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\topmatter
\title
Hamiltonian Systems with Linear Potential \\
and Elastic Constraints.
\endtitle
\rightheadtext{Hamiltonian Systems with Linear Potential}
\author
Maciej P. Wojtkowski
\endauthor
\affil University of Arizona
\endaffil
\address
Maciej P. Wojtkowski,
Department of Mathematics,
University of Arizona,
Tuscon, Arizona 85 721
USA
\endaddress
\email
maciejw@math.arizona.edu
\endemail
\date
March 5, 1997
\enddate
\abstract
We consider a class of Hamiltonian systems with elastic
constraints and arbitrary number of degrees of freedom.
We establish sufficient conditions for complete hyperbolicity of the
system.
\endabstract
\thanks
{\bf
We thank Oliver Knill for valuable discussions
of the subject and for making several comments and corrections
to the early version of this paper.
In December 1996, when completing the paper,
we benefited from the hospitality
of the Erwin Schr\"odinger Institute in Vienna.
We were also partially supported by NSF Grant DMS-9404420.}
\endthanks
\endtopmatter
\document
\vskip.7cm
\subhead \S 0.Introduction \endsubhead
\vskip.4cm
We study a class of
Hamiltonian systems with linear potential and
arbitrary number of degrees of freedom.
The Hamiltonian is given by
$$ H = {\frac 12}\sum_{i,j =1}^{n}k_{ij}\xi_i\xi_j +
\sum_{l=1}^n c_l\eta_l, $$
where $(\eta,\xi)\in \Bbb R^n \times\Bbb R^n$ are ``positions'' and
``momenta'', $K= \{k_{ij}\}$ is a constant symmetric positive
definite matrix giving the kinetic energy. The equations of
motions are
$$
\frac{d^2\eta_i}{dt^2} = -\sum_{j =1}^{n}k_{ij}c_j = const.
$$
We close the system and
couple different degrees of freedom by
restricting it to the positive octant
$$
\eta_1 \geq 0, \eta_2 \geq 0, \dots, \eta_n \geq 0.
$$
When one of the $\eta$ coordinates vanishes the velocity
is changed instantaneously by the rules of elastic collisions,
i.e., the component of the velocity parallel to the face of the
octant is preserved and the component orthogonal to the face is
reversed. Orthogonality is understood with respect to the scalar
product defined by the kinetic energy.
With these elastic constraints the system is closed provided that all the
coefficients, $c_1,\dots,c_n$, are positive. The restriction of this
system to a level set of the Hamiltonian (i.e., we fix the total
energy) has a finite Liouville measure which is preserved by the
dynamics. There are singular trajectories in the system
(hitting the lower dimensional faces of the octant or having zero
velocity on a face of the octant) which are defined for finite time
only but they form a subset of zero measure. Dynamics is well defined
almost everywhere. Moreover the derivative of the flow is also defined
almost everywhere and Lyapunov exponents are well defined for our
system, cf. \cite{O},\cite{R}.
\proclaim{Main Theorem}
If all the off-diagonal entries of the positive definite matrix
$K^{-1}$ are negative then
the Hamiltonian system with elastic constraints restricted to one
energy level is completely hyperbolic,
i.e., it has all but one nonzero Lyapunov exponents almost everywhere,
\endproclaim
By the structural theory of hyperbolic systems with singularities
developed by Katok and Strelcyn \cite{K-S} we can conclude that our
system has at most countably many ergodic components. The mixing
properties of the flow are as usual not readily accessible. But if we
consider the natural Poincar\'e section map (from a face of the octant
to another face) we can apply the results of Chernov and Hasskel \cite
{Ch-H} and Ornstein and Weiss \cite{O-W}
to get Bernoulli property on ergodic components. We are unable
to make rigorous claims about ergodicity because the singularities of
the system are not properly aligned (except for $n =2$) which does not
allow the implementation of the Sinai -- Chernov methods. This point
is discussed in detail in \cite {L-W}. At the same time there is
little doubt that the system is actually ergodic.
There are concrete systems of interacting particles that fall into
the category described in the Main Theorem.
One such system is a variation of
the system of parallel sheets interacting by
gravitational forces, studied recently by Reidl and Miller \cite{R-M}.
Let us consider the system of $n$ point particles in the line
with positions $q_0,q_1,\dots, q_n $ and masses $m_0,...,m_n$.
Their interaction is defined by a linear translationally invariant potential
$U(q) = \sum_{i=1}^n c_i (q_i-q_0)$. The Hamiltonian of the system is
$$
H = \sum_{i=0}^n\frac {p_i^2}{2m_i} + \sum_{i=1}^nc_i(q_i-q_0).\tag0.1$$
We introduce the elastic
constraints
$$
q_1 -q_0 \geq 0, q_2 -q_0 \geq 0, \dots, q_n-q_0 \geq 0, \tag0.2
$$
i.e., the particles go through each other freely except for the
$q_0$-particle which collides elastically with every other particle.
A convenient interpretation of the system is that of a horizontal
floor of finite mass $m_0$ and $n$ particles of masses
$m_1,m_2,\dots, m_n$. The floor and the particles can move only
in the vertical direction and their positions are
$q_0$ and $q_1, \dots, q_n$.
There is a constant force of attraction
between any of the particles and the floor. Moreover the particles
collide elastically with the floor and there are no
collisions between the particles (they move
along different parallel lines perpendicular to the floor).
Hence the particles ``communicate'' with each other only through the
collisions with the floor, which is a rather weak interaction.
Introducing symplectic coordinates $(\eta,\xi)$
$$
\aligned
\eta_0 &= m_0q_0 + m_1q_1 + \dots +m_nq_n\\
\eta_i &= q_i - q_0, \\
p_0 &= m_0\xi_0 - \xi_1 -\dots -\xi_n\\
p_{i} &= m_{i}\xi_0 + \xi_{i} \ \ \ \ \ \ \ \ \ i = 1,2, \dots, n\\
\endaligned \tag{0.3}
$$
and setting the total momentum and the center of mass at zero,
$ \eta_0 = 0, \xi_0 =0$, we obtain the Hamiltonian
$$
H = \frac{(\xi_1 + \dots +\xi_n)^2}{2m_0} +
\sum_{i=1}^n \frac{\xi_i^2}{2m_i} +
\sum_{i=1}^n c_i\eta_i.
$$
This system satisfies the assumptions of the Main Theorem
and hence it is
completely hyperbolic. Note that no conditions on the
masses are required.
We can introduce additional interactions between particles
by stacking groups of them on vertical lines. The particles on
the same vertical line will collide elastically with each other
and only the bottom particle collides with the floor.
Mathematically it corresponds to adding more constraints
to \thetag{0.2}.
We establish that such systems are also completely hyperbolic,
if the masses satisfy certain inequalities.
We must though assume that the accelerations of all
the particles in one stack are proportional to their masses,
with the coefficients of proportionality allowed to be different
for different stacks.
As we add more constraints our conditions on the masses
which guarantee complete hyperbolicity become more
stringent. Which seems somewhat paradoxical: as the
interactions of the particles become richer the ergodicity
of the system (the equipartition of energy) is more likely to fail.
This behavior becomes more intuitive when we modify the original
system of noninteracting particles falling to the floor by splitting
each mass into two or more masses that are stacked on one vertical
line. In the original system the particles have to freely ``share''
their energy with the floor and hence with other particles. In the
modified system the stack of particles acts as ``internal'' degrees of
freedom which may store energy for extended periods of time.
One would expect
that the energy transfer between stacks is less vigorous than in the
case when all the masses in one stack are glued into one particle.
The extremal case is that of one stack, i.e.,
where we introduce the constraints,
$$ q_0 \leq q_1 \leq \dots \leq q_n, \tag{0.4}$$
and $c_i = \alpha m_i, \ i
= 1,\dots,n$.
If $m_1 = m_2 = \dots = m_n$
then the resulting system
is a factor of the system with the constraints \thetag{0.2}
and in particular it is completely
hyperbolic. In general complete hyperbolicity occurs when
the masses
satisfy special inequalities. More precisely,
if there are constants $a_1 < a_2 \leq \dots \leq a_n$ such that
$a_i > i, \ \ i= 1,\dots, n,$ and
$$
m_i = \frac{m_0+m_1+\dots+ m_{i-1}}{a_i-i}, \ \ \ i= 1,\dots, n,
$$
then the system is completely hyperbolic.
These conditions are substantial and not merely technical since
the system is completely integrable, if for some constant $a > n$
$$
m_i = \frac{m_0+m_1+\dots+ m_{i-1}}{a-i}, \ \ \ i= 1,\dots, n.
$$
Let us end this introduction with the outline of the contents of the
paper. In Section 1 we review the notion of flows with collisions
(\cite{W1}), a mixture of differential equations and discrete time
dynamical systems (mappings). We define hyperbolicity (complete and
partial) for flows with collisions
and formulate the criterion of hyperbolicity from \cite{W3}.
In Section 2 we study the geometry of simplicial cones,
which we call wedges. We introduce a special class of wedges,
called simple and discuss their geometric invariants.
As a byproduct we obtain a dual characterization of positive definite
tridiagonal matrices which is of independent interest.
In Section 3 we introduce a Hamiltonian system with linear potential
and elastic constraints which we call a PW system (Particle in a
Wedge). It is defined
by a wedge and an acceleration direction (from the dual wedge).
A point particle is confined to the wedge and accelerated
in the chosen direction (falling down).
We establish that the system of falling particles in a line (FPL
system), introduced and studied in \cite{W1}, is
equivalent to a PW system in a simple wedge with the acceleration
parallel to the first (or last) generator of the wedge. We recast
the conditions of partial hyperbolicity from \cite{W1} in terms
of the geometry of the simple wedge.
In a recent paper Simanyi \cite{S} showed that these conditions
guarantee complete hyperbolicity.
In Section 4 we give a new edition of the results of \cite{W1}, on
monotonicity of FPL and PW systems, in a more geometric language
appropriate for the the present work. The new formulations are
necessary for the proof of the Main Theorem.
In Section 5 we consider two special classes of Hamiltonian systems,
the system \thetag{0.1} with the constraints \thetag{0.4}
and another class. Both classes reduce straightforwardly to
PW systems in simple wedges. We apply the criteria of complete
hyperbolicity and complete integrability and get in particular the
result formulated above.
In Section 6 we introduce wide wedges and we prove the Main Theorem.
In Section 7 we study the system \thetag{0.1}
with arbitrary ``stacking rules''
added to the constraints \thetag{0.2}. We derive the conditions
on the masses
which guarantee complete hyperbolicity of the system,
in terms of the graph of constraints.
Section 8 contains remarks and open problems.
\vskip.7cm
\subhead \S 1. Hamiltonian flows with collisions \endsubhead
\vskip.4cm
A flow with collisions is a concatenation of a flow defined by a vector
field on a manifold and mappings defined on submanifolds
(collision manifolds) of codimension one.
Trajectories of a flow with collisions follow
the trajectories of the flow until they reach one of the collision
manifold where they are glued with another trajectory by the
collision map. More precise description of this simple
concept is somewhat lengthy. We will do it for Hamiltonian flows only.
More detailed discussion can be found in \cite{W1} and \cite{W3}.
Let $(N,\omega)$ be a smooth $2n$-dimensional symplectic manifold with
the symplectic form $\omega$ and $H$ be a smooth function on $N$. By
$\nabla H$ we denote the Hamiltonian vector field defined by the
Hamiltonian function $H$. Let further $M$ be a $2n$-dimensional closed
submanifold of $N$ with piecewise smooth boundary $\partial M$. For
simplicity we assume that $\nabla H$ does not vanish in M. Let $N^h =
\{x\in N| H(x) = h\}$ be a smooth level set of the Hamiltonian.
The Hamiltonian vector field $\nabla H$ is tangent to $N^h$.
We do not require that $M$ is compact, but we do assume that the
restricted level sets of the Hamiltonian, $M\cap N^h$,
are compact for all values of $h$.
In the boundary
we distinguish the regular part,
$\partial M_r$, consisting of points which do not belong to more than
one smooth piece and where the vector field $\nabla H$ is transversal
to $\partial M$. The remaining part of the boundary is called
singular. We assume that the singular part of the boundary has zero
Lebesgue measure in $\partial M$.
The regular part of the boundary is further divided
into $\partial M^-$, the ``outgoing'' part, where $\nabla H$ points
outside of the domain $M$, and $\partial M^+$, the ``incoming'' part,
where $\nabla H$ points inside of the domain $M$.
We assume that a mapping $\Phi :\partial M^- \to \partial M^+$, the
collision mapping, is given and that it preserves the Hamiltonian,
$H\circ \Phi = H$. Any codimension one submanifold of $N^h$
transversal to $\nabla H$ inherits a canonical symplectic structure,
the restriction of the symplectic form $\omega$. Hence $\partial
M_r\cap N^h$
possesses the symplectic structure and we require that the collision
map restricted to $\partial M^- \cap N^h$ preserves this
symplectic structure. The Liouville measure
(the symplectic volume element) defined by the simplectic
structure is thus preserved.
