\documentstyle[pre,aps,preprint]{revtex}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\begin{document}
\draft
\title{Surfing Arnold's Web}
\author{Rupak Chatterjee and A. D. Jackson}
\address{
Department of Physics, State University of New York at Stony Brook,
Stony Brook, New York 11794-3800, USA}
\date{\today}
\maketitle
\centerline{\em Dedicated to Gerald E. Brown on the Occasion of his
Seventieth Birthday}
\begin{abstract}
The free motion of a rigid one-dimensional stick colliding elastically
within an infinitely massive circular wall is first shown to be
equivalent to the three-dimensional motion of a billiard ball within a
spiral column and then mapped onto a two-dimensional billiard problem
with a rotating billiard wall. Indications that such a system has
chaotic orbits and can possess integrable orbits is provided
through the use of projected Poincar\'{e} sections. When chaotic and
integrable orbits co-exist, the chaotic trajectories appear in the form
of Arnold's web. We also consider the limit of a stick of zero length
in which the system becomes integrable.
\end{abstract}
\pacs{PACS number: 05.45.+b}
\section{Introduction}
\label{Intro}
The last ten years of research in classical and quantum chaos
has rekindled interest in a variety of systems in classical and
quantum mechanics which had previously been regarded as elementary.
Complexity in these two theories has often been
associated with large numbers of interacting particles while
simplicity or integrability is connected to systems with small
numbers of weakly interacting particles. This philosophy has,
in turn, defined the traditional scope of statistical mechanics
as being the realm of many-body systems for which the use of such concepts
as the postulate of equal {\em a priori\/} probability, thermodynamic
equilibrium, and the ergodic hypothesis are natural and useful. However,
the ergodic hypothesis, for instance, is not limited to complex systems
but appears in integrable systems such as the simple pendulum which is ergodic
on its energy manifold.
With the introduction of Sinai's billiard in the early seventies \cite{sin},
it was realized that even a non-interacting one particle system in a closed
two-dimensional domain can exhibit many features which are more frequently
encountered in statistical mechanical systems \cite{bun}, \cite{ben}.
In fact, Sinai's interest in this simple billiard model was motivated by
his desire to prove certain ergodic properties of a hard sphere Bose gas.
The traditional billiard problem consists of a point particle
moving in a rectilinear manner and undergoing specular reflection
at the billiard wall. Depending upon the choice of the boundary wall,
the system may be either integrable, KAM-like, or chaotic (K-flow,
C-flow, or Bernoulli). One cannot use initial conditions of finite accuracy
to predict the long-term description of chaotic billiard systems any more
than one can follow the motion of individual particles in a statistical
mechanical system. Rather, physicists turn to probabilistic
concepts such as entropy (thermodynamic
entropy in statistical mechanics or the Kolmorogov-Sinai entropy
for billiards) in order to describe the complex behavior
found in these systems.
Recently, it was realized that the free motion of any rigid body in a
confined domain is equivalent to a well-defined problem of point particle
motion \cite{bcj}, \cite{cjb}. This generalized class of billiard problems
has provided a simple arena for exploring chaos in dynamical systems.
While their immediate physical origin renders these intuitively appealing
systems appealing intuitive, they have been of specific value, {\em e.g.},
in proving the long-standing conjecture that the tossing of a coin
is a (completely chaotic) Bernoulli system \cite{bcj}. The equivalence of
rigid body motion and billiards is of considerable value in the analysis
of numerical simulations using the Poincar{\'e} surface of section technique.
The relative abundance of analytic results available for billiards can
simplify the rigorous examination of rigid body systems
significantly.\footnote{For example, the demonstration that coin tossing is a
Bernoulli system is made elementary once it is recognized that this corresponds
to a billiard which is everywhere convex.} Further, the appearance of chaotic
or integrable orbits can be adjusted with easily tunable parameters (such as
the shape of the boundary wall). This makes billiards appealing dynamical
systems to study.
In this paper, we continue our investigation of billiards
equivalent to rigid body motion. Specifically, we investigate the
free motion of a one-dimensional stick making elastic collisions within
a circle. We shall show that such a system is equivalent to a
three-dimensional billiard problem of a point particle moving inside a
spiral column. (This equivalence is more general and holds for arbitrary
two-dimensional rigid bodies and confining shapes.) The specific example of
a stick inside a circle considered here is further equivalent to a
two-dimensional billiard with a freely-rotating boundary wall. This
problem differs from those investigated previously in two
main aspects. First, the stick here is placed within a closed domain
of constant non-zero curvature as opposed to the earlier papers
where the stick bounced between two infinite lines or infinite planes.
