From 647d192435d36d1155adc8f1a3e56197323bd036 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Wed, 5 Aug 2020 02:54:30 -0400
Subject: Initial commit

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+\documentclass[10pt]{article}
+
+\usepackage{color}
+\usepackage{graphicx}
+\usepackage{amsfonts}
+\usepackage[hidelinks]{hyperref}
+\usepackage{natbib}
+
+\def\Eq#1{\label{#1}}
+
+% colors
+\definecolor{iblue}{RGB}{65,105,225}
+\definecolor{ired}{RGB}{220,20,60}
+\definecolor{igreen}{RGB}{50,205,50}
+\definecolor{ipurple}{RGB}{75,0,130}
+\definecolor{iochre}{RGB}{218,165,32}
+\definecolor{iteal}{RGB}{51,204,204}
+\definecolor{imauve}{RGB}{204,51,153}
+\definecolor{RED}{RGB}{255,0,0}
+
+\def\alertv#1{{\color{green}#1}}
+\def\alertm#1{{\color{magenta}#1}}
+\def\alertb#1{{\color{blue}#1}}
+\def\alertr#1{{\color{red}#1}}
+\def\alertr#1{{\color{RED}#1}}
+\def\alertn#1{{\color{black}#1}}
+
+
+%% symbols
+\let\a=\alpha \let\b=\beta   \let\g=\gamma     \let\d=\delta  \let\e=\varepsilon
+\let\z=\zeta  \let\h=\eta    \let\th=\vartheta \let\k=\kappa  \let\l=\lambda
+\let\m=\mu    \let\n=\nu     \let\x=\xi        \let\p=\pi     \let\r=\rho
+\let\s=\sigma \let\t=\tau    \let\f=\varphi    \let\ph=\varphi\let\ch=\chi
+\let\ch=\chi  \let\ps=\psi   \let\y=\upsilon   \let\o=\omega  \let\si=\varsigma
+\let\G=\Gamma \let\D=\Delta  \let\Th=\Theta    \let\L=\Lambda \let\X=\Xi
+\let\P=\Pi    \let\Si=\Sigma \let\F=\Phi       \let\Ps=\Psi
+\let\O=\Omega \let\Y=\Upsilon
+\def\V#1{{\bf#1}}\def\lhs{{\it l.h.s.}\ }\def\rhs{{\it r.h.s.}\ }
+\def\*{\vskip 3mm}
+\def\0{\noindent}
+\def\be{\begin{equation}}
+\def\ee{\end{equation}}
+\def\bea{\begin{eqnarray}}
+\def\eea{\end{eqnarray}}
+%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
+\def\nn{\nonumber}
+\def\xx{{\V x}}\def\pp{{\V p}}\def\kk{{\V k}}
+\def\AA{{\mathcal A}}
+\def\BB{{\cal B}}\def\CC{{\cal C}}\def\DD{{\mathcal D}}
+\def\EE{{\cal E}}\def\HH{{\cal H}}\def\KK{{\cal K}}\def\LL{{\mathcal L}}
+\def\NN{{\cal N}}\def\FF{{\cal F}}\def\PP{{\mathcal P}}
+\def\QQ{{\mathcal Q}}\def\RR{{\cal R}}\def\TT{{\mathcal T}}
+\let\dpr=\partial\let\fra=\frac
+\def\ie{{\it i.e.}\ }
+\def\eg{{\it e.g.}\ }
+\def\equ#1{(\ref{#1})}
+\def\lis#1{\overline{#1}}
+\def\defi{{\buildrel def\over=}}
+\def\Media#1{{\Big\langle\,#1\,\Big\rangle}}
+\def\media#1{{\Blangle\,#1\,\Brangle}}
+\def\bra#1{{\Blangle#1\Bvert}}\def\ket#1{{\Bvert#1\Brangle}}
+\def\braket#1#2{\Blangle#1\Bvert#2\Brangle}
+\def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,}
+\def\wt#1{\widetilde{#1}}
+\def\wh#1{\widehat{#1}}
+\def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr
+ \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle
+ #1}$\hskip3.pt\crcr}}\,}
+\def\ie{{\it i.e.\ }}\def\etc{{\it etc.\ }}
+\def\FF{{\mathcal F}}
+\def\tto{\Rightarrow}
+\def\ap{{\it a priori}}
+\def\Ba   {{\mbox{\boldmath$ \alpha$}}}
+\def\Bb   {{\mbox{\boldmath$ \beta$}}}
+\def\Bg   {{\mbox{\boldmath$ \gamma$}}}
+\def\Bd   {{\mbox{\boldmath$ \delta$}}}
+\def\Be   {{\mbox{\boldmath$ \varepsilon$}}}
+\def\Bee  {{\mbox{\boldmath$ \epsilon$}}}
+\def\Bz   {{\mbox{\boldmath$\zeta$}}}
+\def\Bh   {{\mbox{\boldmath$ \eta$}}}
+\def\Bthh {{\mbox{\boldmath$ \theta$}}}
+\def\Bth  {{\mbox{\boldmath$ \vartheta$}}}
+\def\Bi   {{\mbox{\boldmath$ \iota$}}}
+\def\Bk   {{\mbox{\boldmath$ \kappa$}}}
+\def\Bl   {{\mbox{\boldmath$ \lambda$}}}
