{u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}}\mathcal{M}_{h}.$

We split our next arguments into some steps.

Step 1: _convergence of minimizing sequences._ Let us consider an arbitrary
minimizing sequence
$\\{u_{n}\\}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$ for the
minimization problem

$\inf_{u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}}\mathcal{M}_{h}.$

By the estimate (3.12), the sequence $\|\dot{u}_{n}\|_{L^{2}}$ is bounded;
using (3.3), it thus follows that $\|u_{n}\|_{L^{\infty}}$ is bounded as well.
Since $\|u_{n}\|_{L^{2}}\leq\|u_{n}\|_{L^{\infty}}$, the sequence
$\\{u_{n}\\}$ is thus bounded in the $H^{1}$-norm. Hence, there exists
$\bar{u}\in H^{1}([0,1])$, with $\bar{u}(0)=\bar{u}(1)$, such that (up to
subsequences, which we do not relabel) $u_{n}$ converges to $\bar{u}$ weakly
in $H^{1}$ and, by compactness, strongly in $L^{\infty}$.

We now claim that $\bar{u}$ does not have collisions, that is,
$\bar{u}(s)\neq\sigma_{i}$ for every $s\in[0,1]$ and $i\in\\{1,\ldots,N\\}$
(and, hence, $\bar{u}\in\Lambda_{h}^{\prime}$). Indeed, let us assume by
contradiction that this is not true. Then, since (3.7) holds true, the same
argument used in [1, Lemma 5.3] implies that

$\int_{0}^{1}Z_{h}(u_{n})\,ds\to+\infty.$

Hence, a contradiction with the fact that $\\{u_{n}\\}$ is a minimizing
sequence is obtained as long as we prove that

$\liminf_{n\to+\infty}\int_{0}^{1}|u^{\prime}_{n}|^{2}\,ds>0.$

To see that this is true, we argue again by contradiction, assuming that (up
to subsequences) $\int_{0}^{1}|\dot{u}_{n}|^{2}\,ds\to 0$. Then, for any
$s_{1},s_{2}\in[0,1]$, passing to the limit in the inequality

$|u_{n}(s_{2})-u_{n}(s_{1})|\leq\left|\int_{s_{1}}^{s_{2}}|u^{\prime}_{n}|\,ds\right|\leq\int_{0}^{1}|u^{\prime}_{n}|^{2}\,ds$

we see that $\bar{u}$ is constant and, thus, $\bar{u}\equiv\sigma_{i^{*}}$ for
some fixed center $\sigma_{i^{*}}$. Moreover, since the derivative of a
constant function is identically zero and $\|\dot{u}_{n}\|_{L^{2}}\to 0$, we
obtain that the convergence of $u_{n}$ to $\sigma_{i^{*}}$ is strong in
$H^{1}$, a contradiction with the fact that
$\|u_{n}-\sigma_{i^{*}}\|\geq\varepsilon$.

Let us notice that, from the just proved fact that $\bar{u}$ does not have
collisions, and since homotopy classes are open w.r.t. the
$L^{\infty}$-topology, the $L^{\infty}$-convergence of $u_{n}$ to $\bar{u}$
implies that $\bar{u}$ lies in the homotopy class $[\gamma]$, as well.

Step 2: _construction of a minimizing sequence $\\{\widetilde{u}_{n}\\}$ away
from the boundary $\|\widetilde{u}_{n}-\sigma_{i}\|\geq\varepsilon$._
Precisely, in this step we show that there exists a minimizing sequence
$\\{\widetilde{u}_{n}\\}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$
such that, for some $\lambda\in(1,2]$, the inequality

$\|\widetilde{u}_{n}-\sigma_{i}\|\geq\lambda\varepsilon,\quad\forall\
i\in\\{1,\ldots,N\\}.$ (3.13)

is satisfied for $n$ large enough.

To prove this, let us consider an arbitrary minimizing sequence
$\\{u_{n}\\}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$. By Step 1,
we can assume that $u_{n}$ converges, weakly in $H^{1}$ and strongly in
$L^{\infty}$, to a limit function $\bar{u}$ which lies in the homotopy class
$[\gamma]$. Since homotopy classes are open w.r.t. the $L^{\infty}$-topology,
we can fix $\rho>0$ such that

$\|u-\bar{u}\|_{L^{\infty}}\leq\rho\;\Longrightarrow\;u\mbox{ lies in the
homotopy class $[\gamma]$}.$ (3.14)

Of course, with such a choice of $\rho$ there exists a corresponding integer
$n_{\rho}\geq 0$ such that

$\|u_{n}-\bar{u}\|_{L^{\infty}}\leq\frac{\rho}{2},\quad\forall\ n\geq
n_{\rho}.$ (3.15)

As a consequence, we deduce that

$\|u_{n}-\sigma_{i}\|_{L^{\infty}}\leq\|u_{n}-\bar{u}\|_{L^{\infty}}+\|\bar{u}-\sigma_{i}\|_{L^{\infty}}\leq\frac{\rho}{2}+\|\bar{u}-\sigma_{i}\|_{L^{\infty}},$

implying that

$\sup_{n\geq n_{\rho}}\max_{i}\|u_{n}-\sigma_{i}\|_{L^{\infty}}<+\infty.$

Hence, we can set

$M=\max\left\\{\sup_{n\geq
n_{\rho}}\max_{i}\|u_{n}-\sigma_{i}\|_{L^{\infty}},\frac{\rho}{2}\right\\}$

and

$\lambda=1+\frac{\rho}{2M}.$

We are now ready to construct the sequence $\\{\widetilde{u}_{n}\\}$ (defined
for $n\geq n_{\rho}$).

So, let us assume that, for some $n\geq n_{\rho}$, the set

$I_{n}=\\{i\in\\{1,\ldots,N\\}:\|u_{n}-\sigma_{i}\|<\lambda\varepsilon\\}$

is non-empty (otherwise, we can simply take $\widetilde{u}_{n}=u_{n}$). Since
$\lambda\leq 2$, by (3.11) we have that there exists exactly one
$i(n)\in\\{1,\ldots,N\\}$ such that

$\|u_{n}-\sigma_{i(n)}\|<\lambda\varepsilon\quad\mbox{ while
}\quad\|u_{n}-\sigma_{j}\|>\lambda\varepsilon,\,\forall\ j\neq i(n).$

Let us set

$k_{n}=\frac{\lambda\varepsilon}{\|u_{n}-\sigma_{i(n)}\|};$

since $\|u_{n}-\sigma_{i(n)}\|\geq\varepsilon$ by the definition of
$\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$, we have

$0<k_{n}-1<\lambda-1=\frac{\rho}{2M}.$ (3.16)