Second, the appearance of a rotating billiard wall is a new
phenomena not previously encountered. Our earlier extension
of the stick problem to three dimensions in \cite{cjb}
led to an interacting billiard system. Again, a relatively
simple change of the original rigid body problem produces a new and interesting
features in the mapped billiard system. Here, the almost surgical separation
of rectilinear and rotational motion offers new ways of thinking about the old
problem of rigid body motion.
We begin this paper with a brief review of earlier results on the billiard
problems related to the motion of a stick in two and three dimensions.
In section \ref{spiral}, we introduce the spiral column billiard, and in
section \ref{rot}, we describe the equivalent two-dimensional rotating
billiard system. We shall present a variety of numerical results. These
will illustrate that this system displays KAM-type motion ({\em i.e.}, a
mixture of chaotic and integrable orbits) when the length of the stick is
larger than the radius of the circle. Shorter sticks appear to lead to
resonance islands and motion which seems to be more chaotic. In general,
the chaotic trajectories exhibit the phenomenon of Arnold diffusion.
We discuss the limiting case of an integrable circular billiard obtained
for a stick of length zero and an associated limiting problem in which the
inertial parameter of the wall is allowed to become infinite. Finally,
some suggestions for future work are mentioned in section
\ref{concl}.
\section{What's the Stick?}
\label{sticks}
We begin by considering the case of a one-dimensional stick of total mass
$M$ which is composed of two equal point masses separated by a rigid rod of
length $2 \ell$ and which makes elastic
collisions between two flat parallel walls
separated by a distance $h$. Let us recall some results from
\cite{bcj} for the case when the stick moves in two dimensions. The
coordinates of the stick will be $z$, the height of its center of mass
above the lower wall, and the angle of rotation $\theta$ from the vertical.
Scaling $z$ with the radius of gyration, $\kappa$, as $\eta =z/\kappa$,
the (scaled) energy of the stick becomes
\be
E = \frac{M}{2} ( {\dot {\theta }}^2 + { \dot {\eta }}^2 ) \ \ .
\ee
This looks formally like the energy of a free point particle
moving in an Euclidean plane parameterized by the dimensionless
coordinates $x_1 = \theta$ and $x_2 =\eta$.
The distance of closest approach of the center of mass to the plane is
$\eta _{\rm min} = (\ell /\kappa )|\cos(\theta )$
so that this point particle moves
between boundaries at the bottom, $b(\theta )$, and the top, $t(\theta )$,
with
\begin{eqnarray}
b(\theta ) & = & \frac{\ell}{\kappa }|\cos(\theta )|, \nonumber\\
t(\theta ) & = & \frac{1}{\kappa} \left ( h - \ell |\cos(\theta )| \right ).
\end{eqnarray}
A collision between the stick and the wall is described by
\begin{eqnarray}
(M\kappa )\Delta \dot{\eta } & = & f_n \tau \ , \nonumber \\
(M \kappa ^{2})\Delta \dot{\theta } & = & \ell \sin(\theta )f_n \tau \ .
\end{eqnarray}
The impulse can be eliminated to obtain a relation
between $\Delta \dot{\eta}$ and $\Delta \dot{\theta}$,
\be
\Delta \dot{\theta}=\frac{\ell}{\kappa} \sin \theta \Delta \dot{\eta},
\ee
and conservation of energy can be used to show that
\be
\Delta \dot{\eta } = \frac{-2(\dot{\eta } + (\ell /\kappa)
\dot{\theta } \sin \theta )}{1 + (\ell /\kappa)^{2}
\sin^2 \theta}.
\ee
The proof of specular reflection is as follows. The tangent to the
lower wall is
\be
\vec{T} =( 1, -\frac{\ell}{\kappa}\sin\theta ).
\ee
Using the condition that $\vec{N} \cdot \vec{T}=0$, the normal vector
is
\be
\vec{N} = (\frac{\ell}{\kappa}\sin\theta , 1).
\ee
Denoting the velocity of the point particle as $\vec{v}=
(\dot{\theta},\dot{\eta})$, the conditions for specular reflection
are
\be
\vec{T} \cdot (\vec{v} + \Delta \vec{v} ) = \vec{T} \cdot \vec{v},
\ee
and
\be
\vec{N} \cdot (\vec{v} + \Delta \vec{v} ) = - \vec{N} \cdot \vec{v}.