In such a setup we define the Hamiltonian flow with collisions
$$
\Psi^t: M \to M, \ \ \ t\in \Bbb R,
$$
by describing trajectories of the flow. So $\Psi^t(x),\ t\geq 0,$
coincides with the trajectory of the original Hamiltonian flow
(defined by $\nabla H$) until we get to the boundary of $M$ at time
$t_c(x)$, the collision time. If $\Psi^{t_c}(x)$ belongs to the
singular part of the boundary then the flow is not defined for $t >
t_c$ (the trajectory 'dies' there). Otherwise the trajectory is
continued at the point $\Phi(\Psi^{t_c}(x))$ until the next collision
time, i.e.,
$$
\Psi^{t_c+t}(x) = \Psi^t\Phi\Psi^{t_c}(x).$$
This flow with collisions may be badly discontinuous but thanks to the
preservation of the Liouville measures by the Hamiltonian flow and the
collision map, the flow $\Psi^t$ is a well defined measurable flow in
the sense of Ergodic Theory (cf.\cite{C-F-S}). Let $\nu = \nu_h$
denote the Liouville measure on the level set of the Hamiltonian
$N^h\cap M$. By the compactness assumption $\nu_h$ is finite for
all smooth level sets $N^h$. We can now study the ergodic properties of
the flow $\Psi^t$ restricted to one level set.
The derivative $D\Psi^t$ is also well defined almost
everywhere in $M$ and for all $t$, except the collision times.
This allows the definition of Lyapunov exponents for our Hamiltonian
flow with collisions, under the integrability assumption
(\cite{O},\cite{R})
$$
\int_{N^h}ln^+||D\Psi^1||d\nu_h < +\infty.
$$
In general the Lyapunov exponents are defined
almost everywhere and they depend on a trajectory of the flow.
Due to the Hamiltonian character of the flow,
two of the $2n$ Lyapunov exponents are automatically zero,
and the others come in pairs of opposite numbers.
Hence there is equal number of positive and negative Lyapunov exponents.
\proclaim{Definition 1.1}
A Hamiltonian flow with collisions is called (nonuniformly) partially
hyperbolic, if it has nonzero Lyapunov exponents almost everywhere,
and it is called (nonuniformly) completely hyperbolic, if
it has all but two nonzero Lyapunov exponents almost everywhere.
\endproclaim
\proclaim{Definition 1.2}
A Hamiltonian flow with collisions is called completely integrable,
if there are $n$ functions $F_1, \dots,F_n$
in involution, with linearly independent
differentials almost everywhere, which are first integrals for both
the flow and the collision map, i.e., $dF_i(\nabla H) = 0$
and $F_i\circ\Phi = F_i,\ i = 1,\dots,n$.
\endproclaim
As usual, completely integrable Hamiltonian flows with collisions
have only zero Lyapunov exponents.
We will outline here a criterion for nonvanishing of Lyapunov
exponents. Complete exposition can be found in \cite{W3}. Note that we
introduce some modifications in the formulations, to facilitate the
applications of this criterion in the present paper.
We choose two transversal subbundles, $L_1(x)$ and $L_2(x), x \in M$,
of Lagrangian subspaces in the tangent bundle of $M$.
We allow these bundles to be
discontinuous and defined almost everywhere. Their measurability is
the only requirement.
An ordered pair of transversal Lagrangian subspaces, $L_1$ and $L_2$,
defines a quadratic form $Q$ by the formula
$$
Q(v) =\omega(v_1,v_2), \ \ \text{where} \ \ v = v_1 + v_2, v_i\in L_i,
i = 1,2.
$$
Further we define the sector $\Cal C$ between $L_1$ and $L_2$
by $\Cal C = \{v| Q(v) \geq 0\}$.
We assume that $\nabla H$ belongs to $L_2$ at almost all points
(we could as well assume that it belongs to $L_1$). This assumption
is very important for the Hamiltonian formalism, it allows to project
the quadratic form $Q$ on the factor of the tangent space to the
level set of the Hamiltonian by the one dimensional subspace
spanned by $\nabla H$. This factor space plays the role of
the ``transversal section'' of the flow restricted to a smooth
level set. Note that in general we do not have an invariant codimension
one subspace transversal to the flow.
\proclaim{Definition 1.3} The Hamiltonian flow with collisions,
$\Psi^t$, is called monotone (with respect to the bundle of sectors
$\Cal C(x), x\in M$),
if for almost all points in $M$
$$
Q(D\Psi^tv) \geq Q(v),
$$
for all vectors $v$ tangent to a smooth level set of the Hamiltonian,
$M\cap N^h$, and all $t\geq 0$ for which the derivative is well defined.
\endproclaim
The monotonicity of the flow does not imply nonvanishing of any
Lyapunov exponents. Actually completely integrable Hamiltonian flows
are typically monotone with respect to some bundle of sectors. To
obtain hyperbolicity one needs to examine what happens to the
``sides'' $L_1$ and $L_2$ of the sector $\Cal C$. Let $\tilde L_1 =
L_1\cap \{v|dH(v) = 0\}$, be the intersection with the tangent space
to the
level set of the Hamiltonian (note that $L_2$ is always tangent to the
level set because we assume that $\nabla H \in L_2$ and hence the
dimension of $\tilde L_1$ is always $n-1$). In a monotone system
there are two possibilities for a vector from $\tilde L_1$ (or from
$L_2$), either it enters the interior of the sector $\Cal C$ at some
time $t > 0$ or it forever stays in $\tilde L_1$ (or in $L_2$).
For a monotone flow we define an $L_1$-exceptional subspace
$\Cal E_1(x) \subset \tilde L_1(x)$ as
$$
\Cal E_1(x) =\tilde L_1(x)\cap
\bigcap_{t\geq 0}D\Psi^{-t}\tilde L_1(\Psi^tx).\tag{1.1}
$$
Similarly we define the $L_2$-exceptional subspace $\Cal E_2(x)$.
The $L_2$-exceptional subspace always contains the Hamiltonian vector
field $\nabla H$.
We call a point $x\in M$
$L_1$-exceptional, if $dim \Cal E_1(x) \geq 1$,
and $L_2$-exceptional, if $dim \Cal E_2(x) \geq 2$.
The following theorem is essentially proven in \cite{W3}
\proclaim{Theorem 1.4}
If the Hamiltonian flow with collisions is monotone and
the sets of $L_1$-exceptional points and
$L_2$-exceptional points have measure zero then
the flow is completely hyperbolic.
\endproclaim
The criterion of partial hyperbolicity is given by the following
(cf.\cite{W3})
\proclaim{Theorem 1.5}
If the Hamiltonian flow with collisions is monotone
then it is also
partially hyperbolic, provided
one of the following conditions is satisfied
\roster
\item
the set of $L_1$-exceptional points has measure zero and
$dim \Cal E_2(x) \leq n-1$
for almost all points $x\in M$,
\item
the set of $L_2$-exceptional points has measure zero and
$dim \Cal E_1(x) \leq n-2$
for almost all points $x\in M$.
\endroster
\endproclaim
\vskip.7cm
\subhead \S 2. Simple wedges
\endsubhead
\vskip.4cm
Let us consider the $n$-dimensional euclidean space $E$.
We define a $k$-dimensional wedge, $k\leq n$, to be a convex cone in $E$
generated by $k$ linearly independent rays.
Hence we have the $k$-dimensional wedge
$W\subset E$, if there is a linearly independent set of $k$ vectors,
$\{e_1,\dots,e_k\}$, such that
$$
W = \{x\in E| x = \lambda_1e_1 + \dots + \lambda_ke_k, \lambda_i \geq
0, i = 1,\dots,k\}.
$$
We call the vectors $\{e_1,\dots,e_k\}$ the generators of the wedge
and we denote the wedge generated by them as $W(e_1,\dots,e_k)$.
The generators are uniquely defined up to positive scalar factors.
We will denote by $S(e_1,\dots,e_k) \subset E$ the linear subspace
generated by the linearly independent vectors $\{e_1,\dots,e_k\}$.
The dual space $E^*$ can be naturally identified with $E$. Thus the
cone $W^*$ dual to the $n$-dimensional wedge $W$ is itself an
$n$-dimensional wedge in $E$.
Let $\{e_1,\dots,e_n\}$ be an ordered basis in $E$ and
$\{f_1,\dots,f_n\}$ be the dual basis, i.e., $\langle f_i,e_j\rangle =
\delta_i^j$, the Kronecker's delta.
\proclaim{Proposition 2.1}
The following properties of an ordered basis $\{e_1,\dots,e_n\}$
of unit vectors and its dual $\{f_1,\dots,f_n\}$ are equivalent
\roster
\item
The orthogonal projection of
$e_l$ on $S(e_{l+1},\dots,e_n)$ is parallel to $e_{l+1}$,
for $l=1,2,\dots,n-1$.
\item
$$\langle e_i,e_j \rangle = \prod_{s=i}^{j-1}\langle e_s,e_{s+1}
\rangle, \ \ \ \
\text{for all} \ \ \ 1\leq i \leq j-1 \leq n-1.$$
\item
$$ \langle f_i,f_j\rangle = 0 \ \ \ \text{for all} \ \ \ 1\leq i,j \leq n,
|i-j| \geq 2.$$
\endroster
\endproclaim
\demo{Proof}
$(1) \Leftrightarrow (2)$
Observe that since $\{e_1, \dots, e_n\}$
are unit vectors, then the condition $(1)$
can be
reformulated as
$\langle e_i,e_j \rangle = \langle e_i,e_{i+1} \rangle \langle e_{i+1},e_j \rangle$ for $i = 1,\dots,k-1,$
and $j = i+2,\dots,k$. We get (2) by induction on the distance
between $i$ and $j$. Clearly the converse is also true.
\noindent $(1) \Rightarrow (3)$
Let us introduce vectors $\tilde f_i = a_ie_{i-1}+ e_i + b_ie_{i+1}$
for $i = 2,\dots,n-1,$ and $\tilde f_1 = e_1 + b_1e_2,
\tilde f_n = b_{n-1}e_{n-1} + e_n$ where the coefficients $a_i, b_i$
are uniquely determined by the condition that $\tilde f_i$ is orthogonal to
$e_{i-1}$ and $e_{i+1}$. We now show that
$\langle \tilde f_i, e_j \rangle = 0$ if $i \neq j$, i.e.,
the vectors $\{\tilde
f_1,\dots, \tilde f_n\}$ differ from the dual basis by
scalar factors only. Indeed if $j <
i-1$, the orthogonal projection of $e_j$ on $S(e_{i-1},e_i,...,e_n)$
is parallel to $e_{i-1}$, and hence $e_j$ is orthogonal to $\tilde f_i$.
Moreover the orthogonal projection of $\tilde f_i$ on $S(e_{i+1},e_i,...,e_n)$
is parallel to $e_{i+1}$, and hence $\tilde f_i$ is orthogonal to
$S(e_{i+1},e_i,...,e_n)$.
Further we have clearly that $\tilde f_i$ is orthogonal to $\tilde
f_j$ for $j \geq i+2$, which proves $(3)$.
\noindent $(3) \Rightarrow (1)$
>From $(3)$ we obtain that
$$\langle f_i,f_i\rangle^{-1} f_i = a_ie_{i-1}+ e_i + b_ie_{i+1}$$
for $i = 2,\dots,n-1,$ and
$$\langle f_1,f_1\rangle^{-1} f_1 = e_1 + b_1e_2,
\langle f_n,f_n\rangle^{-1} f_n = b_{n-1}e_{n-1} + e_n$$
for some coefficients $a_i,b_i$. We prove $(1)$ by induction on
the index $l$. So $e_1 = \langle f_i,f_i\rangle^{-1} f_1 - b_1e_2$
implies that the orthogonal projection of $e_1$ on the subspace
$S(e_2,\dots,e_n)$ is parallel to $e_2$.
Given that the orthogonal projection of $e_{l-1}$ on the subspace
$S(e_l,\dots,e_n)$ is parallel to $e_l$, we conclude that also the
orthogonal projection of
$e_l= \langle f_l,f_l\rangle^{-1} f_l - a_ie_{l-1}-b_le_{l+1}$ on the subspace
$S(e_{l+1},\dots,e_n)$ is parallel to $e_{l+1}$.
\enddemo\qed
We introduce now a special type of a wedge.
\proclaim{Definition 2.2} A $k$-dimensional wedge $W \subset E$ is
called simple, if its generators $\{e_1, \dots, e_k\}$ can be ordered
in such a way that for any $i = 1,\dots,k-1,$
the orthogonal projection of $e_i$ on the $k-i$
dimensional subspace $S(e_{i+1}, \dots, e_k)$
is a positive multiple of $e_{i+1}$. The ordering of the generators
for which this property holds is called distinguished.
\endproclaim
>From Proposition 2.1 we obtain immediately
\proclaim{Proposition 2.3} Let $\{e_1,\dots, e_k\}$ be a set of
linearly independent unit vectors.
The wedge $W(e_1,\dots, e_k)$ is simple and
the ordering of the generators is distinguished if and only if
\roster
\item
$\langle e_i,e_{i+1} \rangle > 0, \ \ \ \text{for} \ \ \
i = 1,\dots,n-1$ and
\item
$\langle e_i,e_j \rangle = \prod_{l=i}^{j-1}\langle e_l,e_{l+1}
\rangle,
\ \ \ \text{for all} \ \ \ 1\leq i \leq j-1 \leq
k-1.$
\endroster
\endproclaim
\proclaim{Corollary 2.4}
Any face of a simple wedge is a simple wedge.