Finally, let us define

$\widetilde{u}_{n}=\sigma_{i(n)}+k_{n}(u_{n}-\sigma_{i(n)}).$ (3.17)

Of course

$\|\widetilde{u}_{n}-\sigma_{i(n)}\|=\lambda\varepsilon$ (3.18)

and so, since $\lambda\leq 2$ and using again by (3.11),

$\|\widetilde{u}_{n}-\sigma_{j}\|>2\varepsilon,\quad\forall\ j\neq i(n).$
(3.19)

Moreover, using (3.15) and (3.16)

$\displaystyle\|\widetilde{u}_{n}-\bar{u}\|_{L^{\infty}}$
$\displaystyle=\|u_{n}-\bar{u}+(1-k_{n})(\sigma_{i(n)}-u_{n})\|_{L^{\infty}}$
$\displaystyle\leq\|u_{n}-\bar{u}\|_{L^{\infty}}+(k_{n}-1)\|u_{n}-\sigma_{i(n)}\|_{L^{\infty}}\leq\frac{\rho}{2}+\frac{\rho}{2M}M=\rho.$

By (3.14), we thus have that $\widetilde{u}_{n}$ has the homotopy class
$[\gamma]$; recalling (3.18) and (3.19), we conclude that
$u_{n}\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$. It thus remains to
show that $\\{\widetilde{u}_{n}\\}$ is still a minimizing sequence: to this
end, we are going to prove that

$\mathcal{M}_{h}(\widetilde{u}_{n})\leq\mathcal{M}_{h}(u_{n}).$ (3.20)

So, let us assume that, for some $n$, $\widetilde{u}_{n}$ is defined as in
(3.17) (otherwise $\widetilde{u}_{n}=u_{n}$ and there is nothing to prove).
Accordingly, we define, for every $t\in[1,k_{n}]$,
$U_{t}=t(u_{n}-\sigma_{i(n)})+\sigma_{i(n)}$ in such a way that $U_{1}=u_{n}$
and $U_{k_{n}}=\widetilde{u}_{n}$. Then, we have

$\frac{d}{dt}\mathcal{M}_{h}(U_{t})=d\mathcal{M}_{h}(U_{t})[u_{n}-\sigma_{i(n)}]=\frac{1}{t}d\mathcal{M}_{h}(U_{t})[t(u_{n}-\sigma_{i(n)})],$
(3.21)

with $t\in[1,k_{n}]$. From (3.18) we deduce that

$\|U_{t}-\sigma_{i(n)}\|\leq
k_{n}\|u_{n}-\sigma_{i(n)}\|=\|\widetilde{u}_{n}-\sigma_{i(n)}\|=\lambda\varepsilon\leq
2\varepsilon,$

where the last inequality follows from the fact that $\lambda\leq 2$. Since

$d\mathcal{M}_{h}(U_{t})\left[U_{t}-\sigma_{i(n)}\right]=d\mathcal{M}_{h}(U_{t})\left[t(u_{n}-\sigma_{i(n)})\right],$

from (3.10) and (3.21) we obtain

$\frac{d}{dt}\mathcal{M}_{h}(U_{t})<0,\quad\forall k\in[1,k_{n}].$

This implies that $\mathcal{M}_{h}(U_{k_{n}})<\mathcal{M}_{h}(U_{1})$ and thus
(3.20).

Step 3: _construction of a minimizing sequence $\\{\hat{u}_{n}\\}$ away from
the boundary $\|\hat{u}_{n}-\sigma_{i}\|\geq\varepsilon$ and such that
$d\mathcal{M}_{h}(\hat{u}_{n})\to 0$._ Precisely, in this step we show that
there exists a minimizing sequence
$\\{\widehat{u}_{n}\\}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$
such that, for some $\lambda^{*}>1$, the inequality

$\|\widehat{u}_{n}-\sigma_{i}\|\geq\lambda^{*}\varepsilon,\quad\forall\
i\in\\{1,\ldots,N\\}.$ (3.22)

is satisfied for $n$ large enough and $d\mathcal{M}_{h}(\widehat{u}_{n})\to
0$.

To this end, we consider a minimizing sequence
$\\{\widetilde{u}_{n}\\}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$
constructed as in Step 2, and thus satisfying (3.13). By Step 1, we can assume
that $\\{\widetilde{u}_{n}\\}$ converges, weakly in $H^{1}$ and strongly in
$L^{\infty}$, to a limit function $u^{*}$ without collisions and lying in the
homotopy class $[\gamma]$. Hence, we can fix $\rho^{*}>0$ so small that the
inequality

$|\widetilde{u}_{n}(s)-\sigma_{i}|\geq\rho^{*},\quad\forall\
s\in[0,1],\,i\in\\{1,\ldots,N\\},$ (3.23)

is satisfied if $n$ is large enough. Moreover, up to some relabeling of the
sequence we can also assume that

$\mathcal{M}_{h}(\widetilde{u}_{n})\leq\inf_{u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}}\mathcal{M}_{h}(u)+\frac{1}{\sqrt{n}},$
(3.24)

for $n$ large enough.

With this in mind, let us consider the set

$\Lambda^{*}=\\{u\in\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}\,:\,|\widetilde{u}(s)-\sigma_{i}|\geq\rho^{*}/2,\quad\forall\
s\in[0,1],\,i\in\\{1,\ldots,N\\}\\}.$

In view of the continuous embedding $H^{1}\subset L^{\infty}$ (and recalling
the fact that homotopy classes are open with respect to the
$L^{\infty}$-topology), $M$ is (strongly) closed in $H^{1}$; moreover, by
(3.23), the sequence $\\{\widetilde{u}_{n}\\}$ belongs to $\Lambda^{*}$ for
$n$ large enough (and it is thus a minimizing sequence also for
$\inf_{\Lambda^{*}}\mathcal{M}_{h}$, since
$\Lambda^{*}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$). Taking
into account (3.24), by Ekeland variational principle (as stated, for
instance, [17, Theorem 4.1 and Remark 4.1]) we can find a further sequence
$\\{\widehat{u}_{n}\\}\subset\Lambda^{*}$ such that

$\mathcal{M}_{h}(\widehat{u}_{n})\leq\mathcal{M}_{h}(\widetilde{u}_{n}),\qquad\|\widehat{u}_{n}-\widetilde{u}_{n}\|\leq\frac{1}{\sqrt{n}}$
(3.25)

and

$\mathcal{M}_{h}(w)-\mathcal{M}_{h}(\widehat{u}_{n})\geq-\frac{1}{\sqrt{n}}\|w-\widehat{u}_{n}\|,\quad\forall\
w\in\Lambda^{*}.$ (3.26)

