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diff --git a/Gallavotti_Jauslin_2020.tex b/Gallavotti_Jauslin_2020.tex new file mode 100644 index 0000000..3d00ce2 --- /dev/null +++ b/Gallavotti_Jauslin_2020.tex @@ -0,0 +1,666 @@ +\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 |