A simple wedge has exactly two distinguished orderings,
one is the reversal of the other.
\endproclaim
\demo{Proof of Corollary 2.4}
It follows from Proposition 2.3 that any face of a simple wedge is
simple and that the reversal of a distinguished ordering is distinguished.
It remains to show that there are no other distinguished orderings.
It follows immediately from the following observation.
Suppose $\{e_1,\dots,e_k\}$ are unit generators of a simple wedge
in a distinguished order. We get that
$ \langle e_1,e_k \rangle < \langle e_i,e_j \rangle,$ for any $1 \leq i < j \leq k, (i,j) \neq
(1,k)$.
\enddemo
Dual characterization of a simple wedge is given by
\proclaim{Proposition 2.4} Let $W(e_1,\dots, e_n)$ be a wedge
in an $n$-dimensional Euclidean space $E$ and
$\{f_1,f_2\dots,f_n\}$ be the dual basis.
$W(e_1,\dots, e_n)$ is a simple wedge and the order of the generators
is distinguished if and only if
\roster
\item
$\langle f_i,f_{i+1} \rangle < 0, \ \ \ \text{for} \ \ \
i = 1,\dots,n-1$ and
\item
$ \langle f_i,f_j\rangle = 0 \ \ \ \text{for all} \ \ \ 1\leq i,j \leq n,
|i-j| \geq 2.$
\endroster
\endproclaim
\demo{Proof}
Assuming without loss of generality that $\{e_1,\dots,e_n\}$ are unit
vectors, we obtain as in the proof of Proposition 2.1 that
$$
\langle f_i,f_i\rangle^{-1}f_i =
a_ie_{i-1} + e_i + b_ie_{i+1}
\ \ \ \text{for} \ \ \
i = 2,\dots,n-1,
$$
where
$$
a_i = -\frac{\langle e_{i-1},e_i\rangle
\left(1- \langle e_{i},e_{i+1}\rangle^2\right)}
{1- \langle e_{i-1},e_{i}\rangle^2\langle e_{i},e_{i+1}\rangle^2},
b_i = -\frac{\langle e_{i},e_{i+1}\rangle
\left(1- \langle e_{i-1},e_{i}\rangle^2\right)}
{1- \langle e_{i-1},e_{i}\rangle^2\langle e_{i},e_{i+1}\rangle^2}.
$$
We conclude that
$\langle f_i,f_i\rangle^{-1}\langle f_i,f_{i-1} \rangle = a_i$
and
$\langle f_i,f_i\rangle^{-1}\langle f_i,f_{i+1} \rangle = b_i$.
Hence indeed the property $(1)$ is equivalent to
the property $(1)$ of Proposition 2.3.
\enddemo\qed
The geometry of a $k$-dimensional
simple wedge is completely determined by the
angles $0<\alpha_i <\frac \pi 2, i = 1,\dots,k-1$, that the vectors
$e_i$ make with the vectors $e_{i+1}$ (or equivalently the subspace
$S(e_{i+1},\dots,e_k)$). Assuming that the generators
$\{e_1,\dots, e_k\}$ are unit vectors we have
$$\cos\alpha_i = \langle e_i,e_{i+1} \rangle,\ \ \ \ \ \ \
i = 1,\dots,k-1.\tag{2.1}$$
We choose to characterize the geometry
of a simple wedge by another set of angles,
$0<\beta_i <\frac \pi 2, i = 1,\dots,k-1$ where
$\beta_{k-1}=\alpha_{k-1}$
and
$$
\cos \beta_i = \langle e_i -\langle e_i,e_{i+2} \rangle e_{i+2},
e_{i+1} -\langle e_{i+1},e_{i+2} \rangle e_{i+2}\rangle,
i = 1,\dots, k-2.
$$
i.e., $\beta_i$ is the angle between two $(k-i)$-dimensional faces,
$S(e_i,e_{i+2},e_{i+3},\dots,e_k)$ and
$S(e_{i+1},e_{i+2},,\dots,e_k)$, of the simple $(k-i+1)$-dimensional
wedge $W(e_i,e_{i+1},\dots,e_k)$.
We have
$$
\tan \beta_i = \frac {\tan \alpha_i}{\sin \alpha_{i+1}},\ \ \ \ \ \ \
i = 1,\dots, k-2.\tag{2.2}
$$
Hence the information contained in the set of $\beta$-angles
determines the simple wedge completely (up to an isometry).
\vskip.7cm
\subhead \S 3. Particle falling in a wedge (PW system)
and the system of falling particles in a line (FPL system)\endsubhead
\vskip.4cm
Given an $n$-dimensional wedge $W$ in an $n$-dimensional Euclidean
space E and a vector $a \in int W^*$, we consider the system of a
point particle falling in $W$ with constant acceleration $a$ and
bouncing off elastically from the $(n-1)$-dimensional faces of the
wedge $W$ (a PW system). In an elastic collision with a face the
velocity vector is instantaneously changed: the component orthogonal to
the face is reversed and the component parallel to the face is
preserved.
The condition that the acceleration vector is in the interior of the
dual cone is equivalent to the system being closed (finite) under the
energy constraint. One can change the acceleration vector by rescaling
time, so that in studying the dynamical properties of such a system
only the direction of acceleration matters.
A PW system is in a natural way a Hamiltonian flow with collisions.
If we choose the generators of an $n$-dimensional wedge $W$ as a basis
in $E$, we can identify $E$ with $\Bbb R^n$ with coordinates
$(\eta_1,\dots, \eta_n)$. The wedge $W$ becomes the positive octant
$W = \{(\eta_1,\dots, \eta_n)\in \Bbb R^n| \eta_i \geq 0,
i=1, \dots, n \}$. Let the scalar product be defined in
these coordinates by a positive definite matrix $L$.
It follows immediately from Proposition 2.4 that
\proclaim{Proposition 3.1}
The wedge $W$ is simple if and only if the matrix $K= L^{-1}$ is
tridiagonal with negative entries below and above the diagonal.
\endproclaim\qed
The PW system in the wedge $W$ with the acceleration
$a\in int W^*$ has the following Hamiltonian
$$
H = {\frac 12}\langle K\xi,\xi \rangle + \langle c,\eta \rangle,\tag 3.2
$$
where $\langle \cdot,\cdot \rangle$ denotes the arithmetic scalar product in
$\Bbb R^n$, $\xi \in \Bbb R^n$ is the momentum of the particle and
$c \in \Bbb R^n$ is a vector with all positive entries, so that
the acceleration vector $a = Kc$ is in the interior of the dual wedge.
This representation of the wedge and the PW system
will be referred to later on as standard.
Let us consider the system of $n$ point particles (or rods) in the line
with positions $0\leq q_1 \leq q_2 \leq \dots \leq q_n $
and masses $m_1,...,m_n$, falling with constant acceleration
(equal to $1$) towards the floor (at $0$).
The particles collide elastically with each other and the floor.
This system of falling particles in a line will be referred to as
an FPL system.
Hence between collisions the motion of the particles is
governed by the Hamiltonian $$ H = \sum_{i=1}^n\left(\frac {p_i^2}{2m_i} +
m_iq_i\right).$$
The configuration space of the system
$W = \{q\in\Bbb R^n| 0\leq q_1 \leq q_2 \leq \dots \leq q_n\}$
with the scalar product determined by the kinetic energy is a simple
$n$-dimensional wedge.
To see this we introduce symplectic coordinates $(x,v)$ in
which the scalar product (and the kinetic energy) have the standard
form,
$$
x_i = \sqrt{m_i} q_i,
v_i = \frac{p_i}{\sqrt{m_i}}, i = 1,\dots,n.\tag 3.3
$$
In these coordinates the Hamiltonian of the system changes
to
$$ H = \sum_{i=1}^n\left(\frac {v_i^2}{2} +
\sqrt{m_i}x_i\right)$$
and we can consider $x$ and $v$ as vectors in the same standard
Euclidean space $\Bbb R^n$. The elastic collisions of the particles
are translated into elastic reflections off the faces of the wedge.
In these coordinates the wedge $W$ is
generated by the unit vectors $\{e_1,\dots,e_n\}$
$$
\sqrt{M_i}e_i = (0,\dots,0,\sqrt{m_i},\dots,\sqrt{m_n}),
$$
where $M_i = m_i+\dots+m_n, i= 1,\dots,n$.
We get that for $1\leq i < j \leq n$
$$
\langle e_i, e_j \rangle = \frac {\sqrt{M_j}}{\sqrt{M_i}},
$$
which immediately yields the properties (1) and (2) of the
Proposition 2.3.
Further using \thetag{2.1} and \thetag{2.2} we get for this simple wedge
$$
\cos^2 \alpha_i = \frac {M_{i+1}}{M_i},\ \ \
\sin^2 \alpha_i = \frac {m_i}{M_i},\ \ \
\tan^2 \beta_i = \frac{m_i}{m_{i+1}}.\tag 3.4
$$
It follows from \thetag{3.4} that every simple wedge can appear as the
configuration space of an FPL system with appropriate masses,
depending on the geometry of the wedge. (Note that the formulas
\thetag{3.4} provide clear justification for the introduction of the
$\beta$-angles in the geometric description of a simple wedge.) The
acceleration vector for an FPL system has the direction of the first
generator of the simple wedge, more precisely the acceleration vector
is $M_1e_1$.
We arrived at the important conclusion that the PW system in a simple
$n$-dimensional wedge with the acceleration along the first (or the
last) generator of the wedge is equivalent to the FPL system with
appropriate masses of the $n$ particles.
Finally we introduce yet another symplectic coordinates $(\eta,\xi)$
for the FPL system in which the configuration wedge becomes
the positive octant (standard representation). Let
$$
\aligned
\eta_1 = q_1,\ \ \eta_{i+1}&=q_{i+1}-q_i, \\
p_{i}&=\xi_i-\xi_{i+1},\ \ p_n = \xi_n,\ \ i = 1,\dots, n-1.
\endaligned
$$
The Hamiltonian of the system becomes
$$
H = \sum_{i=1}^{n-1}\frac {(\xi_i-\xi_{i+1})^2}{2m_i} +
\frac{\xi_n^2}{2m_n} + \sum_1^n
M_i\eta_i.
$$
We get the tridiagonal matrix with negative off-diagonal entries
in the kinetic energy, as required by
the Proposition 3.1.
\vskip.7cm
\subhead \S 4. Monotonicity of the FPL systems
\endsubhead
\vskip.4cm
We will recall now the results about the monotonicity and
hyperbolicity of the FPL systems. This system was introduced and
studied in \cite{W1}, where the reader can find more details.
When the masses of particles are equal the system is completely
integrable. Indeed, if we allow the particles to pass through each
other then the $n$ individual energies of the particles are preserved
and provide us with the $n$ independent integrals in involution. In the
case of elastic collisions of the particles we need to use symmetric
functions of the $n$ individual energies as the first integrals in
involution. It was established in \cite{W1}, that if the
masses are nonincreasing,
$m_1 \geq m_2 \geq \dots \geq m_n$, and are not all equal
then system is partially hyperbolic. In a recent paper Simanyi
\cite{S} showed that, if $m_1 > m_2 \geq \dots \geq m_n$, then the
system is completely hyperbolic.
We will give here a detailed and modified proof that the FPL is
monotone under the above condition, which will be the basis for the
proof of our Main Theorem.
In the phase space of an FPL system we introduce
the Euclidean coordinates $(x,v)$ given by \thetag{3.3}.
We choose two bundles of Lagrangian subspaces $L_1$ and
$L_2$
$$
L_1 = \{dv_1 = \dots = dv_n =0\},\ \ \
L_2= \{dx_i= -\frac{v_1}{\sqrt{m_i}}dv_i, i= 1,\dots,n\}.
$$
The Hamiltonian vector field $\nabla H$ belongs to $L_2$.
The quadratic form $Q$ is given by
$$
Q = \sum_{i=1}^ndx_idv_i + \sum_{i=1}^n\frac{v_i}{\sqrt{m_i}}dv_i^2.
$$
\proclaim{Theorem 4.1}
If $m_1 \geq m_2 \geq \dots \geq m_n$, then the FPL system is
monotone (with respect to the bundle of sectors between $L_1$ and
$L_2$).
\endproclaim
Between collisions the form $Q$ is constant.
Indeed we have
$$
\aligned
\frac{d}{dt}x_i = v_i, \ \ \ \ \ \ \ \ \ \ &\frac{d}{dt}dx_i = dv_i,\\
\frac{d}{dt}v_i = -\sqrt{m_i}, \ \ \ \ \ \ \ \ \ \ &\frac{d}{dt}dv_i
= 0, \ \ i= 1,\dots,n,
\endaligned
$$
which yields $\frac{dQ}{dt}=0$.
The effect of a collision between different particles on the form $Q$
will be obtained with the help of the following important construction
which will play a crucial role in the future. We represent our system
as a PW system in a simple $n$-dimensional wedge $W$, with
geometry determined by the masses, (cf. \thetag{3.4}), and the
acceleration vector parallel to the first generator. A collision of
two particles becomes the collision with an $(n-1)$-dimensional face
of the wedge, containing the first generator.