By the first condition (3.25), we deduce that the sequence
$\\{\widehat{u}_{n}\\}$ is still minimizing. Moreover, taking into account
(3.13), the second condition in (3.25) yields

$\|\widehat{u}_{n}-\sigma_{i}\|\geq\frac{1+\lambda}{2}\varepsilon,\quad\forall\
i\in\\{1,\ldots,N\\}.$ (3.27)

for $n$ large enough, and thus (3.22) for $\lambda^{*}=(1+\lambda)/2>1$.
Finally, again from the second condition (3.25) together with the fact that
$\widetilde{u}_{n}\to u^{*}$ strongly in $L^{\infty}$, we infer that
$\widehat{u}_{n}\to u^{*}$ strongly in $L^{\infty}$ as well: from (3.23), we
thus get

$|\widehat{u}_{n}(s)-\sigma_{i}|\geq\frac{3}{4}\rho^{*}>\frac{\rho^{*}}{2},\quad\forall\
s\in[0,1],\,i\in\\{1,\ldots,N\\},$ (3.28)

for $n$ large enough.

Now, let us observe that property (3.27) and (3.28) together imply that the
sequence $\\{\widehat{u}_{n}\\}$ belongs (at least for $n$ large enough) to
the interior of the set $M$. From this observation, taking into account
(3.26), it easily follows that

$d\mathcal{M}_{h}(\widehat{u}_{n})\to 0.$ (3.29)

Indeed, let us fix an arbitrary $v\in H^{1}([0,1])$ with $v(0)=v(1)$ and
$\|v\|=1$. Then, $\widetilde{u}_{n}+tv\in\Lambda^{*}$ if $|t|$ is small
enough, and thus, using (3.26),

$d\mathcal{M}_{h}(\widehat{u}_{n})[v]=\lim_{t\to
0^{+}}\frac{\mathcal{M}_{h}(\widehat{u}_{n}+tv)-\mathcal{M}_{h}(\widehat{u}_{n})}{t}\geq-\frac{1}{\sqrt{n}}$

and

$d\mathcal{M}_{h}(\widehat{u}_{n})[v]=\lim_{t\to
0^{-}}\frac{\mathcal{M}_{h}(\widehat{u}_{n}+tv)-\mathcal{M}_{h}(\widehat{u}_{n})}{-|t|}\leq\frac{1}{\sqrt{n}}.$

This implies

$\|d\mathcal{M}_{h}(\widehat{u}_{n})\|=\sup_{\|v\|=1}|d\mathcal{M}_{h}(\widehat{u}_{n})[v]|\leq\frac{1}{\sqrt{n}}$

and thus (3.29), as desired.

Step 4: _strong convergence of the sequence $\\{\widehat{u}_{n}\\}$ and
conclusion._ Let us consider the minimizing sequence
$\\{\widehat{u}_{n}\\}\subset\Lambda^{*}\subset\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$
constructed in Step 3; as already observed, $\widehat{u}_{n}$ converges weakly
in $H^{1}$ and strongly in $L^{\infty}$ to a limit function $u^{*}$ without
collisions and lying in the homotopy class $[\gamma]$. We now claim that

$\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\to\int_{0}^{1}|(u^{*})^{\prime}|^{2}\,ds.$
(3.30)

From this, the conclusion follows easily. Indeed, (3.30) implies (by a well-
known property of Hilbert spaces) that $\widetilde{u}_{n}\to u^{*}$ strongly
in $H^{1}$. As a consequence,
$\mathcal{M}_{h}(\widehat{u}_{n})\to\mathcal{M}_{h}(u^{*})$ and thus $u^{*}$
is a minimum point for $\mathcal{M}_{h}$ on
$\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$. Moreover, again by the strong
$H^{1}$-convergence together with (3.27), it holds

$\|u^{*}-\sigma_{i}\|\geq\frac{1+\lambda}{2}\varepsilon>\varepsilon,\quad\forall\
i\in\\{1,\ldots,N\\},$

so that $u^{*}$ belongs to the interior of the set
$\Lambda^{\prime\prime}_{h,\gamma,\varepsilon}$. Hence, $u^{*}$ is a critical
point (in fact, a local minimum) for $\mathcal{M}_{h}$ and, of course, it is
non-constant since the homotopy class $[\gamma]$ is non-trivial. Thus, the
conclusion follows from Proposition 2.2.

So, let us prove (3.30). Since $\\{\widehat{u}_{n}\\}$ is bounded in $H^{1}$,
by (3.29) we have

$d\mathcal{M}_{h}(\widehat{u}_{n})[\widehat{u}_{n}-u^{*}]\to 0.$ (3.31)

On the other hand, recalling formula (3.6) we have

$\displaystyle d\mathcal{M}_{h}(\widehat{u}_{n})[\widehat{u}_{n}]$
$\displaystyle=2\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\left(Z_{h}(\widehat{u}_{n})+2hm\right)\,ds$
$\displaystyle\qquad\qquad+\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(\widehat{u}_{n}),\widehat{u}_{n}\rangle\,ds$
$\displaystyle=2\mathcal{M}_{h}(\widehat{u}_{n})+\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(\widehat{u}_{n}),\widehat{u}_{n}\rangle\,ds$

and

$\displaystyle d\mathcal{M}_{h}(\widehat{u}_{n})[u^{*}]$
$\displaystyle=2\int_{0}^{1}\langle(\widehat{u}_{n})^{\prime},(u^{*})^{\prime}\rangle\,ds\int_{0}^{1}\left(Z_{h}(\widehat{u}_{n})+2hm\right)\,ds$
$\displaystyle\qquad\qquad+\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(\widehat{u}_{n}),u^{*}\rangle\,ds,$

so that

$\begin{split}d\mathcal{M}_{h}(\widehat{u}_{n})[\widehat{u}_{n}-u^{*}]&=2\mathcal{M}_{h}(\widehat{u}_{n})+\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(\widehat{u}_{n}),\widehat{u}_{n}-u^{*}\rangle\,ds\\\
&\qquad\qquad-2\int_{0}^{1}\langle(\widehat{u}_{n})^{\prime},(u^{*})^{\prime}\rangle\,ds\int_{0}^{1}\left(Z_{h}(\widehat{u}_{n})+2hm\right)\,ds.\end{split}$
(3.32)

Now, by the facts that $\\{\widehat{u}_{n}\\}$ is bounded in $H^{1}$ and
converges to $u^{*}$ strongly in $L^{\infty}$, we have

