\input amstex \documentstyle {amsppt} \magnification \magstep1 \openup3\jot \NoBlackBoxes \pageno=1 \hsize 6 truein \catcode\@=11 \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 \Refs \widestnumber\key{XXXX} \ref\key{Ch-H} \by N. I. Chernov, C. 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