Let us consider the wedge $\widetilde W$ obtained by
the reflection in the face. Instead of reflecting the velocity in the
face we can allow the particle to pass through the face to the
reflected wedge $\widetilde W$. Note that the acceleration vector
stays the same (since it lies in the face). What changes is the
quadratic form, it experiences a jump discontinuity. Let $\widetilde
Q$ be the quadratic form associated with the PW system in the
reflected wedge $\widetilde W$. We want to examine the difference of
$Q$ and $\widetilde Q$ at the common face. Actually, if we identify
all the tangent spaces to the common phase space of the two PW
systems (in $W$ and $\widetilde W$) the forms become functions of
tangent vectors from that common space that depend only on velocities
(but not on positions).
For the purpose of future applications we will consider a
generalization of this construction, namely we will not assume that
the two wedges are symmetric, only that they share a common $(n-1)$
dimensional face.
Let us consider two simple $n$-dimensional wedges
$W = W(e_1,\dots, e_n)$ and $\widetilde W = W(\tilde
e_1,\dots, \tilde e_n)$ (we tacitly assume that the
generators are always written in a chosen distinguished order). Let us
assume that the two wedges have isometric $(n-1)$-dimensional faces,
obtained when we drop $e_{l+1}$ and $\tilde e_{l+1}$, respectively, from
the list of generators. We choose to glue the two wedges together
along the isometric faces, i.e., we assume that $e_i = \tilde e_i,$
for $i
\neq l+1$, and that the two wedges are on opposite sides of the
hyperplane containing the isometric faces. Further we consider the PW
systems in these wedges with common acceleration vector parallel to
the first generator $e_1 = \tilde e_1$.
In each of the wedges the PW system is equivalent to
a PFL system with appropriate masses of the particles,
$(m_1,\dots,m_n)$ and $(\widetilde m_1,\dots,\widetilde m_n)$ respectively.
\proclaim{Lemma 4.2}
$$
\aligned
&m_i = \widetilde m_i, \ \ \text{for all} \ \ i \neq l, l+1,\\
&m_{l} + m_{l+1} = \widetilde m_{l} + \widetilde m_{l+1}.
\endaligned
$$
\endproclaim
\demo{Proof}
Since the two systems have acceleration vectors of the same length
it follows that $M_1 = m_1 + \dots +m_n = \widetilde M_1 =
\widetilde m_1 + \dots +\widetilde m_n$. Our claim follows now from
the formulas \thetag{3.4} for the $\alpha$-angles in a simple wedge,
since the isometry of the faces implies
$ \alpha_i = \widetilde\alpha_i$ for $i \neq l, l+1$.
\enddemo\qed
We introduce the standard Euclidean coordinates, \thetag{3.3},
$x\in\Bbb R^n$ and $\tilde x\in\Bbb R^n$ in $W$ and $\widetilde W$
respectively, associated with the PFL systems. The common face of
the two wedges is described by
$$\frac{x_l}{\sin\beta_l} =
\frac{x_{l+1}}{\cos\beta_l}\ \ \ \ \ \ \ \text{and}
\ \ \ \ \ \ \frac{\tilde x_l}{\sin\tilde\beta_l}=
\frac{\tilde x_{l+1}}{\sin\tilde\beta_l}$$
in respective coordinate systems.
These coordinate systems in the configuration space give rise to the
respective coordinates in the phase spaces $(x,v)$ in $W\times\Bbb
R^n$ and $(\tilde x,\tilde v)$ in $\widetilde W\times\Bbb
R^n$. The tangent spaces (bundles) of these phase spaces are
naturally identified because the wedges are contained in the same
Euclidean space.
The two coordinate systems are connected by the
following ``gluing'' transformation
$$
\aligned
&\tilde x_i = x_i, \ \ \ \ \ \ \ \text{for all} \ \ \ \ \ i \neq l-1,l\\
&\tilde x_{l} = -\cos\Theta \ x_{l} + \sin\Theta \ x_{l+1}, \\
&\tilde x_{l+1} = \sin\Theta \ x_{l} + \cos\Theta \ x_{l+1},
\endaligned
\tag 4.2
$$
where $\Theta = \beta_l + \tilde\beta_l$ is defined by the
$\beta$-angles of the respective wedges, i.e.,
$$ \tan^2\beta_l
= \frac{m_l}{m_{l+1}}, \ \ \ \ \
\tan^2\tilde\beta_l = \frac{\widetilde m_l}{\widetilde m_{l+1}}.$$
Let us consider the quadratic forms $Q$ and $\widetilde Q$ associated
with the respective FPL systems. These quadratic forms depend on
velocities (but not on positions), and the space of
velocities of the two models is the same Euclidean space. Hence we
can compare the two quadratic forms as functions on the space of
velocities cross the tangent to the phase space.
\proclaim{Proposition 4.3}
For the velocities of trajectories
leaving $W$ and entering $\widetilde W$ we have
$$
\widetilde Q \geq Q, \ \ \ \ \text{if and only if,}\ \ \ \
\beta_l +\tilde \beta_l \geq
\frac\pi 2.$$
More precisely, we have
$$
\widetilde Q - Q = \frac 1{\sqrt{m_l+m_{l+1}}}
\frac{2\sin(2(\beta_l+\tilde\beta_l))}
{\sin2\beta_l\sin2\tilde\beta_l}
(-\cos\beta_l v_l + \sin\beta_l v_{l+1})
(-\cos\beta_l dv_l + \sin\beta_l dv_{l+1})^2.
$$
\endproclaim
\proclaim{Corollary 4.4}
$Q = \widetilde Q$ if and only if $\beta_l +\tilde \beta_l =
\frac\pi 2$.
\endproclaim
\demo{Proof}
Let us examine the quadratic form
$$
Q = \sum_{i=1}^ndx_idv_i + \sum_{i=1}^n\frac{v_i}{\sqrt{m_i}}dv_i^2.
$$
The first sum is invariant under any coordinate changes which respect
the distinction between positions and velocities. In the second
sum only two terms are affected by the gluing transformation. Hence
we obtain
$$
\widetilde Q - Q = \sum_{i=l}^{l+1}\frac{\tilde v_i}{\sqrt{m_i}}d\tilde v_i^2 -
\sum_{i=l}^{l+1}\frac{v_i}{\sqrt{m_i}}dv_i^2.
$$
Our claim follows now by straightforward calculations. To make them
more transparent we introduce yet another coordinate systems in the planes
$(v_l,v_{l+1})$ and $(\tilde v_l,\tilde v_{l+1})$
$$
\aligned
z_1 &= \sin\beta_lv_l + \cos\beta_lv_{l+1},\\
z_2 &= -\cos\beta_lv_l + \sin\beta_lv_{l+1},
\endaligned
$$
and parallel formulas for $(\tilde z_1,\tilde z_2)$.
We have
$$
\sum_{i=l}^{l+1}\frac{v_i}{\sqrt{m_i}}dv_i^2 =
\frac 1{\sqrt{m_l+m_{l+1}}}(z_1dz_1^2 +
2z_2dz_1dz_2 -2\cot 2\beta_l z_2dz_2^2)
$$
and the gluing map \thetag{4.2} is given by
$$
\aligned
\tilde z_1 &= z_1,\\
\tilde z_2 &= -z_2.
\endaligned
$$
Now we get immediately
$$
\widetilde Q - Q = \frac 2{\sqrt{m_l+m_{l+1}}}
(\cot2\tilde\beta_l-\cot2\beta_l) z_2 dz_2^2.
$$
It remains to observe that crossing from $W$ to $\widetilde W$ corresponds
to $z_2 < 0$.
\enddemo\qed
It follows from Proposition 4.3 that in a PW system in a
simple wedge with the acceleration along the first generator the
value of the $Q$-form does not decrease in a collision with any face
containing the acceleration vector, provided that
$$\beta_i \geq \frac\pi4, \ \ \ i = 1,2,\dots,n-1.$$
Equivalently in the FPL system the $Q$-form does not decrease
in a collision between two particles, if only $m_1\geq m_2 \geq \dots
\geq m_n$, and this condition is necessary.
Let us further examine the change in the
$Q$-form, if in the time interval $[0,t]$ we have the collision with
the floor of the first particle at time $t_c, 0 < t_c < t$ (and no
other collisions). It is clear that the calculation reduces to the
variables $(x_1,v_1)$ alone. Let $x = x_1(0),v=v_1(0), \hat
x=x_1(t), \hat v = v_1(t),a = \sqrt{m_1}$ and $v_c = v_1(t_c^-)<0$.
We have $$
\aligned
\hat x = (t-t_c)v_c -\frac 12 a(t-t_c)^2\\
\hat v = -v_c-a(t-t_c),
\endaligned
$$
where
$$
at_c = v + \sqrt{v^2 + 2ax}, \ \ v_c = -\sqrt{v^2 + 2ax}.
$$
So $(\hat x,\hat v)$ depends on $(x,v)$ smoothly (unless $(x,v) =
(0,0)$) and we can calculate the derivative.
We get $d\hat v = dv -\frac 2{v_c}(vdv +adx)$. From the
preservation of the energy in a collision we conclude that
$\hat v d\hat v +ad\hat x = vdv+adx$. Now the difference of
the quadratic form at time $t$ and time $0$ is
$$
(\hat v d\hat v +ad\hat x)\frac{d\hat v}a -
( v d v +ad x)\frac{d v}a = -\frac 2{av_c}(vdv +adx)^2 \geq 0.
$$
We conclude that in a collision with the floor the derivative of the
Hamiltonian flow is monotone (the $Q$-form does not decrease).
Note that no further conditions on the masses
(on the $\beta$-angles in the PW system) are necessary to assure
the monotonicity in the collision of the first particle with the floor
(collision with the face which does not contain the acceleration
vector). Theorem 4.1 is proven.
Let us now examine the $L_1$ and $L_2$-exceptional subspaces, and
$L_1$ and $L_2$-exceptional points. First we look what happens to
tangent vectors from the two Lagrangian subspaces in a collision with
the floor. Using the formulas developed above we get that for a
vector from $L_1$ either $dx_1 \neq 0$ and then the vector enters the
interior of the sector $\Cal C = \{Q > 0\}$ or $dx_1 = 0$ and then
$d\hat x_1 =0$ and the vector stays in $L_1$, and under the
identification of the tangent spaces the vector does not change.
For a vector from $L_2$ we have $d\hat v_1 = dv_1$ and the vector stays
in $L_2$. If we use $(dv_1,\dots, dv_n)$ as coordinates in $L_2$ these
Lagrangian subspaces become naturally identified and we conclude that
in a collision with the floor a vector from $L_2$ stays in $L_2$ and
is not changed at all.
By Proposition 4.3 in a collision of an
$l$-th particle with the $(l+1)$-st lighter
particle ($m_l > m_{l+1}$) a vector
from $L_2$ either enters the interior of the sector $\Cal
C$ or
$ \frac{dv_l}{\sqrt{m_l}} =\frac{dv_{l+1}}{\sqrt{m_{l+1}}}$
and the $v$-components of the vector are not changed.
(In the language of the PW system this last condition means that the
velocity component of the vector is parallel to the face of the wedge
in which the particle is reflected.)
As a result,
for vectors from $L_2$ we get one equation for each nondegenerate
collision of two particles. Since also no collision with the floor
can change the $v$-components of a vector from $L_2$,
it follows immediately that, if the masses of the particles decrease
(strictly) every vector in $L_2$ enters eventually the interior of
the sector $\Cal C$ except when $\frac{dv_i}{\sqrt{m_i}}$ are all
equal for $i = 1,\dots, n$. This last condition means that the vector is
parallel to the Hamiltonian vector field. It shows that if the masses
decrease there are no $L_2$-exceptional trajectories of the
flow (among regular trajectories).
None of the vectors from $L_1\cap\{dH=0\}$ can enter the interior of
the cone as a result of a collision of two particles. They stay in
$L_1$ but they are changed by the appropriate reflection in a face of
the wedge. Only the collision with the floor can push vectors from
$L_1$ into the interior of the sector $\Cal C$. It happens if
$dx_1\neq 0$ immediately before the collision. Hence in principle
there may be $L_1$-exceptional trajectories on which the collisions
between particles always ``prepare'' some vectors before each collision
with the floor so that $dx_1=0$.
In a recent paper Simanyi \cite{S} showed that the set of
$L_1$-exceptional trajectories is at most a countable union of
codimension 1 submanifolds.
\proclaim{Theorem 4.5(Simanyi \cite{S})}
If $m_1 > m_2 \geq \dots \geq m_n$, then the FPL system is
completely hyperbolic.
\endproclaim
\vskip.7cm
\subhead \S 5. Special examples
\endsubhead
\vskip.4cm
{\bf Capped system of particles}
Let us explore the consequences of the property that a
simple wedge has two distinguished orderings of generators.
An FPL system is equivalent to a PW system with the acceleration
vector parallel to the first generator. Let us modify the FPL system
so that the wedge stays the same but the acceleration becomes parallel
to the last generator. This is accomplished by changing the potential
energy and the resulting Hamiltonian is
$$
H = \sum_{i=1}^n\frac {p_i^2}{2m_i} + m_nq_n.$$
As before the configuration space is $\{q\in\Bbb R^n| 0\leq q_1
\leq q_2 \leq \dots \leq q_n\}$
and the particles collide with each
other and the floor. We will call it the capped system of
particles in a line.
