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\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}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 $R0$, 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 {} % {:{} % {:{} % {:{} % {:{} % {:}}}}} %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 {} % {:{} % {:{} % {:{} % {:{} % {:}}}}} % % every 3:1:2::1024:1 % 2 significa che inizia dal #3