+\def\Bm   {{\mbox{\boldmath$ \mu$}}}
+\def\Bn   {{\mbox{\boldmath$ \nu$}}}
+\def\Bx   {{\mbox{\boldmath$ \xi$}}}
+\def\Bom  {{\mbox{\boldmath$ \omega$}}}
+\def\Bp   {{\mbox{\boldmath$ \pi$}}}
+\def\Br   {{\mbox{\boldmath$ \rho$}}}
+\def\Bro  {{\mbox{\boldmath$ \varrho$}}}
+\def\Bs   {{\mbox{\boldmath$ \sigma$}}}
+\def\Bsi  {{\mbox{\boldmath$ \varsigma$}}}
+\def\Bt   {{\mbox{\boldmath$ \tau$}}}
+\def\Bu   {{\mbox{\boldmath$ \upsilon$}}}
+\def\Bf   {{\mbox{\boldmath$ \phi$}}}
+\def\Bff  {{\mbox{\boldmath$ \varphi$}}}
+\def\Bch  {{\mbox{\boldmath$ \chi$}}}
+\def\Bps  {{\mbox{\boldmath$ \psi$}}}
+\def\Bo   {{\mbox{\boldmath$ \omega$}}}
+\def\Bome {{\mbox{\boldmath$ \varomega$}}}
+\def\BG   {{\mbox{\boldmath$ \Gamma$}}}
+\def\BD   {{\mbox{\boldmath$ \Delta$}}}
+\def\BTh  {{\mbox{\boldmath$ \Theta$}}}
+\def\BL   {{\mbox{\boldmath$ \Lambda$}}}
+\def\BX   {{\mbox{\boldmath$ \Xi$}}}
+\def\BP   {{\mbox{\boldmath$ \Pi$}}}
+\def\BS   {{\mbox{\boldmath$ \Sigma$}}}
+\def\BU   {{\mbox{\boldmath$ \Upsilon$}}}
+\def\BF   {{\mbox{\boldmath$ \Phi$}}}
+\def\BPs  {{\mbox{\boldmath$ \Psi$}}}
+\def\BO   {{\mbox{\boldmath$ \Omega$}}}
+\def\BDpr {{\mbox{\boldmath$ \partial$}}}
+\def\Bstl {{\mbox{\boldmath$ *$}}}
+\def\Brangle {{\mbox{\boldmath$ \rangle$}}}
+\def\Blangle {{\mbox{\boldmath$ \langle$}}}
+\def\Bvert{{\mbox{\boldmath$|$}}}
+\def\Bell   {{\mbox{\boldmath$\ell$}}}
+\let\up\uparrow
+\let\down\downarrow
+\def\({\left(}
+\def\){\right)}
+\let\mc\mathcal
+\let\mrm\mathrm
+
+
+\def\iniz{\setcounter{equation}{0}}
+\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
+\def\inizA{\setcounter{equation}{0}
+\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newdimen\xshift \newdimen\xwidth \newdimen\yshift \newdimen\ywidth%
+\def\ins#1#2#3{\vbox to0pt{\kern-#2pt\hbox{\kern#1pt #3}\vss}\nointerlineskip}
+
+\def\eqfig#1#2#3#4#5{
+\par\xwidth=#1pt \xshift=\hsize \advance\xshift
+by-\xwidth \divide\xshift by 2
+\yshift=#2pt \divide\yshift by 2%
+{\hglue\xshift \vbox to #2pt{\vfil
+#3 \special{psfile=#4.eps}
+}\hfill\raise\yshift\hbox{#5}}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\def\Eqfig#1#2#3#4#5#6{
+\par\xwidth=#1pt \xshift=\hsize \advance\xshift
+by-\xwidth \divide\xshift by 2
+\yshift=#2pt \divide\yshift by 2%
+{\hglue\xshift \vbox to #2pt{\vfil
+#3 \special{psfile=#4.eps}\kern200pt\special{psfile=#5.eps}
+}\hfill\raise\yshift\hbox{#6}}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\eqalign#1{\null\,\vcenter{\openup\jot
+  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
+      \crcr#1\crcr}}\,}
+
+\def\qedsymbol{$\square$}
+\def\qed{\penalty10000\hfill\penalty10000\qedsymbol}
+
+\date{}
+\author{\alertb{Giovanni Gallavotti${}^1$ and
+Ian Jauslin${}^2$}}
+
+\title{\alertr{\bf A Theorem on Ellipses, an Integrable System and a Theorem
+    of Boltzmann}
+}
+
+\begin{document}
+\maketitle
+\kern-8mm
+\centerline{${}^1$ INFN-Roma1 \& Universit\`a ``La Sapienza'', email: giovanni.gallavotti@roma1.infn.it}
+\centerline{${}^2$ Department of Physics, Princeton University, email: ijauslin@princeton.edu}
+
+
+%\kern-1cm
+\begin{abstract}
+%
+We study a mechanical system that was considered by Boltzmann in 1868 in the
+context of the derivation of the canonical and microcanonical ensembles. This
+system was introduced as an example of ergodic dynamics, which was central to