The new feature is that between collisions the particles move
uniformly (with constant velocity) except for the last particle which
is accelerated down (it falls down). It is this last particle (``the
cap'') that keeps the system closed, i.e., the energy surface $\{H =
const\}$ is compact and it carries a finite Liouville measure.
The capped system of particles is equivalent to another
FPL system with different masses. We will calculate these masses
(or equivalently the $\beta$-angles) to establish conditions under
which the capped system is completely hyperbolic or completely
integrable.
Note that the $\beta$-angles are complete Euclidean invariants
of a simple wedge with a chosen distinguished ordering of the
generators, so they do change when we change the distinguished
ordering and the last generator becomes the first.
\proclaim{Theorem 5.1}
The capped system of particles in a line is completely
integrable if
$$
m_k = \frac{n}{k(k+1)}m_n, \ \ \ \text{for}\ \ \ k=1,2,\dots,n-1,
$$
and completely hyperbolic if
$$
m_i(1+\frac{M_i}{m_{i-1}}) \geq M_{i+1},
\ \ \ \text{for}\ \ \ i= 2,\dots,n-2,
$$
and
$$
m_1 \geq M_2, \ \ \ m_n > m_{n-1}\frac{m_{n-2}-m_{n-1}}{m_{n-2}+m_{n-1}}>0.
$$
\endproclaim
\demo{Proof}
We need to calculate the $\alpha$ and $\beta$ angles for the reversed
ordering of the generators of the simple wedge.
Let us denote these angles for the reversed ordering by
$\widehat\alpha_k$ and $ \widehat\beta_k, k= 1,\dots,n-1,$ respectively.
>From \thetag{2.1},\thetag{2.2} and \thetag {3.4}
we obtain for $k= 1,\dots,n-1$
$$
\aligned
\cos^2\widehat\alpha_k = \cos^2\alpha_{n-k}=& \frac {M_{n-k+1}}{M_{n-k}},\\
\sin^2\widehat\alpha_k = &\frac {m_{n-k}}{M_{n-k}},\\
\tan^2\widehat\beta_{n-1} = \tan^2\widehat\alpha_{n-1}=&\frac{m_1}{M_2}
\endaligned
$$
and for $k= 1,\dots,n-2$
$$
\tan^2\widehat\beta_{k} = \frac{\tan^2\widehat\alpha_{k}}{\sin^2\widehat\alpha_{k+1}}
=\frac{m_{n-k}M_{n-k-1}}{m_{n-k-1}M_{n-k+1}}.
$$
Introducing $X_i =\frac 1{M_i}$ for
$i = 1,\dots,n-1$ and $X_0 = 0$ we can rewrite this as
$$
\tan^2\widehat\beta_{n-i} = \frac{X_{i+1}-X_i}{X_i-X_{i-1}},
$$
for $i = 1,\dots,n-1$.
By the results of Section 4,
the condition of complete integrability is that
$\tan^2\widehat\beta_{k} = 1$, for $k= 1,\dots,n-1$. It is equivalent to
the linearity condition
$$
X_{i+1}-X_i = X_i-X_{i-1}, \ \ i = 1,\dots,n-1.
$$
It follows that $X_i = \frac ic$ for some constant c. The claim about
complete integrability follows.
We can apply Theorem 4.5, if
$\tan^2\widehat\beta_{k} \geq 1$ for $k= 1,\dots,n-1$, and
$\tan^2\widehat\beta_{1} > 1$.
This gives us the convexity condition
$$
X_{i+1}-X_i \geq X_i-X_{i-1}, \ \ i = 1,\dots,n-1, \ \ \text{and}
\ \ X_{n}-X_{n-1} > X_{n-1}-X_{n-2},
$$
which translates into the conditions in the Theorem.
\enddemo\qed
{\bf System of attracting particles in a line}
Let us consider the system of $n+1$ point particles in the line
with positions $q_0 \leq q_1 \leq \dots \leq q_n $ and masses
$m_0,...,m_n$. They collide elastically with each other and their
interaction is defined by a linear translationally invariant potential
\hbox{$\sum_{i=1}^n m_i (q_i-q_0)$.} Thus the Hamiltonian of the
system is $$ H = \sum_{i=0}^n\frac{p_i^2}{2m_i} +
\sum_{i=1}^nm_i(q_i-q_0).$$ The total momentum is preserved in this
system. Setting the total momentum to zero and fixing the center of
mass $m_0x_0+ m_1x_1+\dots+m_nx_n=0$ we obtain a PW system
in a simple wedge with acceleration parallel to the first generator
(hence our system is also equivalent to an FPL system).
Indeed, introducing symplectic coordinates $(\eta,\xi)$
$$
\aligned
\eta_0 &= m_0q_0 + m_1q_1 + \dots +m_nq_n\\
\eta_i &= q_i - q_{i-1}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
i = 1,2, \dots, n\\
p_0 &= m_0\xi_0 - \xi_1 \\
p_{i} &= m_{i}\xi_0 + \xi_{i}-\xi_{i+1} \ \ \ \ \ \ \ \ \
i = 1,2, \dots, n-1\\
p_n &= m_n\xi_0 + \xi_n.
\endaligned
$$
and setting the total momentum and the center of mass at zero,
$ \eta_0 = 0, \xi_0 =0$, we obtain the Hamiltonian
$$
H = \frac {\xi_1^2}{2m_0} +
\sum_{i=1}^{n-1} \frac{(\xi_i-\xi_{i+1})^2}{2m_i}+ \frac {\xi_n^2}{2m_n} +
\sum_{i=1}^n M_i\eta_i,
$$
where $M_i = m_i+\dots+m_n,$ for $i = 0,1,\dots,n$.
By Proposition 3.1 the wedge
$W = \{\eta_1 \geq 0, \dots, \eta_n \geq
0\}$ is simple. It can be also checked that
the acceleration is parallel to the first generator. (Acceleration
parallel to the last generator corresponds to the potential
$\sum_{i=0}^{n-1}m_i(q_n-q_i)$, which gives a symmetric system
where the special role is played by $q_n$ rather than $q_0$.)
\proclaim{Theorem 5.2}
The system of attracting particles is completely integrable
if for some $a > n$
$$
m_i = \frac{m_0+m_1+\dots+ m_{i-1}}{a-i}, \ \ \ i= 1,\dots, n,
$$
and it is completely hyperbolic,,
if for some $a_1 < a_2 \leq a_3 \leq \dots \leq a_n$ such that
$a_i > i, i= 1,\dots, n$ we have
$$
m_i = \frac{m_0+m_1+\dots+ m_{i-1}}{a_i-i}, \ \ \ i= 1,\dots, n.
$$
\endproclaim
\demo{Proof}
The $n$-dimensional wedge $W = \{\eta_1 \geq 0, \dots, \eta_n \geq
0\}$ has in the original coordinates the following unit generators
$e_i = (e_i^0,e_i^1,\dots,e_i^n,), i = 1,2,\dots,n$, where
$$
\sqrt{M_0}e_i^k = \cases -\sqrt{\frac{M_i}{M_0-M_i}}\ \ \ \text{if}
\ \ \ k< i,\\
\sqrt{\frac{M_0-M_i}{M_i}}\ \ \ \text{if}
\ \ \ k\geq i.\endcases
$$
The acceleration vector is parallel to $e_1$ and we get that for
$1\leq i X_{1}.$$
This leads to the conditions in the Theorem.
\enddemo\qed
For example, in the case of equal masses $m_1 = m_2 = \dots =m_n$,
we have complete hyperbolicity of the system. Note that we can
rewrite the potential energy as
$$
\sum_{i=1}^nM_i(q_{i}-q_{i-1}).
$$
i.e., we can interpret the interaction of the particles as the attraction
of nearest neighbors, but then the force of attraction decays for
particles further to the right.
\vskip.7cm
\subhead \S 6. Wide wedges and the Main Theorem
\endsubhead
\vskip.4cm
Let $W = W(g_1,\dots,g_k)$ be a $k$-dimensional wedge
in a Euclidean $n$-dimensional space $E$.
\proclaim{Definition 6.1}
A $k$-dimensional wedge $W = W(g_1,\dots,g_k)$ is called
wide, if the angles between the generators exceed $\frac\pi 2$,
i.e., $\langle g_i,g_j\rangle < 0$ for any $1\leq i < j \leq k$.
\endproclaim
Clearly every face of a wide wedge is a wide wedge of lower dimension.
\proclaim{Proposition 6.2} If a $k$-dimensional wedge $W$
is wide then the dual wedge $W^*$ is contained in $W$ and
the inclusion is strict, in the sense that
the only point in the intersection of the boundaries of $W$ and $W^*$
is $0$.
\endproclaim
\demo{Proof} We will prove it by induction on the dimension
of the wedge (the number of generators). For two generators
the Proposition clearly holds. Let $\{g_1,\dots,g_k,g_{k+1}\}$ be
generators of a $k+1$-dimensional wedge $W_{k+1}$. Let us consider the
$k$-dimensional wedge $W_k=W(g_1,\dots,g_k)$ and let $\{h_1,\dots,h_k\}$ be
the generators of the dual wedge $W_k^*$ which also form a basis dual
to the basis $\{g_1,\dots,g_k\}$ (we identify a Euclidean space and its
dual, so that $\{h_1,\dots,h_k\}$ form a basis in the subspace
$S(g_1,\dots,g_k)$. By the inductive assumption
$W_k^* \subset W_k$ and the inclusion is strict. Let us consider
the basis $\{\tilde h_1,\dots,\tilde h_k,\tilde h_{k+1}\}$ dual to
$\{g_1,\dots,g_k,g_{k+1}\}$, i.e., $\langle \tilde h_i,g_j\rangle =
\delta_i^j$, the Kronecker's delta.
It is quite clear that for $i =1, \dots, k$
$$
\tilde h_i = h_i - \langle h_i,g_{k+1}\rangle \tilde h_{k+1}. \tag6.1
$$
We know (by the inductive assumption) that $h_i \in int W_k$
for $i =1, \dots, k$. We need to show that for $i =1, \dots, k+1,$
$\tilde h_i\in int W_{k+1}$, which by \thetag{6.1} will follow from
$\langle h_i,g_{k+1} \rangle< 0$ and $\tilde h_{k+1}\in int
W_{k+1} $.
By the inductive assumption $h_i = \sum_{i=1}^kh_i^jg_j$ with $h_i^j > 0$
for $j= 1,2,\dots,k$. It follows that for $i =1, \dots, k$
$$
\langle h_i, g_{k+1} \rangle
= \sum_{i=1}^kh_i^j\langle g_j,g_{k+1} \rangle < 0,\tag6.2
$$
because the wedge is assumed to be wide.
Let finally $\tilde h_{k+1} = \sum_{i=1}^{k+1}a_ig_i$.
We have $a_{k+1} = \langle \tilde h_{k+1},\tilde h_{k+1}\rangle > 0$.
For $i =1, \dots, k$
$$
0 = \langle \tilde h_{k+1},h_i\rangle = a_i + a_{k+1}\langle
h_i,g_{k+1} \rangle,
$$
which in view of \thetag{6.2} shows that $a_i >0$.
\enddemo\qed
\proclaim{Corollary 6.3}
If the $n$-dimensional wedge $W(g_1,\dots,g_n)$ is wide then
the angle between any two of its codimension $1$ faces exceeds
$\pi\over 2$.
\endproclaim
\proclaim{Corollary 6.4}
If the $n$-dimensional wedge $W(g_1,\dots,g_n)$ is wide and
$\{f_1,\dots,f_n\}$ is the basis dual to $\{g_1,\dots,g_n\}$
then
$\langle f_i,\,f_j \rangle > 0$, for any $1\leq i,j \leq n$.
\endproclaim
Note that the converse of Proposition 6.2 (or of any of the
Corollaries) does not hold for $k\geq 3$.
\proclaim{Proposition 6.5}
If a wedge is wide, then the orthogonal projection
of the interior of the dual wedge onto any of its faces
is contained in the interior of that face.
\endproclaim
In particular, orthogonal
projections on
any face of the wedge of any vector from the interior of the
dual wedge are nonzero.
\demo{Proof} Let $\{g_1,\dots,g_k\}$ be generators of the wedge
and let $\{h_1,\dots,h_k\}$ be the dual basis (and hence
also the generators of the dual wedge). Orthogonal projections
of $h_1,\dots,h_l$ on the subspace $S(g_1,\dots,g_l)$
form the basis dual to the basis $\{g_1,\dots,g_l\}$ in this subspace.
It follows that the orthogonal projection of the dual wedge onto
a face is the dual wedge of that face. Since all the faces are also
wide, by Proposition 6.2 this projection is contained in the face
and the inclusion is strict.
\enddemo\qed
We can now formulate and prove our
\proclaim{Main Theorem 6.6}
The PW system in a wide wedge with arbitrary
acceleration vector from the interior of the dual wedge
is completely hyperbolic.