\ee
Using equation (4), one has
\be
\vec{T} \cdot ( \Delta \vec{v} ) = \Delta \dot{\theta} -
\frac{\ell}{\kappa} \sin \theta \Delta \dot{\eta} ~=~ 0,
\ee
whereas the normal constraint gives
\be
\vec{N} \cdot (2\vec{v} + \Delta \vec{v} ) = (2 \dot{\theta}
+ \Delta \dot{\theta}) (\frac{\ell}{\kappa}\sin \theta)
+ (2 \dot{\eta} + \Delta \dot{\eta}).
\ee
>From the conservation of energy and (4), one finds that
\begin{eqnarray}
{} & {} & \Delta \dot{\theta} (2 \dot{\theta} + \Delta \dot{\theta})
+ \Delta \dot{\eta} (2 \dot{\eta} + \Delta \dot{\eta}) \nonumber \\
& = & \Delta \dot{\eta} [(2 \dot{\theta} + \Delta \dot{\theta})
(\frac{\ell}{\kappa}\sin \theta) + (2 \dot{\eta} + \Delta \dot{\eta})]
~=~0,
\end{eqnarray}
and thus, (11) is satisfied.
We now allow the stick to move in three dimensions and make
elastic collisions with two flat walls which are the planes $z=0$ and
$z=h$. We orient the stick using the usual polar angles, $\theta$ and
$\phi$. The corresponding energy can be written as
\be
E = \frac{1}{2} M \ell^2 [ {\dot \theta}^2 + \sin^2 \theta \ {\dot \phi}^2 ]
+ \frac{1}{2} M {\dot z}^2 \ \ ,
\ee
and the angular momentum of the stick by
\begin{eqnarray}
L_x & = & M \ell^2 [ -\sin \phi \ {\dot \theta} - \sin \theta \
\cos \theta \ \cos \phi \ {\dot \phi} ] \nonumber \\
L_y & = & M \ell^2 [ \cos \phi \ {\dot \theta} - \sin \theta \
\cos \theta \ \sin \phi \ {\dot \phi} ] \nonumber \\
L_z & = & M \ell^2 [ \sin^2 \theta \ {\dot \phi} ] \ \ .
\end{eqnarray}
This problem initially appears to be five-dimensional. However, it is
clear that the $x$ and $y$ motion of the center of mass of the stick are
trivial and can be ignored. It is also clear that the force-free motion
of the stick does not result in $\theta$ and $\phi$ being linear functions
of time except for geometrical accidents.
Now consider a collision with the wall which imparts some impulse,
$f \tau$, in the $z$-direction.
\be
\Delta (M {\dot z} ) = f \tau \ \ .
\ee
There is a corresponding change in the angular momenta:
\begin{eqnarray}
\Delta L_x & = & - \ell \sin \theta \ \sin \phi \ (f \tau) \nonumber \\
\Delta L_y & = & \ell \sin \theta \ \cos \phi \ (f \tau) \nonumber \\
\Delta L_z & = & 0 \ \ .
\end{eqnarray}
As a consequence of the third of these equations, we see that
\be
\Delta {\dot \phi} = 0 \ \ .
\ee
Using this fact, the equations for $L_x$ and $L_y$, and the equations
for $\Delta L_x$ and $\Delta L_y$, we find that
\be
\Delta {\dot \theta} = \frac{1}{\ell} \sin \theta \ \Delta {\dot z} \ \ .
\ee
Finally, we can use the fact that the collision is strictly elastic and
equate kinetic energies before and after the collision. This leads us to
a quadratic equation with a trivial solution $\Delta {\dot z} = 0$
and a non-trivial solution of
\be
\Delta {\dot z} = \frac{-2({\dot z} + \ell {\dot \theta}
\sin \theta )}{1 + \sin^2 \theta} \ \ .
\ee
We are now in a position to draw all desired conclusions about this special
problem. Since ${\dot \phi}$ does not change during the collisions, the
coordinate $\phi$ is quite passive. It serves only to ``complicate'' the
motion in $\theta$. Equation (19) is {\em identical\/} to what was found
in the above two-dimensional problem. There is no $\phi$-dependence in
this equation. Furthermore, there can be no $\phi$-dependence in the
wall function. The wall function is also exactly what we had in
the two-dimensional case. Thus, with the scaling of variables
described previously, we again find that we have specular reflection in the
$(\ell \theta ,z)$ plane for every collision.