$\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds\int_{0}^{1}\langle\nabla
Z_{h}(\widehat{u}_{n}),\widehat{u}_{n}-u^{*}\rangle\,ds\to 0,$

while from the weak convergence in $H^{1}$ we obtain

$\displaystyle
2\int_{0}^{1}\langle(\widehat{u}_{n})^{\prime},(u^{*})^{\prime}\rangle\,ds\int_{0}^{1}\left(Z_{h}(\widehat{u}_{n})+2hm\right)\,ds\rightarrow$
$\displaystyle\
2\int_{0}^{1}|(u^{*})^{\prime}|^{2}\,ds\int_{0}^{1}\left(Z_{h}(u^{*})+2hm\right)\,ds$
$\displaystyle=\mathcal{M}_{h}(u^{*}).$

Therefore, (3.31) and (3.32) yields

$\mathcal{M}_{h}(\widehat{u}_{n})\to\mathcal{M}_{h}(u^{*})$

and thus, again by the convergence $\widehat{u}_{n}\to u^{*}$ in $L^{\infty}$,

$\displaystyle\int_{0}^{1}|(\widehat{u}_{n})^{\prime}|^{2}\,ds$
$\displaystyle=\mathcal{M}_{h}(\widehat{u}_{n})\left(\int_{0}^{1}\left(Z_{h}(\widehat{u}_{n})+2hm\right)\,ds\right)^{-1}$
$\displaystyle\qquad\rightarrow\
\mathcal{M}_{h}(u^{*})\left(\int_{0}^{1}\left(Z_{h}(u^{*})+2hm\right)\,ds\right)^{-1}=\int_{0}^{1}|(u^{*})^{\prime}|^{2}\,ds$

finally proving (3.30). ∎

######  Remark 3.2.

Let us recall, as already observed at the beginning of Section 2, that the
difference $h$ between the relativistic energy $\mathfrak{E}$ and the rest
energy converges to the classical mechanical energy in the non-relativistic
limit (i.e. when $c\to+\infty$). Indeed, a simple computation shows that

$\lim_{c\to+\infty}\left(\dfrac{mc^{2}}{\sqrt{1-|y|^{2}/c^{2}}}-mc^{2}\right)=\dfrac{1}{2}\,m|y|^{2},\quad\forall\
y\in\mathbb{R}^{n}.$

From this point of view, the classical system

$m\ddot{x}=\nabla V(x)$ (3.33)

can formally be seen as the non-relativistic limit of (1.2). A natural problem
is then to study the asympotic behaviour of the solutions given by Theorem 3.1
in the non-relativistic limit, in order to investigate whether they converge
to solutions of (3.33). This is a very delicate question and it seems to be
difficult to give a general answer. We refer to the study of the model problem
of Section 4 (see Remark 3.2) for a partial result in this direction when
$N=1$ and $W\equiv 0$.

######  Remark 3.3.

As well known since the pioneering paper by Gordon [13], a condition on the
strength of the singularity is typically needed when applying variational
methods to problems with singular potentials. In the classic (i.e., non-
relativistic) case, for potentials of the type (3.2) the usual “strong force”
condition reads as $\alpha\geq 2$; in our relativistic setting, instead, it is
enough to assume the weaker condition $\alpha>1$. This is a consequence of the
fact that the expression $Z_{h}$, defined in (3.5) and appearing in the
Maupertuis functional (3.4), behaves near the singularity as the square of the
potential $V$: as a consequence, the potential $Z_{h}$ satisfies the property
(3.7) in the proof of Theorem 3.1. Notice, however, that for (3.7) to be true
$\alpha\geq 1$ would be enough, while Theorem 3.1 does not cover the
(physically relevant) Newtonian case $\alpha=1$. The reason for this is that
the further property (3.8), required as well in the proof of Theorem 3.1, is
valid only when $\alpha>1$.

######  Remark 3.4.

As observed in Remark 3.1, Theorem 3.1 in the case $N=1$ just provides the
existence of _one_ periodic solution of energy $\mathfrak{E}$, for any
$\mathfrak{E}>mc^{2}$. Actually, this conclusion is a direct consequence of
[1, Theorem 15.1] (based on the result in [19]). Indeed, as recalled in the
proof of Proposition 2.2, critical points of the Maupertuis functional give
rise (via a linear reparameterization of time) to periodic solutions of the
usual second order system

$z^{\prime\prime}=\nabla Z_{h}(z)$

with energy $2hm>0$. Since, as observed in the proof of Theorem 3.1, the
potential $Z_{h}$ satisfies assumptions (3.7)- (3.8), and, moreover,
$Z_{h}(u)>0$ and $\nabla Z_{h}(u)\to 0$ for $|u|\to+\infty$, all the
assumptions required in [1, Theorem 15.1] are satisfied (by a translation, we
can assume in this case $\sigma_{1}=0$), and thus the conclusion follows. It
is worth noticing that the arguments used in the proof of [1, Theorem 15.1],
relying on min-max critical point theory, are suitable also to treat the
analogous problem in arbitrary space dimension.

######  Remark 3.5.

Let us observe that by means of a direct application of standard results it is
possible to prove the existence of periodic solutions with fixed energy for
the equation (2.1) when $V$ is a globally defined smooth potential. Indeed,
from [5, Theorem 1.4] applied to the energy functional $\mathcal{E}_{h}$ (see
also Theorem 1.1 in [1]) we immediately deduce the validity of the following
result:

> Theorem 1. Let $V:\mathbb{R}^{n}\to\mathbb{R}$, $V\in
> C^{1}(\mathbb{R}^{n})$, $h\in\mathbb{R}$ and $Z_{h}$ as in (2.9). Assume
> that at least one connected component of the set
>
> $\\{x\in\mathbb{R}^{n}:\ Z_{h}(x)+2hm\geq 0\\}$
>
> is non-empty and compact and
>
> $\nabla Z_{h}(x)\neq 0,\quad\forall\ x\in Z_{h}^{-1}(-2hm).$
>
> Then, there exists a periodic solution $q:[0,1]\to\mathbb{R}^{n}$ of the
> geodesic equation (2.12).

Recalling Proposition 2.1, Theorem 1 implies the existence of a periodic
solution of (2.1) with relativistic energy $\mathfrak{E}=h+mc^{2}$.