+Boltzmann's derivation. It consists of a single particle in two dimensions,
+which is subjected to a gravitational attraction to a fixed center. In
+addition, an infinite plane is fixed at some finite distance from the center,
+which acts as a hard wall on which the particle collides elastically. Finally,
+an extra centrifugal force is added. We will show that, in the absence of this
+extra centrifugal force, there are two independent integrals of motion.
+Therefore the extra centrifugal force is necessary for Boltzmann's claim of
+ergodicity to hold.
+%
+\end{abstract}
+\*
+%
+\0Keywords: {\small Ergodicity, Chaotic hypothesis, Gibbs
+distributions, Boltzmann, Integrable systems}
+\*
+%\end{document}
+
+In 1868, \cite{Bo868a} laid the foundations for our modern understanding of the
+behavior of many-particle systems by introducing the ``microcanonical
+ensemble'' (for more details on this history, see \cite{Ga016}). The principal
+idea behind this ensemble is that one can achieve a good understanding of
+many-particle systems by focusing not on the dynamics of each individual
+particle, but on the statistical properties of the whole. More precisely, the
+state of the system becomes a random variable, chosen according to a
+probability distribution on phase space, which came to be called the
+``microcanonical ensemble''. An important assumption that was made implicitly
+by Boltzmann is that the dynamics of the system be ergodic. In this case,
+time-averages of the dynamics can be rewritten as averages over phase space,
+and the qualitative properties of the dynamics can be formulated as statistical
+properties of the microcanonical ensemble.
+
+To support this assumption, Boltzmann presented a mechanical system that
+very same year (\cite{Bo868b}) as an example of an ergodic system. This
+system consists of a particle in two dimensions that is attracted to a
+fixed center via a gravitational potential $-\frac{\alpha}{2r}$. In
+addition, he added an extra centrifugal potential $\frac g{2r^2}$. As was
+known since at least the times of Kepler, this system is subjected to a
+central force, and is therefore integrable. In order to break the
+integrability, Boltzmann added an extra ingredient: a rigid infinite planar
+wall, located a finite distance away from the center (see figure
+\ref{trajectory}). Whenever the particle hits the wall, it undergoes an elastic
+collision and is reflected back. Boltzmann's argument was, roughly, that in
+the absence of the wall, the dynamics is quasi-periodic, so the particle
+should intersect the plane of the wall at points which should fill up a
+segment of the wall densely as the dynamics evolves, and concluded that the
+region of phase space in which the energy is constant must also be filled
+densely. As we will show, this is not the whole story; following a
+conjectured integrability for $g=0$, \cite[p.150]{Ga013b},
+and first tests
+%(by (GG) proposing a relation with KAM theory for $g>0$ and by (IJ) proposing chaotic motions at large $g$)
+in
+\cite[p.225--228]{Ga016}, we have found that, in the absence of the
+centrifugal term $g=0$, the dynamics (which has two degrees of freedom) still admits
+two constants of motion even in presence of the hard wall.  This
+suggests that, if a suitable KAM analysis could be carried out, the system
+would not be ergodic for small values of $g$.