This apparently three-dimensional problem is really
a two-dimensional problem in the $(\ell \theta ,z)$ plane. The
only difference is that, as a consequence of the more complicated equations
of motion, the trajectories between consecutive wall hits are no longer
straight lines. Although the time-dependence of $\theta$ is not linear, it
is not complicated. We simply consider the free motion in
a rotated coordinate
system such that the angular momentum vector lies along the $z'$-axis.
In this frame the angular velocity $\omega_{z'}$ will be a constant. It
is then easy to transform back to the original $\theta z$-coordinates.
The energy (13) can be rewritten as
\be
E= \frac{1}{2} M[(\ell \dot \theta )^2 + \dot z ^2] + \frac {L_z ^2}{
2M \ell^2 \sin ^2 \theta } \ \ .
\ee
Since all reference to $ \phi $ has disappeared, this is the total
energy of the billiard ball in the reduced $(\ell \theta, z)$ plane.
The third term, $ L_z ^2 / 2M \ell^2 \sin ^2 \theta $, can be interpreted
as the potential energy for the two-dimensional billiard system.
This explains the non-linear time dependence of the $\theta$ variable. Thus, a
one-dimensional stick bouncing elastically between two flat walls
is equivalent to an interacting billiard problem (with suitable walls) on a
flat two-dimensional manifold (with a specific form of the interacting field).
\section{The Spiral Column Billiard}
\label{spiral}
Consider the usual stick of length $2\ell$ with point masses of
$M/2$ at the ends. This time, it will move inside a circle of
radius $R$ making elastic collisions as usual. Let us mark
the two ends of the stick as $P_1$ and $P_2$. Locate the center of mass
of the stick with polar angles $(r,\theta)$ and orient the stick
according to the angle $\alpha$ which $P_1$ makes with the $x$-axis.
The symmetry between $P_1$ and $P_2$ is simply $\alpha \rightarrow \alpha
+ \pi $. Thus, we will derive all the necessary equations for $P_1$
where the $P_2$ equations are obtained easily through this symmetry.
The energy of this system is
\be
E= \frac{M}{2}(\ell ^2 \dot{\alpha} ^2 + \dot{x} ^2 + \dot{y} ^2) \ .
\ee
The contact point of the stick with the circle is given by
\begin{eqnarray}
x_c &=& r_c \cos \theta + \ell \cos \alpha , \nonumber \\
y_c &=& r_c \sin \theta + \ell \sin \alpha ,
\end{eqnarray}
with the constraint that
\be
x_c ^2 + y_c ^2 = R ^2 = r_c ^2 + \ell ^2 + 2 r_c \ell \cos
(\theta - \alpha) \ .
\ee
Solving for $r_c$, the wall function for the billiard is
\be
b(\theta, \alpha ) \equiv r_c(\theta, \alpha) \nonumber \\
= -\ell |\cos (\theta - \alpha)| + (R^2 - \ell ^2
\sin ^2 (\theta - \alpha)) ^{1/2} .
\ee
We are now considering a configuration space of $(x,y,\ell \alpha)$
for our billiard system ({\em i.e.}, the location of the center of mass
of the stick plus its orientation). The boundary wall is given by
\be
(b(\theta, \alpha ) \cos \theta,~b(\theta, \alpha ) \ \ .
\sin \theta,~\ell \alpha)
\ee
This is a spiral column.
Let us now construct the dynamics of this problem. The polar angle
locating the physical point of contact will be denoted as
$\Phi$, {\em i.e.},
\begin{eqnarray}
R \cos \Phi &=& b(\theta, \alpha ) \cos \theta
+ \ell \cos \alpha \ , \nonumber \\
R \sin \Phi &=& b(\theta, \alpha ) \sin \theta + \ell \sin \alpha \ .
\end{eqnarray}
The normal at this point is strictly in the radial direction,
\be
\hat{n} = (\cos \Phi , \sin \Phi) \ ,
\ee
and the corresponding tangent vector is
\be
\hat{t} = (-\sin \Phi, \cos \Phi ) \ .