\endproclaim
In the standard representation
$W = \{(\eta_1,\dots, \eta_n)\in \Bbb R^n| \eta_i \geq 0,
i=1, \dots, n \}$ and the scalar product in the $\eta$ coordinates
is defined by a positive definite matrix $L=(l_{ij})$. The assumption that
$W$ is wide translates to the assumption that $l_{ij} < 0$, for
$i\neq j$. By Corollary 6.4 it follows that the inverse matrix $K =
L^{-1}$ has all positive entries. The Hamiltonian of the system is
$$
H = {\frac 12}\langle K\xi,\xi \rangle + \langle c,\eta \rangle,
$$
where $\langle \cdot,\cdot \rangle$ denotes the arithmetic scalar product in
$\Bbb R^n$, $\xi \in \Bbb R^n$ is the momentum of the particle and
$c \in \Bbb R^n$ is a vector with all positive entries, so that
the acceleration vector equal to $Kc$ is in the interior of the dual wedge.
Now the Main Theorem can be reformulated as the Main Theorem from the
Introduction.
\demo{Proof of the Main Theorem}
Let the wide wedge be $W = W(g_1,\dots,g_n)$. We will divide it
into $n!$ simple wedges. For that purpose let us note that
a simple wedge is uniquely defined by a choice of the first
generator $e_1$ and a flag of subspaces
$$
S_1 \supset S_{2}\supset \dots \supset S_n
$$ such that
$e_1$ is not orthogonal to any of the subspaces.
Indeed, given such a flag we define $e_k$ as the orthogonal projection
of $e_1$ on $S_{k}$. Clearly the wedge
$W(e_1,\dots,e_n)$ is simple. Conversely, for a simple wedge
$W(e_1,\dots,e_n)$ we obtain the flag of subspaces by considering
$S_{k} = S(e_k,\dots,e_n), k = 1,\dots,n$.
The first generator of all our simple wedges will be
the acceleration vector.
We define a simple wedge $W_\sigma$, for any permutation
$\sigma$ of the $n$ indices $\{1,2,\dots,n\}$, by the flag
$$
S_{k} = S(g_{\sigma(k)},g_{\sigma(k+1)}\dots,g_{\sigma(n)}),\ \
\ \ k = 1,\dots,n.
$$
By Proposition 6.5 the acceleration vector is not orthogonal to any of
the faces of the wide wedge,
so that these $n!$ flags define indeed simple wedges
and moreover
$$
\bigcup_{\sigma} W_\sigma = W(g_1,\dots,g_n),
$$
and the interiors of these simple wedges are mutually disjoint.
The intersection of all these wedges is the ray spanned by their
first generator $e_1$ (the acceleration vector).
Let us consider two adjacent wedges, $W_{\sigma_0}$ and
$W_{\sigma_1}$,
i.e., two wedges which
have a common $(n-1)$ dimensional face. Without loss of generality we
can assume that $\sigma_0$ is the identity permutation. Then by
necessity $\sigma_1$ is a transposition of two adjacent indices,
say $l$ and $l+1$. Let us consider the $\beta$-angles for
$W_{\sigma_0}$ and $W_{\sigma_1}$, $\beta_1,\dots, \beta_{n-1}$ and
$\tilde\beta_1,\dots, \tilde\beta_{n-1}$, respectively.
\proclaim{Lemma 6.7}
For the two adjacent wedges $\beta_l +\tilde \beta_l >
\frac\pi 2$.
\endproclaim
\demo{Proof of Lemma 6.7}
The two adjacent wedges have the same first $l$ generators,
$e_1,\dots,e_l$, and the same last $n-l-1$ generators $e_{l+2},\dots,e_n$.
Let $ e_{l+1}, \tilde e_{l+1}$ be the two different generators for
the wedges $W_{\sigma_0}$ and $W_{\sigma_1}$, respectively.
The angles $\beta_l$ and $\tilde \beta_l$ are equal to the angles between
codimension $1$ subspaces of $S_{l} = S(g_l,g_{l+1},\dots,g_n)$.
$\beta_l$ is the angle between the subspace
$S(e_{l}, g_{l+2},\dots, g_n)$ and the subspace
$S(e_{l+1}, g_{l+2},\dots, g_n) = S(g_{l+1}, g_{l+2},\dots, g_n)$.
$\tilde\beta_l$ is the angle between the subspace
$S(e_{l}, g_{l+2},\dots, g_n)$ and the subspace
$S(\tilde e_{l+1},g_{l+2},\dots,g_n) = S(g_{l},g_{l+2},\dots,g_n)$.
Since these three subspaces of $S_l$
have in common the codimension $2$ subspace
$S(g_{l+2},\dots,g_n)$ we conclude that $\beta_l + \tilde \beta_l$
is the angle between two subspaces of $S_l$,
$S(g_{l+1},g_{l+2},\dots,g_n)$ and $S(g_{l},g_{l+2},\dots,g_n)$.
Observing that these are two faces of the wide wedge
$W(g_{l},g_{l+1},\dots,g_n)$ we obtain the Lemma from Corollary 6.3.
\enddemo\qed
We introduce in each of the simple wedges the form $Q$ furnished
by the canonical isomorphism with an FPL system.
We obtain a piecewise continuous $Q$-form in the tangent bundle
of the phase space of our system. This form is defined by
two Lagrangian bundles $L_1$ and $L_2$. Note that the bundle
$L_1$ is continuous (with the natural identification of the tangent
spaces to the phase space $W\times \Bbb R^n$ it is actually
constant) while
$L_2$ experiences jump discontinuities when we cross from one simple
wedge to another (cf. Corollary 4.4).
By Proposition 4.3 and Lemma 6.7 our system is $Q$-monotone. To apply
Theorem 1.4 it remains to examine $L_1$ and $L_2$-exceptional
trajectories.
Let us consider Euclidean
coordinates $(x,v)\in \Bbb R^n\times\Bbb R^n$
in which
the kinetic energy has the standard form ($\langle v,\,v\rangle/2$),
where $x\in\Bbb R^n$
represents a position of the particle in the wedge
$W$ and $v\in \Bbb R^n$
represents its velocity. In these coordinates $L_1 =\{dv_1 = \dots =
dv_n =0\}$ and the $Q$-form is
$$Q = \sum_{k=1}^ndx_kdv_k +
\sum_{i,j=1}^nz_{ij}dv_idv_j,$$
where the symmetric matrix $\{z_{ij}\}$
depends on the simple wedge $W_\sigma$ in which $x$ is located and the
velocity $v$. Vectors from $L_1$ are not changed by the derivative of
the flow unless there is a collision with one of the faces of the wide
wedge. This collision considered in the respective FPL system
becomes the collision with the floor.
The effect of such a collision on vectors from $L_1$
was discussed in Section 4. In
our present language the conclusion is the following. Either a vector
from $L_1\cap \{dH =0\}$ is parallel to the face of the wedge where
the collision is occurring and then it is not changed by the derivative
of the flow, or it is transversal to the face and then it enters the
interior of the sector
$\Cal C = \{Q(v)\geq 0\}$ as a result of the collision. Hence the only
vectors from $L_1$ which do not ever enter the interior of the
sector $\Cal C$
are the vectors parallel to all the faces with which the particle
collides
in the future. There are no vectors parallel to all the
$(n-1)$-dimensional faces of the wide wedge. It follows
that the only $L_1$-exceptional trajectories could be those for which
the particle does not collide in the future
with one (or several) of the faces.
\proclaim{Claim 1} There are no nondegenerate trajectories avoiding in
the future collisions with one of the faces.
\endproclaim
A degenerate trajectory is the one which hits two faces
simultaneously or has velocity with zero orthogonal component to the
face at the time of collision, i.e., it is a trajectory
for which there is no
natural continuation of the dynamics.
To prove the claim let us note that, if the avoided face is
$W(g_2,\dots,g_n)$, then the component of the velocity in the
direction of $g_1$ is preserved at all other collisions, since all
the faces with which our orbit collides are parallel to $g_1$.
Between collisions we have
$$
\frac{d}{dt}\langle v,g_1\rangle = -\langle a,g_1 \rangle < 0
$$ because the acceleration vector $a$ is taken from the interior
of the dual cone.
We obtain the contradiction that on our orbit $\langle v,g_1\rangle$
goes to $-\infty$. Hence there are no such orbits.
(Note that we did not need to use the Simanyi's method \cite{S}
and we have established more. It is quite plausible that also in the
case of a simple wedge there are no $L_1$ exceptional orbits. It is
indeed so in the case obtained from the symmetric wide wedge,
when every pair of adjacent simple wedges $W_{\sigma_1}$ and
$W_{\sigma_2}$ is symmetric with respect to the common face
(cf. Section 7).)
It remains to show that there are only few $L_2$-exceptional
trajectories. We identify all the $L_2$ subspaces with the tangent
velocity space by the natural projection. Further the tangent velocity
space can be naturally identified with the tangent configuration
space. Hence we can use $(d\eta_1,\dots,d\eta_n)$ as coordinates in the
spaces $L_2$.
With this identification vectors from $L_2$ are not changed
in a collision with the faces of the wide wedge. Neither they are
changed between collisions. But when the trajectory crosses from one simple
wedge to an adjacent one a tangent vector from $L_2$
is likely to get into the interior of the sector
$\Cal C$ because $L_2$ experiences a jump discontinuity.
Again the results of Section
4 (Proposition 4.3) can be translated into our current language as the
following alternative. When the trajectory crosses transversally from one
simple wedge to another, a vector from $L_2$ either enters immediately
the interior of the sector $\Cal C$ or it is parallel to the common face.
The trajectories with
velocities which are not transversal to the common face at the time of
crossing can be dropped from considerations, they form a set of zero
measure in the phase space. Let us call such trajectories
degenerate. We are going to prove
\proclaim{Claim 2} There are no nondegenerate $L_2$-exceptional
trajectories.
\endproclaim
We need to prove that along any nondegenerate trajectory the
intersection of all subspaces of codimension $1$
containing the faces of the
wedges that the trajectory crosses is equal to the one
dimensional subspace spanned by the acceleration vector. The task of
tracing these subspaces along a given trajectory is made cumbersome
by the fact that they depend in general on the geometry of the wide
wedge. We avoid this difficulty by focusing on very special common
faces, which in particular lie in subspaces that do not depend on the
geometry of the wedge.
As observed earlier, for a pair of adjacent simple wedges,
$W_{\sigma_1}$ and $W_{\sigma_2}$, we have that
the permutation
$\sigma_2\sigma_1^{-1}$ is a transposition of two adjacent indices. We
consider only those pairs of adjacent simple wedges for which this
transposition is the transposition of $1$ and $2$. It is not hard to
see that the common face of such a pair must be contained in the
subspace of the form
$$S(a,g_{\sigma_1(3)},g_{\sigma_1(4)}\dots,g_{\sigma_1(n)}).$$
This subspace is given by the equation $$
\frac{\eta_k}{a_k} = \frac{\eta_l}{a_l},
$$
where $\sigma_1(1) = k, \sigma_1(2) = l$.
Hence the crossing of such a common face forces the respective
relation on the $L_2$-exceptional subspace
$$
\frac{d\eta_k}{a_k} = \frac{d\eta_l}{a_l}.\tag6.3
$$
It remains to show that there are enough of the relations \thetag{6.3}
along every nondegenerate trajectory to force
$$
\frac{d\eta_1}{a_1} = \frac{d\eta_2}{a_2} = \dots = \frac{d\eta_n}{a_n}.
$$ It follows readily from Claim 1. Indeed, according to Claim 1, any
trajectory will collide with the face $W(g_2,\dots,g_n)$ and then
after some time it will collide with every other face of the wide
wedge. Hence, for any $s = 2,\dots,n$, the trajectory must go from a
simple wedge $W_{\sigma_1}$ to $W_{\sigma_2}$, where $\sigma_1(1) = 1$
and $\sigma_2(1) = s$, (needless to say these simple wedges are not
adjacent in general). Let us trace the crossings from one simple wedge
to the adjacent one, on the way from $W_{\sigma_1}$ to $W_{\sigma_2}$,
by the transpositions required to get from $\sigma_1$ to $\sigma_2$.
Among these transposition we must have enough transpositions of the
first two indices to change $1$ into $s$. Independent of how many
times and when this special transposition occurs, the respective
equalities \thetag{6.3} will force $$ \frac{d\eta_1}{a_1} =
\frac{d\eta_s}{a_s}, $$ which proves the Claim 2, and ends the proof of
the Main Theorem.
\enddemo\qed
\vskip.7cm
\subhead \S 7. Systems of attracting particles with
arbitrary constraints
\endsubhead
\vskip.4cm
Let us consider the system of particles falling to the floor of finite
mass described in the
Introduction, with the Hamiltonian
$$H = \sum_{i=0}^n\frac {p_i^2}{2m_i} + \sum_{i=1}^n
c_i(q_i-q_0) \tag{7.1}$$
and the elastic constraints
$$
q_1 -q_0 \geq 0, q_2 -q_0 \geq 0, \dots q_n-q_0 \geq 0.\tag{7.2}
$$
It satisfies the conditions
of the Main Theorem. Indeed
by the change of variables \thetag{0.3} and
the Hamiltonian reduction $ \eta_0 = 0, \xi_0 =0$,
we obtain the Hamiltonian
$$H = \frac {(\xi_1 + \dots +\xi_n)^2}{2m_0} +
\sum_{i=1}^n \frac{\xi_i^2}{2m_i} +
\sum_{i=1}^n c_i\eta_i, \tag{7.3}$$
and the system is constrained to the wedge
$W = \{\eta_1 \geq 0, \dots, \eta_n \geq 0\}$.