+
+\begin{figure}[ht]
+\hfil\includegraphics[width=6cm]{trajectory.pdf}
+
+\caption{A trajectory. The large dot is the attraction center $O$, and the line
+  is the hard wall $\LL$. In between collisions, the trajectories are ellipses.
+  The ellipses are drawn in full, but the part that is not covered by the
+  particle is dashed.
+  }
+\label{trajectory}
+\end{figure}
+
+
+\setcounter{section}{0}
+\section{Definition of the model and main result}
+\label{sec1}
+\iniz
+
+
+Let us now specify the model formally, and state our main result more
+precisely. We fix the gravitational center to the origin of the $x,y$-plane and
+let $\LL$ be the line $y=h$. The Hamiltonian for the system in between
+collisions is
+%
+\be H=\frac{p_x^2+p_y^2}2 -\frac{\a}{2r}+\frac{g}{2r^2}
+\Eq{e1.1}\ee
+%
+where $\a>0,g\ge0,r=\sqrt{x^2+y^2}$ and the particle moves following
+Hamilton's equations as long as it stays away from the obstacle $\LL$. When
+an encounter with $\LL$ occurs the particle is reflected elastically and
+continues on.
+
+\cite{Bo868b}, considered this system on the hyper-surface $A={\V
+p}^2-\frac\a r+\frac{g}{r^2}$. The intersection of this hyper-surface with
+$y=h$ is the region $\FF_A$ enclosed within the curves
+%
+\be \pm\sqrt{(A-\frac{g}{x^2+h^2} +\frac{\a}{\sqrt{x^2+h^2}})},\qquad
+x_{min}<x<x_{max}\Eq{e1.2}\ee
+%
+with $x_{min}$ and $x_{\max}$ the roots of
+$A=\frac{g}{x^2+h^2} -\frac{\a}{\sqrt{x^2+h^2}}$. He argued that
+all motions (with few exceptions) would cover densely the surfaces of
+constant $A<0$ if $\a,g>0$. 
+\bigskip
+
+From now on, unless it is explicitly stated otherwise, we will assume that
+$g=0$.
+\bigskip
+
+In this case, the motion between collisions takes place at constant
+energy $\frac12A$ and constant angular momentum $a$, and traces out an
+ellipse. One of the foci of the ellipse is located at the origin, and we
+will denote the angle that the aphelion of the ellipse makes with the
+$x$-axis by $\theta_0$. Thus, the ellipse is entirely determined by the triplet
+$(A,a,\theta_0)$. When a collision occurs, $A$ remains unchanged, but $a$ and
+$\theta_0$ change discontinuously to values $(a',\theta_0')=\FF(a,\theta_0)$, and thus
+the Kepler ellipse of the trajectory changes. In addition, the semi-major
+axis $a_M$ of the ellipse is also fixed to $a_M=-\frac\a{2A}$ (Kepler's
+law): so the successive ellipses have the same semi-major axis, while the
+eccentricity varies because at each collision the angular momentum changes:
+$e^2=1+ \frac{4 A a^2}{\a^2}$. Thus, the motion will take place on arcs of
+various ellipses $\EE$, which all share the same focus and the same semi-major
+axis, but whose angle and eccentricity changes at each collision.
+
+Our main result is that the (canonical) map $(a',\f'_0)=\FF(a,\theta_0)$, which
+maps the angular momentum and angle of the aphelion before a collision to their
+values after the collision, admits a constant of motion. This follows from the
+following geometric lemma about ellipses.