The assumptions of Theorem 1 can be written in term of the original potential
$V$: indeed, recalling (2.9), it is easy to see that

$\\{x\in\mathbb{R}^{n}:\ Z_{h}(x)+2hm\geq 0\\}=\\{x\in\mathbb{R}^{n}:\
V(x)+h\geq 0\\}\cup\\{x\in\mathbb{R}^{n}:\ V(x)+h\leq-2mc^{2}\\}$

and that

$\nabla Z_{h}(x)=\dfrac{2}{c^{2}}\,(V(x)+h+mc^{2})\,\nabla V(x),\quad\forall\
x\in\mathbb{R}^{n}.$

Now, let us recall that solutions $x$ of (2.1) with relativistic energy
$h+mc^{2}$ are confined in the Hill’s region $\Omega_{h}$ defined in (2.5).
Hence, if there exists $h\in\mathbb{R}$ are such that the set

$\\{x\in\mathbb{R}^{n}:\ V(x)+h\geq 0\\}$

is non-empty and compact and

$\nabla V(x)\neq 0,\quad\forall\ x\in V^{-1}(-h),$

then the assumptions of Theorem 1 are satisfied and the existence of a
periodic solution of (2.1) with energy $\mathfrak{E}=h+mc^{2}$ is proved.

Let us notice that when $\beta>1$ and $h>0$ this result applies in particular
to the pure power potential $V:\mathbb{R}^{n}\to\mathbb{R}$ defined by

$V(x)=-\frac{|x|^{\beta}}{\beta},$

for every $x\in\mathbb{R}^{n}$.

##  4 Remarks on the optimality of the condition $\mathfrak{E}>mc^{2}$

With the aim of establishing the sharpness of the assumption
$\mathfrak{E}>mc^{2}$ (equivalently, $h>0$) in Theorem 3.1, in this section we
focus on the model equation

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=-\frac{\kappa
x}{|x|^{\alpha+2}},\qquad x\in\mathbb{R}^{2}\setminus\\{0\\},$ (4.1)

with $\kappa>0$ and $\alpha>1$, which corresponds to (3.1) with a single
centre at the origin (that is, $N=1$ and $\sigma_{1}=0$) and $W\equiv 0$. Let
us recall that the energy of solutions $x$ of (4.1) in this case is given by

$E(x,\dot{x})=\frac{mc^{2}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}-\frac{\kappa}{\alpha|x|^{\alpha}}.$
(4.2)

Obviously, equation (4.1) can have two different types of periodic solutions:
the circular ones and the non-circular ones. The following result holds true.

######  Proposition 4.1.

Let $\alpha>1$. Then:

  1. 1.

equation (4.1) has circular periodic solutions of energy $\mathfrak{E}$ if and
only if

$\mathfrak{E}>\eta(\alpha),$ (4.3)

where $\eta:(1,+\infty)\to\mathbb{R}$ is defined by

$\eta(\alpha)=\left\\{\begin{array}[]{ll}\displaystyle
2mc^{2}\,\dfrac{\sqrt{\alpha-1}}{\alpha}&\quad\mbox{if $\alpha\in(1,2)$,}\\\
&\\\ mc^{2}&\quad\mbox{if $\alpha\geq 2$.}\end{array}\right.$

  2. 2.

equation (4.1) does not admit non-circular periodic solutions if $\alpha\geq
2$.

######  Remark 4.1.

Proposition 4.1 immediately implies that the condition $\mathfrak{E}>mc^{2}$
assumed in Section 3 is optimal when $\alpha\geq 2$. Let us also observe that,
in this case, the arguments used along the proof allow us to conclude that for
every $\mathfrak{E}>mc^{2}$ there exists a unique (up to time-translation and
time-inversion) periodic solution of energy $\mathfrak{E}$, which, thus, is
necessarily the solution provided by Theorem 3.1 (recall also Remark 3.1). As
a consequence, we can infer that circular solutions are local minimizers for
the Maupertuis functional whenever $\alpha\geq 2$. Notice that for
$\alpha\in(1,2)$, instead, we cannot obtain the same conclusion since non-
circular solutions could exist.

Moreover, again for $\alpha\in(1,2)$, we do not know if the existence of
periodic solutions when $2mc^{2}\sqrt{1-\alpha}/\alpha<\mathfrak{E}\leq
mc^{2}$ can be ensured also for the more general problem (3.1).

###### Proof of Proposition 4.1.

1\. The study of the existence of circular solutions of (4.1) can be carried
out by passing to polar coordinates, writing

$x(t)=re^{i\omega(t)},\quad\forall\ t\in\mathbb{R},$

where $r>0$. It is possible to see that $\dot{\omega}$ is constant. Indeed,
for solutions $x$ of (4.1) the relativistic angular momentum

$x\wedge\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}$

(see also (4.10) and the discussion at the beginning of the proof of the
second statement) is conserved. Hence

$\frac{mr^{2}\dot{\omega}}{\sqrt{1-|\dot{\omega}|^{2}r^{2}/c^{2}}}$

is constant, thus implying that $\dot{\omega}$ is constant. We can then assume
that

$x(t)=re^{i\omega t},\quad\forall\ t\in\mathbb{R},$

for some $\omega\in\mathbb{R}$. We immediately deduce that

$\frac{\rm d}{{\rm
d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right)=-\frac{cm\omega^{2}r}{\sqrt{c^{2}-\omega^{2}r^{2}}}\,e^{i\omega
t},$

for every $t\in\mathbb{R}$. Hence, $x$ is a solution of (4.1) if and only if

$-\frac{cm\omega^{2}r}{\sqrt{c^{2}-\omega^{2}r^{2}}}\,e^{i\omega
t}=-\frac{\kappa}{r^{\alpha+1}}\,e^{i\omega t},$ (4.4)

i.e. if and only if $r$ and $\omega$ are related by

$\omega^{2}=\frac{-\kappa^{2}+\kappa\sqrt{\kappa^{2}+4c^{4}m^{2}r^{2\alpha}}}{2m^{2}c^{2}r^{2\alpha+2}}.$

From this condition we deduce that the energy given in (4.2) can be rewritten
as

$\begin{split}E(x,\dot{x})=\dfrac{c^{2}\kappa}{r^{\alpha+2}\omega^{2}}-\dfrac{\kappa}{\alpha
r^{\alpha}}=\kappa\left(\dfrac{2m^{2}c^{4}r^{\alpha}}{-\kappa^{2}+\kappa\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}}}-\dfrac{1}{\alpha
r^{\alpha}}\right)=\kappa\,f(r^{\alpha}),\end{split}$ (4.5)

where

$f(t)=\frac{2m^{2}c^{4}t}{-\kappa^{2}+\kappa\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}}}-\frac{1}{\alpha
t},\quad\forall\ t>0.$

We now study the admissible values of this energy, depending on the radius
$r>0$. It is immediate so see that

$\lim_{t\rightarrow+\infty}f(t)=\frac{mc^{2}}{\kappa}.$ (4.6)