\ee
Since the impulsive force is assumed to be normal to the surface
at the physical point of contact, the linear impulse equations
are
\begin{eqnarray}
M \Delta \vec{v} \cdot \hat{n} &=& f \tau \ ,\nonumber \\
M \Delta \vec{v} \cdot \hat{t} &=& 0 \ ,
\end{eqnarray}
or
\be
M (\cos \Phi \Delta \dot{x} + \sin \Phi \Delta \dot{y} )
= f \tau \ ,
\ee
and
\be
-\sin \Phi \Delta \dot{x} + \cos \Phi \Delta \dot{y} = 0 \ .
\ee
The angular impulse equation
\be
I \Delta \omega = |\vec{\ell}||\vec{f}| \sin (\chi) \tau
\ee
gives
\be
M \ell ^2 \Delta \dot{\alpha} = \ell f \sin (\Phi - \alpha) \tau \ .
\ee
Equating (30) and (33) appropriately, one finds
\be
\cos \Phi \Delta \dot{x} + \sin \Phi \Delta \dot{y}
= \frac{\ell \Delta \dot{\alpha}}{\sin (\Phi - \alpha)}
\ee
and using (31),
\be
\Delta \dot{\alpha} = \left(\frac{\sin(\Phi - \alpha)}{\cos \Phi}\right)
\frac{\Delta \dot{x}}{\ell} \ .
\ee
Eliminating the angle $\Phi$ in favour of $\theta$ and $\alpha$
{\em via\/} equations (26), we have
\be
\Delta \dot{\alpha} = \left( \frac{b(\theta, \alpha)
\sin(\theta - \alpha)}{b(\theta, \alpha) \cos \theta + \ell
\cos \alpha} \right) \frac{\Delta \dot{x}}{\ell} \ ,
\ee
whereas (31) gives
\be
\Delta \dot{y}=\left( \frac{b(\theta, \alpha) \sin \theta + \ell
\sin \alpha}{b(\theta, \alpha) \cos \theta + \ell
\cos \alpha} \right) \Delta \dot{x} \ .
\ee
The value of $\Delta \dot{x}$ is obtained from energy conversation,
\be
\ell ^2 \dot{\alpha}^2 +\dot{x} ^2 + \dot{y} ^2 =
\ell ^2 (\dot{\alpha} + \Delta \dot{\alpha})^2 +(\dot{x}
+\Delta \dot{x})^2 +(\dot{y}+\Delta \dot{y})^2
\ee
resulting in
\be
\Delta \dot{x} = \frac{[\dot{y}(b \sin \theta + \ell
\sin \alpha) + \dot{x} (b \cos \theta + \ell
\cos \alpha) + \ell \dot{\alpha} (b \sin (\theta - \alpha))]
(b \cos \theta + \ell \cos \alpha )}{R^2 + b^2 \sin ^2 (\theta
-\alpha)} \ .
\ee
The analogous equations which apply when the other end of the stick, $P_2$,
collides with the circle are obtained by letting $\alpha \rightarrow
\alpha + \pi$.
We now have all the information needed in order to construct the proof
of specular reflection ((8) and (9)). The two (unnormalized) tangents
to the billiard wall (25) are
\be
\vec{T} _{\theta} = (b_{\theta} \cos \theta - b \sin \theta ,~
b_{\theta} \sin \theta + b \cos \theta, 0 )
\ee
and
\be
\vec{T} _{\alpha} = (b_{\theta} \cos \theta ,~
b_{\theta} \sin \theta , -\ell) \ ,
\ee
where
\be
b_{\theta} = \frac{\partial b}{\partial \theta} =
\frac{b \ell \sin(\theta - \alpha)}{b + \ell \cos (\theta - \alpha)} \ .
\ee
Simple manipulations will show that
\be
\vec{T} _{\theta} \cdot \Delta \vec{v} = 0
\ee
produces (37) whereas
\be
\vec{T} _{\alpha} \cdot \Delta \vec{v} = 0
\ee
results in (36). Finally, the (unnormalized) surface $\vec{N}$ derived from
$\vec{T} _{\theta} \cdot \vec{N} = 0 $ and $\vec{T} _{\alpha} \cdot \vec{N}
= 0 $ is
\be
\vec{N} = (b_{\theta}\sin \theta + b \cos \theta ,
-b_{\theta} \cos \theta + b \sin \theta , b b_{\theta} /\ell ) .
\ee
It can be shown that using this vector,
\be
\vec{N} \cdot (2 \vec{v} + \Delta \vec{v}) = 0
\ee
is equivalent to the energy conservation condition (38).