+\bigskip
+
+\0{\bf Lemma 1:} {\it Given an ellipse $\mathcal E$ with a focus at $O$ that
+intersects $\LL$ at a point $P$. Let $Q$ denote the orthogonal projection of
+$O$ onto $\LL$ (see figure \ref{fig1}).  The distance $R_0$ between $Q$ and
+the center of $\mathcal E$ depends solely on the semi-major axis $a_M$, the
+distance $r$ from $O$ to $P$, and $\cos(2\lambda)$ where $\lambda$ is the angle
+between the tangent of the ellipse at $P$ and $\LL$ (to define the direction of
+the tangent, we parametrize the ellipse in the counter-clockwise direction):
+\be
+R_0=\sqrt{\frac14 r^2+\frac14(2a_M-r)^2 +\frac12 r (2a_M-r)\cos(2\l)}
+.
+\Eq{e1.3}\ee
+}
+\*
+
+\begin{figure}[ht]
+\hfil\includegraphics[width=100pt]{fig1.pdf}
+
+\caption{The attractive center is $O$, hence it is the focus of the
+  ellipse in absence of centrifugal force $g=0$. $Q$ is the projection of
+  $O$ on the line $\LL$ and $P$ is a collision point. The arrow
+  represents the velocity of the particle after the collision.}
+\label{fig1}
+\end{figure}
+
+\underline{Proof}: We switch to polar coordinates
+$p=(r\cos\f,r\sin\f)$.
+
+Let $O'$ denote the other focus of the ellipse, and $C$ denote its center. The
+first step is to compute the vector $\protect\overrightarrow{O'P}$, which in polar
+coordinates is
+\be
+\protect\overrightarrow{O'P}=((2a_M-r)\cos\f',(2a_M-r)\sin\f')\Eq{e1.4}\ee
+%
+Let $\psi:=\pi+\f-\lambda$ denote the angle between the tangent of the
+ellipse at $P$ and the vector $\protect\overrightarrow{PO}$ (see figure \ref{ellipse}),
+and $\psi':=\pi+\f'-\lambda$ denote the angle between the tangent of the
+ellipse at $P$ and the vector $\protect\overrightarrow{PO'}$.  
+
+\begin{figure}[ht]
+  \hfil\includegraphics[width=8cm]{ellipse.pdf}
+
+\caption{An ellipse with foci $O$ and $O'$ and center $C$. The thick line is
+$\LL$, which intersects the ellipse at $P$, and $Q$ is the projection of $O$
+onto $\LL$. The dashed line is the tangent at $P$. $\lambda$ is the angle
+between $\LL$ and the tangent, $\f$ is the polar coordinate, $\f'$ is the angle
+between $\LL$ and $\protect\overrightarrow{O'P}$. $\psi$ is the angle between the
+tangent and $\protect\overrightarrow{PO}$, which is equal to the angle between the
+tangent and $\protect\overrightarrow{PO'}$. $R_0$ is the distance between $Q$ and $C$.}
+\label{ellipse}
+\end{figure}
+
+By the focus-to-focus
+reflection property of ellipses, we have $\psi'=\pi-\psi$. Thus
+$\f'=2\lambda-\pi-\f$ and we find;
+
+
+\begin{figure}[ht]
+  \hfil\includegraphics[width=8cm]{fig2.pdf}
+
+\caption{Two ellipses, before and after a collision. The collision line $\LL$
+is the line at $y=1$, $P$ is the collision point; $Q$ is the projection of $O$
+onto $\LL$; the two ellipses $\EE$ and $\EE'$ have a common focus $O$, and
+$O,O'$ are the foci of $\EE$, whereas $O,O''$ are the foci of $\EE'$; $C$ and
+$C''$ are the centers of $\EE$ and $\EE'$ respectively; the ellipses are drawn
+completely although the trajectory is restricted to the parts above $y=h=1$.
+The distance from $C''$ to $Q$ is the same as that from $C$ to $Q$.  The upper
+ellipse $\EE$ contains the trajectory that starts at the collision point $P$
+following the other ellipse $\EE'$ which has undergone reflection.}
+\label{fig2}
+\end{figure}
+
+\be
+  R_0^2=|Q-C|^2
+  =
+  \frac14\left(r^2+(2a_M-r)^2+2r(2a_M-r)\cos(2\lambda)\right)
+  .