It is a straightforward calculation that the inverse of the matrix
$K$ giving the kinetic energy has only negative off-diagonal elements.
Hence the wedge $W$ is wide. We will denote by $\{e_1,\dots, e_n\}$
the generators of $W$.
In the case $n = 3$ by taking different masses
we can obtain all possible wide wedges. For $n \geq 4$ there are many
more wide wedges then covered by these Hamiltonians.
Let us take the special potential
function with $c_i = \alpha m_i$. The Hamiltonian equations become
$$
\aligned
\frac{d\eta_i}{dt} &= \frac{\xi_1 + \dots \xi_n}{m_0} +
\frac{\xi_i}{m_i} = u_i\\
\frac{d\xi_i}{dt} &= - \alpha m_i, \ \ \ \ \ \ \ \ i = 1, \dots, n.
\endaligned
\tag{7.4}
$$
We have further
$$
\frac{d^2\eta_i}{dt^2} = \frac{du_i}{dt} =
-\alpha\frac{m_0 +m_1 + \dots m_n}{m_0} = -\alpha \frac{M}{m_0},
$$
i.e., the accelerations of all $\eta$ coordinates are equal.
This choice of the
acceleration vector and the particular geometry
of the wide wedge given by the kinetic energy in
\thetag{7.3} implies that the partition into simple wedges,
introduced in the proof of the Main Theorem, is given by
$$
W_\sigma = \{\eta \in \Bbb R^n\big| 0 \leq \eta_{\sigma(1)} \leq
\eta_{\sigma(2)} \leq \dots \leq \eta_{\sigma(n)}\}.
$$
We consider now a system obtained by adding more constraints
of the form $q_k \leq q_l$ to the wide
constraints \thetag{7.2}. These additional constraints
constitute the ``stacking rules'' as explained in the Introduction.
They define a convex polyhedral cone $T$ contained in the wide wedge $W$.
In our list of constraints some constraints are the consequence
of others. We can naturally introduce the minimal set of constraints.
Clearly the minimal set of constraints
is in one to one correspondence with the faces
of the cone $T$.
A convenient way of describing the minimal set of constraints
is by an oriented graph
$\Cal G$ with $n+1$ vertices labeled by the masses $m_0, m_1,\dots,
m_n$. The graph contains an edge from $m_k$ to $m_l$,
if the constraint $q_k \leq q_l$ belongs to the minimal set of
constraints. The resulting graph is connected. We will refer to
$m_0$ as the floor of the graph. Every vertex can be reached by a path
from the floor. We will call such a
graph a { \it graph of constraints}. In Fig 1. we give
all possible graphs of constraints for $4$ masses (up to permutations
of the masses). If the graph of constraints is a tree, the cone $T$
is a wedge, which is in general neither simple nor wide.
The leftmost graph
corresponds to the wide wedge, and the rightmost graph to a
simple wedge.
For $5$ masses there are $16$ possible graphs of
constraints, out of which $8$ graphs define $T$ which is a wedge.
\topinsert
\vskip 2in
\hsize= 4.5in
\raggedright
\noindent{\bf Figure 1}
Possible graphs of constraints for 3 particles.
\endinsert
The edges starting at the floor of a graph of constraints
correspond to possible collisions of particles with
the floor. All the other edges correspond to possible collisions
between two particles.
A graph of constraints defines naturally a partial order
of the vertices (masses) which we will denote by $\prec$.
Let us fix a graph of constraints $\Cal G$.
A vertex $m_l$ is a {\it successor} of the vertex $m_k$,
if $m_k \prec m_l$, in particular $m_k$ is its own successor.
We call a vertex $m_l$ an { \it immediate successor} of $m_k$,
if there is an edge in the graph from $m_k$ to $m_l$. If
$m_l$ is an immediate successor of $m_k$, then
$m_k$ is an {\it immediate predecessor} of $m_l$.
Only immediate successors of the floor can collide with it.
Let
$$
\Cal N(m_k) = \sum_{m_k \prec m_l}m_l
$$
be the total mass of all successors of $m_k$.
Let as before $M = \Cal N(m_0) = \sum_{l=0}^nm_l$ be the total mass.
We denote by $\Cal P(m_k) = M - \Cal N(m_k)$.
\proclaim {Theorem 7.1}
The system
with the Hamiltonian
$$H = \frac {(\xi_1 + \dots +\xi_n)^2}{2m_0} +
\sum_{i=1}^n \frac{\xi_i^2}{2m_i} +
\sum_{i=1}^n \alpha_i m_i\eta_i, $$
with $\alpha_i >0,$ $i =1,\dots,n$,
and a given graph of
constraints $\Cal G$ is completely hyperbolic,
if for every edge in the graph $\Cal G$ from $m_k$ to $m_l$, $k > 0$,
we have
$$ \alpha_k = \alpha_l. \tag{7.5}$$
and
$$
\frac{m_l}{m_k} < 1 + \frac{m_k + m_l}{\Cal P(m_k)}.\tag{7.6}
$$
\endproclaim
\demo{Proof}
We follow the proof of the Main Theorem. We split the wide wedge
$W$ into $n!$ simple wedges $W_\sigma$ indexed by all permutations
$\sigma$ of $\{1,2,\dots,n\}$.
We will first prove that the cone $T$ is the union of some of
these simple wedges, i.e.,
$$
T = \bigcup_{W_\sigma \cap int T \neq \emptyset} W_\sigma.\tag7.7
$$
We have that the acceleration vector $a = (a_1,\dots, a_n)$
is equal to
$$
a_k = -\frac{d^2\eta_k}{dt^2} =
\frac{\alpha_1m_1 + \dots+\alpha_n m_n}{m_0} + \alpha_k,
$$
and by \thetag{7.5} it is parallel to all the faces of $T$ which are not
the faces of the wide wedge $W$. Moreover due to the special
geometry of the wide wedge any such face is orthogonal to
most of the faces of the wide wedge $W$. More precisely the
faces of $T$ which correspond to a collision of $m_k$ and $m_l$,
$k, l > 0$ (i.e., the face defined by the equation $\eta_k = \eta_l$)
is orthogonal to any of the faces of $W$
with the exception of $\eta_k = 0$ and $\eta_l = 0$.
The proof of \thetag{7.7} is accomplished now by the induction on
the dimension $n$. When $n=2$ the claim is obvious. (When $n = 3$
we can convince ourselves about the validity of \thetag{7.7}
by straightforward geometric considerations.) Let us
assume that \thetag{7.7}
holds for $n \leq N, N\geq 2$ and all possible graphs of constraints.
We will be proving \thetag{7.7} for $n = N+1$.
If $m_{\sigma(1)}$ is not an immediate successor of the floor
then $W_\sigma$ is disjoint from the interior of the cone
$T$. Hence the simple wedges having nonempty intersections
with the interior of $T$ can be split according to $\sigma(1)$,
and $m_{\sigma(1)}$ must be one of the immediate successors
of the floor.
Let us consider only the simple wedges $W_\sigma$ with a fixed
allowed value of $\sigma(1)$, say $\sigma(1) = N+1$.
By intersecting $T$ with $\{\eta_{N+1} = 0\}$ we obtain
a convex cone $\widehat T$ corresponding to the graph of constraints
$\widehat{\Cal G}$ obtained from $\Cal G$ by collapsing the
edge from the floor to $m_{N+1}$.
Clearly the orthogonal projection $\hat a$ of the acceleration vector
$a$ on $\{\eta_{N+1} = 0\}$
lies in all the faces of $\widehat T$ corresponding to the edges of the
graph $\widehat{\Cal G}$ with the exception of the edges starting at the
floor. Using the inductive assumption we conclude that
the cone $\widehat T$ is the union of some $N$ dimensional simple wedges
defined by the acceleration vector $\hat a$.
It follows now from the convexity of $T$ that the simple wedges
$W_\sigma$ with $\sigma(1) = N+1$ are all contained in $T$,
which proves \thetag{7.7}.
As in the proof of Main Theorem
in each of the simple wedges we introduce the quadratic form
$Q$ furnished by the canonical isomorphism with an FPL system.
We are going to prove that with this choice of the
quadratic form (or equivalently the two fields of Lagrangian
subspaces) our system is monotone. Indeed the form $Q$ is
conserved as long as the trajectory stays in one simple wedge and
as shown in the proof of Main Theorem it does not
decrease when the orbit crosses to an adjacent simple wedge
or collides with one of the faces of the wide wedge.
It remains to study the conditions of monotonicity
when the trajectory hits a face of $T$ which does not lie
in the face of the wide wedge. It corresponds to
a collision of two masses, $m_k$ and $m_l$, $ k, l> 0$,
and hence also to an edge
of the graph which does not start at the floor.
The appropriate conditions were calculated in Section 4,
they are formulated in terms of $\beta$-angles of the simple wedge.
The problem is to translate them into the conditions on the
masses in our system.
For the purpose of clarity
we will first find appropriate $\beta$-angles
under the assumption that all the
coefficients $\alpha_i, i = 1,\dots,n,$ are equal. (Note that
this is forced anyway by \thetag{7.5}, if there is only one immediate
successor of the floor
in the graph $\Cal G$.)
In such a case we have that the simple wedge
$$
W_\sigma = \{\eta \in \Bbb R^n\big| 0 \leq \eta_{\sigma(1)} \leq
\eta_{\sigma(2)} \leq \dots \leq \eta_{\sigma(n)}\}.
$$
We call a permutation $\sigma$
{\it compatible} with the graph of constraints $\Cal G$, if
$\sigma(k) \leq \sigma(l)$ whenever $m_k \prec m_l$.
Clearly the configuration space $T$ of our
system is the union of simple wedges $W_\sigma$ for all
permutations $\sigma$ compatible with the constraints.
Let us consider the collision of the two particles $m_k$ and $m_l$,
occurring in the simple wedge $W_\sigma$.
We put $k = \sigma(s)$, then by necessity
$l = \sigma(s+1)$.
The condition of monotonicity
in such a collision is, according to the results of Section 4,
that the angle $\beta_s$ in the simple wedge $W_\sigma$ is not less
than $\pi \over 4$. It was established in the proof of Theorem 5.2
that
$$
\tan^2\beta_s = \frac{m_k}{m_l}\left(1 + \frac{m_k + m_l}
{m_0 + m_{\sigma(1)} + \dots + m_{\sigma(s-1)}}\right).
$$
This angle is the angle in the space
$S\left(e_{\sigma(s)},e_{\sigma(s+1)},
\dots, e_{\sigma(n)}\right)$ between two subspaces of codimension $1$,
$S\left(e_{\sigma(s+1)}, \dots, e_{\sigma(n)}\right)$ and
$\{\eta_k = \eta_l\}$.
Hence the condition of monotonicity reads
$$
\frac{m_l}{m_k} \leq 1 + \frac{m_k + m_l}
{m_0 + m_{\sigma(1} + \dots + m_{\sigma(s-1)}}. \tag{7.8}
$$
This condition is most restrictive when the
denominator in the right hand side of the inequality
is the largest possible. After a moment of reflection it becomes
apparent that this denominator does assume the value
of $\Cal P(m_k)$ in one of the simple wedges of our configuration
space and it cannot be bigger than that. This shows that
\thetag{7.8} follows by necessity from the assumption \thetag{7.6}.
We conclude that our system is $Q$-monotone at least in the case of
the special acceleration vector. In the general case we observe that
although the simple wedges are
changed when the acceleration vector is changed,
the $\beta$-angles that appear above remain the same.
Indeed the angle $\beta_s$
is equal to the angle in the space
$S\left(e_{\sigma(s)},e_{\sigma(s+1)},
\dots, e_{\sigma(n)}\right)$ between two subspaces of codimension $1$,
$S\left(e_{\sigma(s+1)}, \dots, e_{\sigma(n)}\right)$ and the intersection
of $S\left(e_{\sigma(s)},e_{\sigma(s+1)},
\dots, e_{\sigma(n)}\right)$ with the face with which our trajectory
collides. This face is given by the equation $\eta_k = \eta_l$
independent of the acceleration vector.
To apply Theorem 1.4
we still need to examine $L_1$ and $L_2$-exceptional trajectories.
As in the proof of the Main Theorem we consider the Euclidean
coordinates $(x,v)\in \Bbb R^n\times\Bbb R^n$ in which
the kinetic energy has the standard form ($\langle v,\,v\rangle/2$),
where $x\in\Bbb R^n$
represents a position in the wedge
$W$ and $v\in \Bbb R^n$
represents a velocity. In these coordinates $L_1 =\{dv_1 = \dots =
dv_n =0\}$ and we can identify all of these Lagrangian subspaces
with the tangent to the configuration space.
Vectors from $L_1$ are not changed by
the derivative of the flow, if there are no collisions with the
faces of $T$ in the time interval. Collisions with the faces of
$T$ do change vectors in $L_1$. We need to distinguish between the
faces of $T$ which lie in the faces of the wide wedge $W$
(collisions with the floor) and
those which do not (collisions between particles).