The results of this section have been obtained for the special case of a stick
moving inside a circle. They are, however, of materially greater generality.
It is possible to show that the motion of any rigid body inside a
two-dimensional confining wall of arbitrary shape is also equivalent
to a spiral billiard in three dimensions.
\section{The Rotating Billiard Wall}
\label{rot}
Our goal here is to reinterpret the previous billiard problem in such
a way that we can reduce it to a two-dimensional system. This can be
achieved as follows. In the present special case of the motion of a
rigid body inside a circle, the cross-section of the spiral column,
(25), has a fixed shape which rotates uniformly with $\alpha$. This
cross-section can be promoted to a rigid body which is allowed to rotate
about its (fixed) center with an angular orientation given by $\alpha$.
Furthermore, we associate the inertial parameter $M \ell ^2 $ with this
new rotating rigid wall. (In distinction to most other billiard
problems, the wall is no longer infinitely massive.) The motion of the
center of mass of the original stick is now described by the motion
of a point particle with a mass $M$ which moves inside the new
rotating wall. The billiard ball therefore sees an (instantaneous)
cross-section of the spiral column wall at every moment in time.
Our configuration space has been reduced from $(x,y,\ell \alpha)$
to $(x,y)$ by creating a rotating billiard wall of angular velocity
$\dot{\alpha}$ and moment of inertia $I=M \ell ^2$.
Once again, the point particle makes collisions with the rotating wall
in which the impulsive force is perpendicular to the wall surface
at the point of impact. However, since the wall is {\em not\/} infinitely
massive, it will recoil at every collision
such that the total energy of the point particle $M(\dot{x}^2
+\dot{y} ^2)/2$ plus the rotating wall $M\ell \dot{\alpha}^2 /2 $
will be conserved. Therefore, energy conservation is exactly as before (38).
The final step is to show that with every
collision, the billiard ball velocity
changes by equations (37) and (39) while the walls angular momentum
is shifted by (36).
The local tangent and local normal vectors at the collision point
are
\be
\vec{T} = (t_x , t_y )/(t_x ^2 + t_y ^2 )^{1/2}
\ee
and
\be
\vec{N} = (-t_y , t_x )/(t_x ^2 + t_y ^2 )^{1/2}
\ee
where
\be
t_x = b_{\theta} \cos \theta - b \sin \theta \ ,
\ee
and
\be
t_y = b_{\theta} \sin \theta + b \cos \theta \ .
\ee
Since the impulsive force is strictly normal, we have
\be
\vec{T} \cdot \Delta \vec{v} = \frac{(t_x \Delta \dot{x} +
t_y \Delta \dot{y})}{(t_x ^2 + t_y ^2 )^{1/2}} = 0 \ ,
\ee
which is equivalent to the tangential condition (43) of section
\ref{spiral}. Thus, equation (37) is satisfied.
Now, the linear impulsive force equation is
\be
\Delta \vec{p} = M(\Delta \dot{x},\Delta \dot{y})=-f \tau \hat{n} \ ,
\ee
while the angular impulse is
\be
M \ell ^2 \Delta \dot{\alpha} = \frac{f \tau b b_{\theta}}
{(t_x ^2 + t_y ^2 )^{1/2}} \ .
\ee
>From the two equations above, we have
\be
(t_y \Delta \dot{x} - t_x \Delta \dot{y})=
\frac{\ell ^2}{b b_{\theta}} \Delta \dot{\alpha} (t_x ^2 + t_y ^2) \ ,
\ee
which results in (36) after some simple manipulations.
The advantage of this billiard problem over the equivalent problem
of section \ref{spiral} is two-fold. We have accomplished a very clean
separation of linear and rotational motion. Further, the problem has
effectively been reduced to two spatial dimensions and lends itself
to convenient numerical analysis. The present argument permits some
generalization. Specifically, similar results can be obtained for a rigid body
of arbitrary shape. However, as the arguments above indicate, the original
static wall must be circular if the equivalent rotating wall is to be rigid
with a shape which is independent of its orientation.
A sample trajectory is depicted in
Fig.\,1 along with the rotating boundary wall. Even though the phase
space of the billiard ball alone is three-dimensional ($(x,y,p_x ,p_y )$
plus energy conservation), the phase space of the total billiard system
is actually five-dimensional ($(x,y,p_x ,p_y ,\ell \alpha , p_{\alpha})$
plus energy conservation). It is evidently not possible to depict the
complete, four-dimensional Poincar{\'e} section for this billiard problem.