+\Eq{e1.5}\ee
+See figures \ref{ellipse} and \ref{fig2}.\qed
+
+\*
+\0{\bf Theorem 1}: {\it The quantity
+%
+\be R= a^2+h\a e \sin\theta_0\Eq{e1.6}
+\equiv\frac\alpha{2a_M}(h^2+a_M^2-R_0^2)
+\Eq{e1.7}\ee
+%
+where $e$ is the eccentricity $e=\sqrt{1+\frac{4 A a^2}{\a^2}}$, is a constant
+of motion.}
+\*
+
+\underline{Proof}:
+During a collision, the value of $\l$ changes from $\l$ to $\p-\l$, while
+$r$ and $a_M$ stay the same. By lemma 1, this implies that the distance $R_0$
+between $Q$ and the center of the ellipse is preserved during a collision.
+Furthermore, the position of the center $C$ of the ellipse is given by
+$C=a_Me(\cos\theta_0,\sin\theta_0)$
+so
+\be
+  R_0^2=|Q-C|^2=a_M^2e^2-2a_Meh\sin\theta_0+h^2.\Eq{e1.8}
+\ee
+Furthermore, the angular momentum is equal to
+$a^2=\frac12a_M\alpha(1-e^2)$ 
+so
+\be
+  -R_0^2+h^2+a_M^2
+  =
+  \frac{2a_M}\alpha(a^2+e\alpha h\sin\theta_0)\Eq{e1.9}
+\ee
+is a conserved quantity.  \qed
+\bigskip
+
+\0{\bf Remark:} Some useful inequalities are
+%
+\be
+\eqalign{
+    &r_{max}<{2}{a_M}; \ x_{max}=\sqrt{r_{max}^2-h^2};\
+      R_0^2\in ((a_M-r)^2,a_M^2);\cr
+      &\frac{\a h^2}{2a_M}\,<\,R\,<\,
+      (1+\frac{a_M^2}{h^2}-\Big(\frac{a_M}{h}-
+      \frac{r}h\Big)^2)\frac{\a h^2}{2a_M}\cr}
+\Eq{e1.10}\ee
+%
+hence in the plane $(x,\l)$ the rectangle $(-x_{max},x_{max})\times(0,\p)$
+(recall that $x_{max}$ is the largest $x$ accessible at energy $\frac12A$)
+is the surface of energy $\frac12A$ and the trajectories are the curves of
+constant $R$ inside this rectangle.
+
+\def\SEC{Conjectures on action angle variables}
+\section{\SEC}
+\label{sec2}
+\iniz
+
+In the previous section, we exhibited a constant of motion, which, along
+with the conservation of energy, brings the number of independent conserved
+quantities to two. In a continuous Hamiltonian system, this would imply the
+existence of action-angle variables, which are canonically conjugate to the
+position and momentum of the particle, in terms of which the dynamics
+reduces to a linear evolution on a torus. In this case, the collision
+with the wall introduces some discreteness into the problem, and the
+existence of the action angle variables is not guaranteed by standard
+theorems. Indeed, in the presence of the collisions, we no longer have a
+Hamiltonian system, but rather a discrete symplectic map (or a
+non-differentiable Hamiltonian), which describes the change in the state of
+the particle during a collision. In this section, we present some
+conjectures pertaining to the existence of action angle variables for this
+problem.
+\bigskip
+
+The first step is to change to variables which are action-angle variables for
+the motion in between collision. We choose the {\it Delaunay} variables, whose
+angles are the argument of the aphelion $\theta_0$ defined above, the {\it mean
+anomaly} $M$, and whose actions are the angular momentum $a$, and another
+momentum usually denoted by $L$ and related to the semi-major axis $a_M$ and
+to the energy $E=\frac12 A$:
+\be L:=-\sqrt{\frac{\alpha}2a_M},\quad a_M:=-\frac\alpha{2A}
+,\quad
+A:=p^2+\frac{a^2}{r^2}-\frac\alpha{r}\equiv-\frac{\alpha^2}{4L^2}
+\Eq{e2.1}\ee
+It is well known that this change of variables is canonical. In between
+collisions, the dynamics of the particle in the variables
+$(M,\theta_0;L,a)$ is, simply,
+\be
+  \dot M=\frac{\alpha^2}{4L^3}
+  ,\quad
+  \dot\theta_0=0
+  ,\quad
+  \dot L=0
+  ,\quad
+  \dot a=0
+  .\Eq{e2.2}
+\ee
+These variables are thus action-angle variables in between collisions, but when
+a collision occurs, $\theta_0$ and $a$ will change.