On the other hand, from the Taylor expansion

$\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}}=\kappa\left(1+2\frac{m^{2}c^{4}}{\kappa^{2}}t^{2}-2\frac{m^{4}c^{8}}{\kappa^{4}}t^{4}\right)+o(t^{4}),\quad
t\to 0,$

we deduce

$f(t)=\dfrac{2m^{2}c^{4}t}{2m^{2}c^{4}t^{2}-2\frac{m^{4}c^{8}}{\kappa^{2}}t^{4}+o(t^{4})}-\frac{1}{\alpha
t}=\left(1-\frac{1}{\alpha}\right)\frac{1}{t}+o\left(\frac{1}{t}\right),\quad
t\to 0.$

Therefore, since $\alpha>1$, we obtain

$\lim_{t\rightarrow 0^{+}}f(t)=+\infty.$ (4.7)

As far as the monotonicity of $f$ is concerned, we have

$f^{\prime}(t)=\dfrac{2m^{2}c^{4}\left(-\kappa^{2}+k\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}}\right)-8m^{4}c^{8}\kappa
t^{2}\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}}}{\left(-\kappa^{2}+\kappa\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}}\right)^{2}}+\dfrac{1}{\alpha
t^{2}},$ (4.8)

for every $t>0$. Now, let us define

$g(t)=\sqrt{\kappa^{2}+4m^{2}c^{4}t^{2}},$

for every $t>0$. From (4.8) we infer that

$\displaystyle f^{\prime}(t)$
$\displaystyle=\dfrac{2m^{2}c^{4}[(-\kappa^{2}+\kappa
g(t))g(t)-4m^{2}c^{4}\kappa
t^{2}]}{g(t)(-\kappa^{2}+g(t))^{2}}+\dfrac{1}{\alpha t^{2}}$
$\displaystyle=-\dfrac{2m^{2}c^{4}\kappa^{2}}{g(t)(g(t)-\kappa)}+\dfrac{1}{\alpha
t^{2}}=\dfrac{-2\alpha t^{2}m^{2}c^{4}\kappa^{2}+g(t)(g(t)-\kappa)}{\alpha
t^{2}g(t)(g(t)-\kappa)},$

for every $t>0$. A simple computation shows that

$f^{\prime}(t)\geq 0\quad\Longleftrightarrow\quad\kappa
g(t)\leq\kappa^{2}+2(2-\alpha)m^{2}c^{4}t^{2}\quad\Longleftrightarrow\quad\kappa
g(t)-\kappa^{2}\leq 2(2-\alpha)m^{2}c^{4}t^{2}.$ (4.9)

Now, if $\alpha\geq 2$, observing that $\kappa g(t)-\kappa^{2}>0$, for every
$t>0$, we deduce that $f^{\prime}(t)<0$, for every $t>0$; taking into account
(4.5), (4.6) and (4.7), we can conclude that in this case the range of the
energy is $(mc^{2},+\infty)$. As a consequence, for every
$\mathfrak{E}\in(mc^{2},+\infty)$ there exists $r_{\mathfrak{E}}>0$ such that

$E(x,\dot{x})=\mathfrak{E},$

where $x(t)=r_{\mathfrak{E}}e^{i\omega_{\mathfrak{E}}t}$, with
$\omega_{\mathfrak{E}}$ as in (4.4). This proves the validity of (4.3) for
$\alpha\geq 2$.

On the other hand, if $\alpha\in(1,2)$, from (4.9) we deduce that

$\begin{array}[]{ll}f^{\prime}(t)\geq
0&\displaystyle\Longleftrightarrow\kappa^{2}g^{2}(t)\leq(\kappa^{2}+2(2-\alpha)m^{2}c^{4}t^{2})^{2}\\\
&\\\
&\displaystyle\Longleftrightarrow(2-\alpha)^{2}m^{2}c^{4}t^{2}+(1-\alpha)\kappa^{2}\geq
0.\end{array}$

Hence, we have

$\begin{array}[]{ll}f^{\prime}(t)\geq 0\displaystyle\Longleftrightarrow
t\geq\frac{\kappa}{mc^{2}}\frac{\sqrt{\alpha-1}}{(2-\alpha)}=:t_{\min}.\end{array}$

As a consequence, $f$ has the unique mininum point $t_{\min}$ with
corresponding minimum

$f(t_{\min})=\dfrac{2m^{2}c^{4}t_{\min}}{-\kappa^{2}+\kappa
g(t_{\min})}-\dfrac{1}{\alpha
t_{\min}}=\dfrac{1}{t_{\min}}\left(\dfrac{1}{2-\alpha}-\frac{1}{\alpha}\right)=\dfrac{2mc^{2}}{\kappa}\,\dfrac{\sqrt{\alpha-1}}{\alpha}$

Recalling (4.5), we conclude that the minimum value of the energy is

$2mc^{2}\,\dfrac{\sqrt{\alpha-1}}{\alpha}.$

Arguing as above, taking into account (4.7), we conclude that there exists a
circular solution with energy $\mathfrak{E}$ if and only if
$\mathfrak{E}>2mc^{2}\sqrt{1-\alpha}/\alpha$.

2\. In order to study non-circular periodic solutions, we follow the approach
of [6, Section 2.2] (which deals with the Kepler/Coulomb case $\alpha=1$). Let
us first observe that, according to the discussion of Section 2, equation
(4.1) has a Hamiltonian structure. The Hamiltonian given in (2.3) in this case
is

$H(x,p)=mc^{2}\sqrt{1+\frac{|p|^{2}}{m^{2}c^{2}}}-\dfrac{\kappa}{\alpha|x|^{\alpha}},$

where $p$ is defined in (2.2). We introduce the angular momentum

$\mathcal{L}=\langle
x,Jp\rangle,\quad\hbox{where}\;\,J=\begin{pmatrix}0&\;1\\\ -1&\;0\\\
\end{pmatrix}.$ (4.10)

Now, let us define

$\Upsilon=(0,+\infty)\times\mathbb{T}^{1}\times\mathbb{R}^{2}$

and the symplectic diffeomorphism

$\Psi\colon\Upsilon\to(\mathbb{R}^{2}\setminus\\{0\\})\times\mathbb{R}^{2},\qquad(r,\vartheta,l,\Xi)\mapsto(x,p),$

given by

$x=re^{i\vartheta},\qquad p=le^{i\vartheta}+\dfrac{\Xi}{r}ie^{i\vartheta}.$

In the new variables the Hamiltonian and the angular momentum, which are first
integrals, can be written as

$\mathcal{L}_{0}(r,\vartheta,l,\Xi)=\Xi$

and

$H_{0}(r,\vartheta,l,\Xi)=mc^{2}\sqrt{1+\dfrac{l^{2}+\Xi^{2}/r^{2}}{m^{2}c^{2}}}-\dfrac{\alpha}{r}.$