Rather, we have obtained a projected Poincar\'{e} section of the actual orbits
in phase space in order to indicate the stochastic behaviour of the
system under consideration. This type of section
is simply a projection of the complete four-dimensional Poincar\'{e}
manifold onto a two-dimensional plane in phase space. On this
two-dimensional cross-section, orbits will appear to overlap
each other even though they are spatially distinct.
Fig.\,2 shows such a section for the billiard
with $R=2$ and $\ell = 1.9$. We first note that we have made
a canonical transformation from the coordinates $(x,y,p_x ,p_y,
\ell \alpha , p_{\alpha})$ to $(L_z , xp_x + yp_y , \tan ^{-1}
(p_y / p_x ), (1/2)\ln (p_x ^2 + p_y ^2 ), \ell \alpha , p_{\alpha})$
where $ L_z = yp_x - xp_y $. At each collision with the wall,
the angular momentum $L_z $ was plotted along with the angle
of incidence $\phi _c = \tan ^{-1} (p_y / p_x )$ at the point
of contact. Fig.\,2 contains two such orbits: one integrable
and one chaotic. The chaotic orbit is an example of Arnold
diffusion \cite{suz}, \cite{lal}. This diffusive process
for chaotic orbits is not trapped by integrable KAM tori
as is the case in lower-dimensional systems ({\em i.e.}, systems
with two degrees of freedom) and thus, the whole of phase space is
permeated by a network of stochastic trajectories. It was shown
by Arnold that chaotic diffusion proceeds along a `web' of dense
overlapping resonances and, therefore, the chaotic structure
of Fig.\,2 is called an {\em Arnold Web}.\footnote{The chaotic orbit in Fig.\,2
was terminated after $10^4$ collisions in order to make the web structure
apparent.}
We have followed the stability of these structures for fixed $R=2$ as a
function of the length of the rod, $\ell$. The KAM tori shown in Fig.\,2
deform as $\ell$ is reduced. When $\ell=1.1$, these same tori are
transformed into five double resonant islands located roughly in the area
between the
original tori shown in Fig.\,2. For even small values of $\ell$, these
resonant islands seem to disappear leaving apparently chaotic
motion.\footnote{It is always possible to demonstrate the absence of
chaos by offering a numerically determined periodic trajectory as a counter
example. Evidently, positive proof of chaotic motion cannot be made
numerically.} The qualitative result of this investigation is the motion
becomes ``more chaotic'' as the stick grows shorter with KAM tori first
evolving into resonance islands which subsequently dissolve into chaos.
On the other hand, the motion of a stick with length exactly zero is an
integrable system. (This limit is simply a point particle moving in a
circle, which is obviously integrable.) This provides an indication that
the $\ell \rightarrow 0$ limit is somewhat delicate and might even suggest
that the transition from the non-integrable motion found for $\ell \neq 0$
to the integrable circle at $\ell = 0$ is, in some sense, a phase transition.
The mathematical puzzle can be resolved by physical thinking. Consider
a plot of the motion using the present canonical co-ordinates, $L_z$ and
$\phi_c$. For fixed energy the system will explore a decreasing range of
$L_z$ as $\ell \rightarrow 0$. Motion within this range will become
increasingly chaotic, as indicated. However, for sufficiently small $\ell$,
the width of this range will become comparable to our ability to resolve
the details of the structure within it. We will feel comfortable in declaring
the particle a point particle, and we will not be bothered by an underlying
chaotic motion which we cannot actually observe. A similar observation
could be made regarding our earlier work \cite{bcj} on an
ellipse bouncing between two parallel walls. As the distance between the
walls is increased, the system becomes more chaotic and, in fact,
undergoes a transition from a KAM-system to a K-system. (Increasing
the distance between the walls is equivalent to making the
ellipse smaller and entirely analogous to letting $\ell \rightarrow 0$
as above).