+\bigskip
+
+The following conjecture states that there exists an action-angle variable
+during the collisions.
+\bigskip
+
+\0{\bf Conjecture 1:} {\it There exists a variable $\gamma$ and an integer
+  $k$ such that, every $k$ collisions, the change in $\gamma$ is
+  \begin{equation}
+    \gamma'=\gamma+\o(L,R)\Eq{e2.3}
+  \end{equation}
+  in which case $\gamma$ is an angle that rotates on a circle of radius depending
+  on $L,R$. The function $\o(L,R)$ has a non zero derivative with respect to
+  $R$ at constant $L$, \ie the motion on the energy surface is quasi periodic
+  and anisochronous.}
+\*
+
+We will now sketch a construction of this variable $\gamma$, which we obtain
+using a generating function $F(L,R,M,\theta_0)$.
+\bigskip
+
+First of all, by theorem 1, the angular momentum $a(\theta_0)$ is a solution of
+\begin{equation}
+  a^2=R-h\a\sin\theta_0\sqrt{1-\fra{a^2}{L^2}}\Eq{e2.4}
+\end{equation}
+that is, if $\e=\pm$,
+%
+\be
+a^2=R-\frac{h^2\alpha^2}{2L^2}\sin^2\theta_0+
+\e\sqrt{\frac{h^4\alpha^2}{4L^4}\sin^4\theta_0+h^2\a^2\sin^2\theta_0-
+\frac{R\alpha^2h^2}{L^2}\sin^2\theta_0}\Eq{e2.5}\ee
+%
+and $a=\h\sqrt{a^2}$, so that there may be four possibilities for the value of
+$a$ denoted $a=a_{\e,\h}(\theta_0,R,L)$ with $\e=\pm,\h=\pm$. The choice of the
+signs $\e=\pm1$, and $\h$ must be examined carefully.
+\bigskip
+
+We then define the generating function
+\be F(L,R,M,\theta_0)=LM+\int_0^{\theta_0} a(L,R,\ps)d\ps
+%-\int_0^{2\pi} a(L,R,\ps)d\ps
+\Eq{e2.6}\ee
+%
+which yields the following canonical transformation:
+
+\be\eqalign{
+  \g=&\dpr_R\int^{\theta_0}_0 a_{\e,\h}(L,R,\ps)\,d\ps
+  %-\dpr_R\int^{2\pi}_0 a_{\e,\h}(L,R,\ps)\,d\ps
+  \cr
+  M'=&M+\dpr_L\int^{\theta_0}_0 a_{\e,\h}(L,R,\ps)\,d\ps
+  %-\dpr_L\int^{2\pi}_0 a_{\e,\h}(L,R,\ps)\,d\ps
+  \cr
+}\Eq{e2.7}\ee
+%
+It is natural, if Boltzmann's system is integrable (at $g=0$), that the new
+variables are its action angle variables and $M',\gamma$ rotate uniformly in spite of
+the collisions.
+\bigskip
+
+However, in this case, the signs $\e$ and $\h$ may change from one collision to
+the next, complicating the situation. A careful numerical study of the system
+has led us to the following conjecture (see figure \ref{action_angle}).
+\bigskip
+
+%
+\0{\bf Conjecture 2:} {\it If $R>h\a$ (which is the case in which the
+  circle, of radius $R_0$, of the centers encloses the focus $O$), when the
+  motion collides for the $n$-th time, the angular momentum is proportional
+  to $(-1)^n$, and, thus, $\epsilon=(-1)^n$. The sign $\eta$ is fixed to
+  $+$. The increment $\Delta_2\gamma$ in $\gamma$ between the $n$-th and
+  the $n+2$-th collision is independent of $n$.  } \*
+\bigskip
+
+\begin{figure}
+  \hfil\includegraphics[width=8cm]{action-angle.pdf}
+  \caption{A plot of the increment in $\gamma$ between the $n$-th and the $n+2$-nd collision as a function of $n$. The blue `+' signs correspond to even $n$, and the red `$\times$' to odd $n$. The variation of $\Delta_2\gamma$ is as small as 1 part per million, thus supporting conjecture 2.}
+  \label{action_angle}
+\end{figure}
+
+\0{\it Remark:} The change of variables over the variables $a,\theta_0$ to
+$R,\g$ at fixed $L$ is {\it remarkably} essentially the same as the one
+(\ap\ unrelated) to find action-angle variable for the auxiliary
+Hamiltonian $R=R(a,\theta_0)$. This might remain true even when $R<h\a$:
+interpretable as a kind of auxiliary pendulum motion.  \*
+
+At the time of publication, it has been
+brought to our attention that G. Felder has proved that the orbits are all
+either periodic or quasi-periodic, which would be implied from conjecture
+1.