Hence, every solution $(r,l,\vartheta,\Xi)=(r(t),l(t),\vartheta(t),\Xi(t))$ of
the Hamiltonian system satisfies

$mc^{2}\sqrt{1+\dfrac{l(t)^{2}+L^{2}/(r(t))^{2}}{m^{2}c^{2}}}-\dfrac{\kappa}{\alpha
r(t)}\equiv\mathfrak{E},$

for some $\mathfrak{E},L\in\mathbb{R}$. This relation can be written as

$\ell^{2}=\frac{1}{c^{2}}\left(\frac{\kappa^{2}}{\alpha^{2}r^{2\alpha}}-\frac{c^{2}L^{2}}{r^{2}}+2\mathfrak{E}\frac{\kappa}{\alpha
r^{\alpha}}+\mathfrak{E}^{2}-m^{2}c^{4}\right).$ (4.11)

Let us now fix $L>0$ and a value $\mathfrak{E}\in\mathbb{R}$ of the energy and
define

$\Phi_{\mathfrak{E},L}(r)=\frac{1}{c^{2}}\left(\frac{\kappa^{2}}{\alpha^{2}r^{2\alpha}}-\frac{c^{2}L^{2}}{r^{2}}+2\mathfrak{E}\frac{\kappa}{\alpha
r^{\alpha}}+\mathfrak{E}^{2}-m^{2}c^{4}\right),$

for every $r>0$. According to (4.11), non-constant trajectories in the phase-
plane $(r,\ell)$ are confined in the region $\\{(r,\ell):\
\Phi_{\mathfrak{E},L}(r)\geq 0\\}$. Moreover, closed orbits exist if and only
if $\Phi_{\mathfrak{E},L}$ has two consecutive zeros and it is striclty
positive between them. This cannot occur when $\alpha\geq 2$, as it is clear
by studying the behaviour of the function $\Phi_{\mathfrak{E},L}$.

Indeed, assume first that $\alpha=2$. In this case the function
$\Phi_{\mathfrak{E},L}$ can be written has

$\Phi_{\mathfrak{E},L}(r)=\frac{1}{c^{2}r^{4}}\,P_{2}(r^{2}),\quad r>0,$

where $P_{2}$ is the second order polynomial defined by

$P_{2}(x)=\frac{\kappa^{2}}{\alpha^{2}}-\left(c^{2}L^{2}+2\mathfrak{E}\frac{\kappa}{\alpha}\right)\,x+(\mathfrak{E}^{2}-m^{2}c^{4})\,x^{2},$

for every $x\geq 0$. Since $P_{2}(0)>0$, the polynomial $P_{2}$ (and then
$\Phi_{\mathfrak{E},L}$) cannot have two consecutive positive zeros, being
positive between them. Hence, no bounded non-costant orbits exist.

Now, let us consider the case $\alpha>2$. We first observe that

$\lim_{r\rightarrow
0^{+}}\Phi_{\mathfrak{E},L}(r)=+\infty,\quad\lim_{r\rightarrow+\infty}\Phi_{\mathfrak{E},L}(r)=\dfrac{\mathfrak{E}^{2}-m^{2}c^{4}}{c^{2}}.$
(4.12)

and

$\Phi_{\mathfrak{E},L}^{\prime}(r)=\dfrac{2}{c^{2}r^{2\alpha+1}}\left(c^{2}L^{2}r^{2\alpha-2}-\dfrac{\kappa^{2}}{\alpha}-2\mathfrak{E}\kappa
r^{\alpha}\right)=\dfrac{2}{c^{2}r^{2\alpha+1}}\,\widetilde{\Phi}_{\mathfrak{E},L}(r),$
(4.13)

where $\widetilde{\Phi}_{\mathfrak{E},L}:(0,+\infty)\to\mathbb{R}$ is defined
by

$\widetilde{\Phi}_{\mathfrak{E},L}(r)=c^{2}L^{2}r^{2\alpha-2}-\dfrac{\kappa^{2}}{\alpha}-2\mathfrak{E}\kappa
r^{\alpha},$

for every $r>0$. It is immediate to see that

$\lim_{r\rightarrow
0^{+}}\widetilde{\Phi}_{\mathfrak{E},L}(r)=-\dfrac{\kappa^{2}}{\alpha}\quad\hbox{and}\quad\lim_{r\rightarrow+\infty}\widetilde{\Phi}_{\mathfrak{E},L}(r)=+\infty.$
(4.14)

Moreover, we have

$\widetilde{\Phi}^{\prime}_{\mathfrak{E},L}(r)=2r^{2\alpha-3}\,\left((\alpha-1)c^{2}L^{2}-\alpha\mathfrak{E}\kappa
r^{2-\alpha}\right),$

for every $r>0$. As a consequence, we obtain

$\begin{array}[]{ll}\widetilde{\Phi}^{\prime}_{\mathfrak{E},L}(r)>0,\quad\forall\
r>0&\;\text{ if }\mathfrak{E}\leq 0,\\\
\widetilde{\Phi}^{\prime}_{\mathfrak{E},L}(r)>0\quad\Longleftrightarrow\quad\displaystyle
r>\left(\dfrac{\alpha-1}{\alpha}\dfrac{c^{2}L^{2}}{\mathfrak{E}\kappa}\right)^{1/(2-\alpha)}&\;\text{
if }\mathfrak{E}>0.\end{array}$ (4.15)

Therefore, from (4.14) and (4.15) we deduce that, in both cases,
$\widetilde{\Phi}_{\mathfrak{E},L}$ has exactly one zero. Taking into account
(4.12) and (4.13), we conclude that ${\Phi}_{\mathfrak{E},L}$ has exactly one
critical point, which is a global minimum point. This proves that
${\Phi}_{\mathfrak{E},L}$ cannot have two consecutive zeros being strictly
positive between them. Hence, no bounded non-costant orbits exist. ∎

######  Remark 4.2.

We point out that in the Keplerian/Coulombian case $\alpha=1$ periodic
solutions of energy $\mathfrak{E}$ exist if and only if
$0<\mathfrak{E}<mc^{2}$. Indeed, in [6] it is proved that this condition is
necessary and sufficient for the existence of non-circular periodic solutions.
On the other hand, using the notation in the proof of the first statement of
Proposition 4.1, it can be checked that the function $f$ is increasing, with
$\lim_{t\to 0^{+}}f(t)=0$ and $\lim_{t\to+\infty}f(t)=mc^{2}/\kappa$:
therefore, the existence of circular periodic solutions is guaranteed if and
only if $0<\mathfrak{E}<mc^{2}$. The value $\alpha=1$ is then in some sense a
threshold for the range of admissible energies.