In the context of the equivalent problem of point particle and rotating wall,
there are two consequences of changing the length of the stick. Both the
shape of the rotating wall and its moment of inertia, $I = M \ell^2$, change
with the length. Thus, it is in some sense more natural to consider changes in
the inertial parameter of the wall while maintaining its shape. This can be
realized by regarding the stick as a rigid, massless rod of length $2 \ell$
with point masses $M/2$ located at some adjustable point along its length (or
extension). Specifically, it is useful to consider the somewhat artificial
limit $I \rightarrow \infty$ in which the wall is stationary. In this limit,
the phase space of the system is reduced to
$(L_z , xp_x + yp_y , \tan ^{-1}(p_y / p_x ), (1/2)\ln(2ME))$. Comparing
these with the canonical coordinates for the general case given above,
we see that the final coordinate is now reduced to an obvious integral
of the motion. The phase space structure of such a system (with $R=2$
and $\ell = 1.5$) is depicted in Fig.\,3. Since the dimensionality of
this system is so low ($2N=4$), Fig.\,3 describes a complete Poincar\'{e}
section rather than the projection of such a section. As a result,
tori and chaotic regions are cleanly separated in a familiar fashion, and
the phenomena of Arnold diffusion cannot appear. The chaotic regions
can never penetrate the areas bounded by KAM tori. This would suggest
that the system is KAM for all $I \ge M \ell^2$. Numerical studies confirm
this expectation. If more chaotic motion is to be found, it must require
$I < M \ell^2$. Tori and the web seem to persist if the original moment of
inertia is reduced by a factor of $100$. All remnants of these indicators
of a KAM-system vanish when the original moment of inertia is reduced by a
factor of $200$.
The important observation is that we can control the qualitative nature of the
motion --- from integrable to KAM to (probably) K --- by the tuning of simple
physical parameters ({\em i.e.}, the length of the rod and the
distribution of its mass).
\section{Conclusion}
\label{concl}
This work represents a natural extension of our earlier
work \cite{bcj}, \cite{cjb} by investigating rigid body systems in closed
rather than open domains. This has led to the new phenomena of a rotating
billiard wall with a finite moment of inertia. This feature arose naturally
from the mapping of the rigid body problem and was not put in by hand in some
{\em ad hoc\/} manner. This is in the same spirit as in our previous
work where the introduction of curvilinear motion was exclusively
dictated by the three-dimensional rigid body under consideration
and was not constructed {\em a priori\/} as is usually the case in the
standard billiards possessing curvilinear trajectories such as
the gravitational billiard or the Aharonov-Bohm billiard. There are
positive consequences of having such complications arise from the introduction
of new physics rather than by artifice. Here, we have constructed three
distinct physical systems which can be mapped onto each other and which have
precisely the same content. Each physical system comes with its own
intuitive understanding and complementary ``natural'' limits, which lead to
a richer understanding of them all. The variant described in section
\ref{rot} of a point particle and a
rotating wall is particularly appealing since it makes an exceptionally clean
separation between the translational and rotational aspects of the problem.
The appearance of Arnold's web is extremely satisfying
since virtually all aspects of Hamiltonian chaotic systems
have now appeared in our rigid body billiard models. We have therefore
introduced a rich class of two- and three-dimensional dynamical
systems in which there exist both integrable and chaotic phenomena
and in which the interesting features of curvilinear motion,
rotating billiard walls, and Arnold diffusion can be probed by tuning
physically meaningful parameters.
\vskip 1.0cm
{\em It is both a professional and personal pleasure to dedicate this work
to Gerald E. Brown on the occasion of his seventieth birthday. Through
more than three decades, Gerry Brown has continued to be a primary source
of new and exciting ideas in nuclear physics. Time and again, he displays
the courage to make major changes in his intellectual focus in order to follow
new insights and enthusiasms. Life with Gerry is never dull. We await what
will come next with admiration and affection.}
\vskip 1.0cm
We would like to acknowledge helpful discussions with A. Halasz and N.L.
Balazs.
This work was supported in part by the US Department of Energy under grant
No.~DE-FG02-88ER 40388.
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\begin{figure}
\caption{A trajectory in configuration space for the stick system
with $R=2$ and $\ell = 1$. The rotating billiard wall at the point
of contact is explicitly shown.}
\end{figure}
\begin{figure}
\caption{A projected Poincar\'{e} section for the stick system
with $R=2$ and $\ell = 1.9$ where $\phi _c$ is plotted
in units of $\pi$.}
\end{figure}
\begin{figure}
\caption{A projected Poincar\'{e} section for the billiard system
with $M_{wall} \rightarrow \infty$, $R=2$ and $\ell = 1.5$.}
\end{figure}
\end{document}