In a collision with a face of the wide wedge,
say $\{\eta_1 = 0\}$, a vector from $L_1\cap\{dH = 0\}$ will enter the
interior of the sector $\Cal C$ unless it is parallel to the face,
i.e., $d\eta_1 = 0$; in which case the vector will not be changed
in the collision.
Furthermore in a collision between particles no vector
from $L_1$ can be pushed into the interior of the sector,
but all of them are changed by the orthogonal reflection in the
face. Thus we have to address
the possible presence of $L_1$-exceptional
trajectories on which the collisions between particles
always ``prepare''
some vectors before each collision with a face $\{\eta_i = 0\}$,
so that $d\eta_i = 0$. We will now apply the method of Simanyi
\cite{S} to prove
\proclaim{Claim 1}
The set of $L_1$-exceptional
trajectories is contained in a countable union of submanifolds of
codimension at least one.
\endproclaim
\demo{Proof of Claim 1}
Let us consider an exceptional trajectory for which there are some
vectors in $L_1\cap\{dH=0\}$ which never enter the interior
of the sector. We denote the subspace of these vectors
(the $L_1$-exceptional subspace) by $\Cal E_1
\subset L_1\cap\{dH=0\}$. We will establish that the $L_1$-exceptional
subspaces depend only on
the combinatorics of finitely many collisions along the trajectory but
not on the velocities. Indeed, let $R_1,R_2,\dots$ be the sequence of
orthogonal reflections in the faces of $T$ corresponding to
consecutive collisions of the particles. (In the graph
of constraints these collisions are represented by the edges which do
not start at the floor.) Let the consecutive collisions with the
floor of the particles $m_{k_1}, m_{k_2},
\dots$, etc.,
occur exactly after
$t_1,t_2,\dots$,etc., collisions between particles.
We have (cf. \thetag{1.1})
$$
\Cal E_1 = \bigcap_{i=1}^{+\infty} R_1^{-1}R_2^{-1}\dots R_{t_i}^{-1}
\{d\eta_{k_i} = 0\}.
$$
Clearly this infinite intersection must be equal to a finite
intersection, say $t_N$ reflections determine
$\Cal E_1$. As a consequence
there are at most countably many possible $L_1$-exceptional
subspaces.
We will establish that the vectors in an $L_1$-exceptional subspace
satisfy
$$
\langle v,\, dx \rangle = v_1dx_1 + \dots + v_ndx_n = 0\tag{7.9}
$$
i.e., the velocity $v$ must be orthogonal to $\Cal E_1$.
Given \thetag{7.9} we obtain the claim by observing that
for a fixed
$L_1$-exceptional subspace the relation \thetag{7.9} describes
a submanifold in the phase space of codimension $d$.
We conclude that the set of $L_1$-exceptional points
is contained in a countable union of
submanifolds of codimension at least one.
To show \thetag{7.9} we observe that for vectors in
$L_1$-exceptional subspaces
$\langle v,\, dx \rangle $ is constant in time. Indeed it does not
change in collisions because both the velocity $v$ and the tangent
vector are changed by the same orthogonal reflection.
Between collisions we have
$$
\frac{d^2}{dt^2} \langle v,\, dx \rangle
=
-\frac{d}{dt} \langle a,\, dx \rangle
= 0,
$$
where $a = -\frac{dv}{dt}$ is the acceleration vector.
Hence between collisions $\langle v,\, dx \rangle $ could change
linearly with the constant rate $\langle a,\, dx \rangle $. This rate
would not change in a collision. Hence it must be zero or else
$|\langle v,\, dx \rangle |$ would grow unboundedly which is
impossible (velocity must be bounded due to energy conservation
and the tangent vector is changed only by orthogonal reflections).
Further we observe that for vectors in
an $L_1$-exceptional subspace
also $\langle x,\, dx \rangle = x_1dx_1 + \dots + x_ndx_n$ is not
changed in collisions and between collisions it has constant
rate of change equal to $\langle v,\, dx \rangle $.
We conclude again that this rate of change has to be zero or
else $|\langle x,\, dx \rangle|$ would grow unboundedly,
which is impossible.
The idea to use \thetag{7.9} and its proof belongs to Simanyi
\cite{S}.
\enddemo\qed
Finally let us examine the $L_2$-exceptional subspace along a
nondegenerate trajectory. As in the proof of Main Theorem we use
projection of the $L_2$ subspaces on the tangent velocity space as a
way to identify all of these spaces. Moreover the tangent velocity
space can be naturally identified with the tangent configuration
space. With this identification, by the results of Section 4, the action
of the derivative of the flow on vectors from $L_2$ is the following.
The vectors stay in $L_2$ and are unchanged except at crossings from
one simple wedge to another or in collisions of particles. At a
crossing from one simple wedge to another a vector from $L_2$ is
pushed inside the sector $\Cal C$ except for vectors parallel to the
common face of the two simple wedges, which stay unchanged.
Since
we assumed the strict inequalities in \thetag{7.6}, the respective
$\beta$-angles are always strictly bigger than $\frac{\pi}4$.
As a consequence
in a collision of two particles
a vector from $L_2$ is pushed inside the sector $\Cal C$ except for
vectors parallel to the corresponding face of $T$, which are not changed.
Hence an $L_2$-exceptional subspace consists of vectors which are
parallel to all the faces of the simple wedges in which the trajectory
is reflected, with the exception of the faces of the wide wedge, or
which are crossed by the trajectory. By the assumption \thetag{7.5}
the acceleration vector is parallel to all these faces. We will prove
that there are no other vectors with this property.
\proclaim{Claim 2} For a nondegenerate trajectory the $L_2$-
exceptional subspace is spanned by the acceleration vector.
\endproclaim
\demo{Proof od Claim 2}
Since the $L_2$ subspace is identified with the tangent configuration
space
we can use $(d\eta_1, \dots, d\eta_n)$ as coordinates.
Our goal is to show that there are enough collisions and crossings
on every nondegenerate trajectory to insure that the intersection of
all the faces is spanned by the acceleration vector. The task of
bookkeeping is facilitated by the graph of collisions $\Cal G$.
Let $m_{s_1}, \dots, m_{s_r}$ be the $r$ immediate successors of the
floor.
Since every mass has by necessity to collide with one of its immediate
predecessors (but not necessarily with any of its immediate successors)
we conclude that for every mass there is a chain of collisions which
connects it to one of the immediate successors of the floor.
A collision between the particles $m_l$ and $m_p$, forces
the equality $ d\eta_l = d\eta_p$. Hence for every particle $m_l$ there is
$m_{s_j}$, an immediate successor of the floor, such that
$d\eta_{l} = d\eta_{s_j}$ must hold for all vectors in the
$L_2$-exceptional
subspace.
Further, every immediate successor of the floor must have infinitely many
collisions with the floor. The immediate successor $m_{s_j}$ can
collide with the floor only in the simple wedge $W_\sigma$
for which $\sigma(1) = s_j$.
Consider the permutation $\sigma_1$
such that $\sigma_1(i)= s_i, i=1,2,$ and $W_{\sigma_1} \subset T$,
and the permutation $\sigma_2$ differing from $\sigma_1$ by the
transposition of the first two elements, i.e.,
$$\sigma_2(1) = s_2, \sigma_2(2) = s_1, \sigma_2(i)= \sigma_1(i), i
\neq 1,2.$$ Clearly $W_{\sigma_2} \subset T$ and the common face
of $W_{\sigma_1}$ and $W_{\sigma_2}$ is
(cf. the proof of Claim 2 in Section 6)
$$\frac{\eta_{s_1}}{a_{s_1}} = \frac{\eta_{s_2}}{a_{s_2}}.$$
The crossing of this common face forces
$$\frac{d\eta_{s_1}}{a_{s_1}} = \frac{d\eta_{s_2}}{a_{s_2}}.$$
As in the proof of Claim 2 in Section 6 we can conclude that
there are enough of these crossings to force
$$\frac{d\eta_{s_1}}{a_{s_1}} = \frac{d\eta_{s_2}}{a_{s_2}} =
\dots = \frac{d\eta_{s_r}}{a_{s_r}}.$$
Combining with the equalities above we conclude that
the $L_2$-exceptional subspace contains only
vectors parallel to the acceleration.
\enddemo\qed
Our Theorem is proved. \enddemo\qed
Let us apply Theorem 7.1 to the problem of ``splitting and stacking''
the masses described in the Introduction. We start with the
system \thetag{7.1} with the elastic constraints \thetag{7.2}.
This system is completely hyperbolic.
Further we split each of the masses $m_1,\dots, m_n$ into two
$$
m_i = (1-\kappa_i)m_i + \kappa_i m_i,\ \ \ \ \ \text{for}
\ \ \ \ \ 0 < \kappa_i < 1,
$$
and we assume that $m_0 \prec (1-\kappa_i)m_i \prec \kappa_i m_i$,
$i = 1,\dots,n$,
i.e., we have $n$ stacks with two particles. By Theorem 7.1
this system is completely hyperbolic, if we assume
additionally that for $i = 1,\dots,n$
$$
\frac1{\kappa_i} + \frac{m_i}M > 2,\tag{7.9}
$$
where $M = m_0 + m_1 + \dots + m_n$.
\vskip.7cm
\subhead \S 8. Remarks and open problems
\endsubhead
\vskip.4cm
1. In the case $n=2$ the Main Theorem was proven in the Appendix
in \cite{W1}.
The billiard in a wedge symmetric with respect to the acceleration
direction was studied by Miller and Lehtihet
\cite{M-L}, and they discovered numerically the sharp transition from the
mixed behavior to complete hyperbolicity as the angle increases past
$90$ degrees.
\bigskip
2. The system \thetag{7.1} with the constraints \thetag{7.2}
in the special case of equal masses $m_1 = m_2 = \dots = m_n$
and $c_1 = \dots = c_n$ reduces to an FPL system. More
precisely it is a finite extension of the system with
the elastic constraints
$ q_0 \leq q_1 \leq \dots \leq q_n,$ which
was determined to be equivalent to a completely hyperbolic
FPL system (Theorem 5.2). Hence in this special case the Main Theorem
follows from \cite{W1} and the work of
Simanyi \cite{S} (Theorem 4.5).
\bigskip
3. When choosing the bundles of Lagrangian
subspaces (the quadratic form $Q$) in the proof of Main Theorem
we relied on the canonical isomorphism of PW systems in simple
wedges with FPL systems. We are unable to write down the quadratic form
$Q$ explicitly in the general case. We can do it for the
system \thetag{7.1}, if we take the special potential
function $c_i = \alpha m_i$. For such a system the quadratic form $Q$
is given in $W_\sigma$ by
$$
Q=\sum_{i=1}^n\left(d\eta_id\xi_i +\frac{u_i}{\alpha m_i}(d\xi_i)^2\right)
-\sum_{k < l} u_{\sigma(k)}\frac{m_{\sigma(k)}m_{\sigma(l)}}{\alpha M}
\left(\frac{d\xi_{\sigma(k)}}{m_{\sigma(k)}} -
\frac{d\xi_{\sigma(l)}}{m_{\sigma(l)}}\right)^2,
$$
where $u_i$ are defined in \thetag{7.4}.
Independently of the isomorphism with an FPL system one can check
that the form $Q$ is constant in the absence of
collisions. It is also straightforward
to see that $Q$ is not decreased
when the trajectory crosses from one simple wedge to another (only one
term in the second sum is changed). The monotonicity of reflections in
the faces is also not hard to check. Indeed, for the reflection in the
face $\{\eta_1 = 0\}$ we have
$$
\aligned
\xi_1^+ = -\xi_1^- - \frac{2m_1}{m_0+m_1}\sum_{i = 2}^n\xi_i,
\ \ \ \ \ \ & u_1^+ = -u_1^-,\\
\xi_k^+ = \xi_k^-,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&u_k^+ = - \frac{2m_1}{m_0+m_1}u_1^- +u_k^-,\ \ \ \ \ \ \ \
k = 2,\dots,n.
\endaligned
$$
\bigskip
4. We expect that the Main Theorem remains
valid, if some of the entries in $L$
(or $K$) are zero. When $L$ (or $K$) is diagonal
we get of course a completely integrable
system. We conjecture that if $L$ has equal and positive
off-diagonal elements then elliptic periodic orbits
are present, excluding hyperbolicity of the system.
It is suggested by the results of \cite {Ch-W},
where it was established for the FPL system
that beyond the completely integrable case elliptic periodic points
appear.
\bigskip
5. It is an interesting question, if
the ``splitting and stacking'' of the masses, with or
without violation of
the sufficient conditions \thetag{7.9}, produces systems
with slower mixing.
\bigskip
5. It is of considerable interest to find completely hyperbolic system of
arbitrary number of particles with nonlinear potential of interaction.
For FPL systems it was established in \cite{W2} that the system is
completely hyperbolic for the potential function from a large class of
convex functions. Translation of this result into other classes of
systems considered in this paper produces only ``unnatural''
interactions.
\bigskip
6. The setup in the Main Theorem allows to introduce
infinite dimensional limits of our systems. It remains an open
and intriguing question which limit should be taken and what are
its properties.
\bigskip
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\enddocument