+
+\section{Conclusion and outlook}
+In this brief note, we have shown that the system considered by Boltzmann in
+1868, in the case $g=0$, admits two independent constants of motion.
+This indicates that it should be possible to compute action
+angle variables for this system, which is not entirely trivial because of the
+discontinuous nature of the collision process. If such a construction could be
+brought to its conclusion, then it would show that the trajectories are either
+periodic or quasi-periodic, a fact which is consistent with the numerical
+simulations we have run.
+
+This is not a contradiction of Boltzmann's claim that this model is ergodic,
+since Boltzmann considered the model at $g\neq 0$. However, we expect that a
+KAM-type argument can be set up for this model, to show that the system cannot
+be ergodic, even if $g>0$, provided $g$ is sufficiently small. However it
+may still have invariant regions of positive volume where the motion is ergodic. 
+
+\*
+
+
+\0{\bf Acknowledgements}: The authors thank G. Felder for giving us the impetus
+to write this note up in its current form, and to publish it. I.J. gratefully
+acknowledges support from NSF grants 31128155 and 1802170.
+
+\bibliographystyle{plainnat}
+%\bibliographystyle{alpha}
+%\bibliographystyle{amsref}
+%\bibliographystyle{apsrmp}
+%\bibliographystyle{spmpsci}
+%\bibliographystyle{annotate}
+%\bibliography{0Bib}
+\begin{thebibliography}{3}
+\providecommand{\natexlab}[1]{#1}
+\providecommand{\url}[1]{\texttt{#1}}
+\expandafter\ifx\csname urlstyle\endcsname\relax
+  \providecommand{\doi}[1]{doi: #1}\else
+  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
+
+\bibitem[Boltzmann(1868a)]{Bo868a}
+L.~Boltzmann.
+\newblock Studien \"uber das Gleichgewicht der lebendingen Kraft zwischen bewegten materiellen Punkten.
+\newblock \emph{Wiener Berichte}, {\bf 58}, 517--560, (49--96), 1868.
+
+\bibitem[Boltzmann(1868b)]{Bo868b}
+L.~Boltzmann.
+\newblock L{\"o}sung eines mechanischen problems.
+\newblock \emph{Wiener Berichte}, {\bf 58}, (W.A.,\#6):\penalty0 1035--1044,
+  (97--105), 1868.
+
+\bibitem[Gallavotti(2014)]{Ga013b}
+G.~Gallavotti.
+\newblock \emph{Nonequilibrium and irreversibility}.
+\newblock Theoretical and Mathematical Physics. Springer-Verlag, 2014.
+
+\bibitem[Gallavotti(2016)]{Ga016}
+G.~Gallavotti.
+\newblock Ergodicity: a historical perspective. equilibrium and nonequilibrium.
+\newblock \emph{European Physics Journal H}, {\bf 41}, 181--259, 2016.
+\newblock \doi{DOI: 10.1140/epjh/e2016-70030-8}.
+
+\end{thebibliography}
+
+\*\*
+\end{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
+
+%plot "lambdak-0.25-3.4--0.3333" u 1:2 every 2::1:::2000
+% plot 'file' every {<point_incr>}
+%                           {:{<block_incr>}
+%                             {:{<start_point>}
+%                               {:{<start_block>}
+%                                 {:{<end_point>}
+%                                   {:<end_block>}}}}}
+%plot "grafk-0.326753-2--0.3333" u 3:4 every 2:1:0:0:5:0 w l  
+%plot "gammak-0.3-1--0.2" u 1:3 every 2:1:0:0:2:0 w l  
+
+% Syntax:
+%       plot 'file' every {<point_incr>}
+%                           {:{<block_incr>}
+%                             {:{<start_point>}
+%                               {:{<start_block>}
+%                                 {:{<end_point>}
+%                                   {:<end_block>}}}}}
+%
+%                           every 3:1:2::1024:1
+%                           2 significa che inizia dal #3 
-- 
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