######  Remark 4.3.

We conclude this section with the asymptotic study for $c\to+\infty$ of the
circular solutions of energy $\mathfrak{E}>mc^{2}$ (i.e. $h>0$) whose
existence has been proved in Proposition 4.1. We refer to Remark 3.2 for the
motivation and the general discussion. In the case of the model problem
studied in this section, the limit problem when $c\to+\infty$ is

$m\ddot{x}=-\frac{\kappa x}{|x|^{\alpha+2}}.$

It is well known that circular solutions of energy $h>0$ of this problem exist
if and only if $\alpha>2$ and that in this case their radius $R_{h}$ is given
by

$R^{\alpha}_{h}=\dfrac{\kappa}{h}\,\dfrac{\alpha-2}{2\alpha}.$ (4.16)

On the other hand, we recall from the proof of Proposition 4.1 that the radius
$r=r_{h}(c)$ of these circular solutions satisfies the relation

$\varphi(r,c)=h+mc^{2},$ (4.17)

where $\varphi:(0,+\infty)\times(0,+\infty)\to\mathbb{R}$ is defined by

$\varphi(r,c)=\kappa\left(\dfrac{2m^{2}c^{4}r^{\alpha}}{-\kappa^{2}+\kappa\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}}}-\dfrac{1}{\alpha
r^{\alpha}}\right),\quad\forall\ (r,c)\in(0,+\infty)\times(0,+\infty)$ (4.18)

(cf. (4.5)). More precisely, the discussion in the proof of Proposition 4.1
shows that for every $h>0$ equation (4.17) can be uniquely solved for $r$ as a
function of $c$, and

$\partial_{r}\varphi(r,c)<0,\quad\text{ for every $(r,c)$ such that
}\varphi(r,c)=h+mc^{2}.$ (4.19)

Let us notice, from a different point of view, that the function $r_{h}$ can
be seen as implicitely defined by equation (4.17). The implicit function
theorem implies then that $r_{h}$ is differentiable. A simple computation
proves that

$\displaystyle\partial_{c}\varphi(r,c)$
$\displaystyle=\displaystyle-\dfrac{4mc}{\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}}}\,\dfrac{1}{(-\kappa+\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}})^{2}}\cdot$
$\displaystyle\quad\displaystyle\cdot\Big{(}\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}}\left(2m^{2}c^{4}r^{2\alpha}+2\kappa
mc^{2}r^{\alpha}+\kappa^{2}\right)$
$\displaystyle\quad\displaystyle-4m^{3}c^{6}r^{3\alpha}-4m^{2}c^{4}\kappa
r^{2\alpha}-2mc^{2}\kappa^{2}r^{\alpha}-\kappa^{3}\Big{)},$

for every $(r,c)\in(0,+\infty)\times(0,+\infty)$. It is then easy to see that

$\partial_{c}\varphi(r,c)<0,\quad\forall\
(r,c)\in(0,+\infty)\times(0,+\infty).$ (4.20)

Hence, differentiating (4.17) with respect to $c$ and taking into account
(4.19) and (4.20), we deduce that

$r^{\prime}_{h}(c)=\dfrac{2mc-\partial_{c}\varphi(r_{h}(c),c)}{\partial_{r}\varphi(r_{h}(c),c)}<0,\quad\forall\
c>0.$

Ad a consequence, $r_{h}$ decreases and there exists

$\lim_{c\to+\infty}r_{h}(c)=r_{h}(\infty)\geq 0.$

We now show that $r_{h}(\infty)=R_{h}$, with $R_{h}$ as in (4.16), when
$\alpha>2$ and that $r_{h}(\infty)=0$ when $\alpha\in(1,2]$, thus proving that
for $h>0$ there is convergence to the non-relativistic circular solutions when
$\alpha>2$, while relativistic circular solutions disappear in the non-
relativistic limit when $\alpha\in(1,2]$. To this end, let us observe that,
using (4.18), equation (4.17) can be written as

$\dfrac{2m^{2}c^{4}r^{2\alpha}}{-\kappa+\sqrt{\kappa^{2}+4m^{2}c^{4}r^{2\alpha}}}-mc^{2}r^{\alpha}=hr^{\alpha}\,+\dfrac{\kappa}{\alpha},$

i.e.

$\psi(mc^{2}r^{\alpha})=\Theta(r^{\alpha}),$ (4.21)

where $\psi,\Theta:(0,+\infty)\to\mathbb{R}$ are defined by

$\psi(x)=\dfrac{2x^{2}}{-\kappa+\sqrt{\kappa^{2}+4x^{2}}}-x,\quad\Theta(x)=hx+\dfrac{\kappa}{\alpha},$

for every $x>0$, respectively. It is easy to prove that $\psi$ is a decreasing
function with range $(\kappa/2,\kappa)$.

Now, from (4.21) we deduce that

$\psi(mc^{2}r^{\alpha}_{h}(c))=\Theta(r^{\alpha}_{h}(c)),$ (4.22)

for every $c>1$. Let us notice that if $r_{h}(\infty)>0$ then
$mc^{2}r_{h}(c)\to+\infty$ for $c\to+\infty$; as a consequence, passing to the
limit in (4.22) and recalling the definition of $\Theta$, we obtain

$\dfrac{\kappa}{2}=hr^{\alpha}_{h}(\infty)+\dfrac{\kappa}{\alpha},$

i.e.

$\kappa\left(\dfrac{1}{2}-\dfrac{1}{\alpha}\right)=hr^{\alpha}_{h}(\infty),$
(4.23)

which is impossible if $h>0$ and $\alpha\in(1,2]$. This shows that in the case
$\alpha\in(1,2]$ the limiting radius is $0$. On the other hand, if $\alpha>2$
and $r_{h}(\infty)=0$, then

$\lim_{c\to+\infty}\Theta(r_{h}^{\alpha}(c))=\dfrac{\kappa}{\alpha}<\dfrac{\kappa}{2}.$

Hence, there exists $c_{0}>0$ such that

$\Theta(r_{h}^{\alpha}(c))<\dfrac{\kappa}{2},\quad\forall\ c>c_{0}.$

Thus, for every $c>c_{0}$, the value $\Theta(r_{h}^{\alpha}(c))$ does not
belong to the range of $\psi$, contradicting (4.22). We then conclude that if
$\alpha>2$ necessarily $r_{h}(\infty)>0$ and, as a consequence, (4.23) holds
true. This implies that $r_{h}(\infty)=R_{h}$ and the circular relativistic
solutions converge to the circular classical solution of energy $h>0$.

Data availability statement. Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

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