Murphy-cohen Conjecture And Billiards Nature

The thesis mainly concerns hard ball systems on a line, billiard trajectories in polyhedral angles and period billiard trajectories in a plane convex domain. We obtain a series of new results on these classical objects and show some mysteries which need further study.First, we study a conjecture of T.J. Murphy and E.G.D. Cohen: consider a system of n+1 hard balls, assume multiple collisions do not occur, if no interior ball has mass less than the arithmetic mean of the masses of its immediate neighbors, then the maximum number of collisions of the system is n(n+1)/2. We get a proof of this conjecture with weaker conditions. By the way, a numbers game appeared in the proof is intimately related to Coxeter groups.Then we discuss billiard trajectories in polyhedral angles, since it is well known that any hard ball system on a line is isomorphic to a billiard in an appropriate polyhedral angle. One can obtain interesting results on hard ball systems by the study of billiard trajectories using elementary geometry. For instance, consider a system of three hard balls with masses m₀, m₁, m₂ on a line, we deduce that the number of collisions of the system is less thanÏ€(1+M)^1/2+1, where M=min{m₀/m₁, m₂/m₁}, m₁ is the mass of the interior ball. The same conclusion holds for the system on a semi-line: m₀ or m₂ is positive infinity. We also obtain a result parallel to the conjecture of Murphy and Cohen: for a hard ball system on a semi-line of masses m₁, m₂, ..., m_n from the neighbor of the wall to the others in turn, if m₁≥3m₂, m_i≥(m_i-1m_i+1)^1/2, 2＜i＜n-1, then the number of collisions does not exceed n(n+1)/2.We improve the method of Ya.G. Sinai and M.B. Sevryuk to get a completely explicit estimate of the number of collisions of billiard trajectories in a polyhedral angle. And by a critical principle it follows an exact estimate of the number of collisions for hard ball systems on a line, which implies an old result of G.A. Gal'perin. Our treatment deals with hard ball systems on a semi-line as a particular case. Fur- thermore, we realize that hard ball systems, are setvalued dynamics: the output of a multiple collision is not unique nor finite but all the possibilities obey the laws of conservation of momentum and energy. For the generalized setvalued definition, we have proved that our estimate is still valid. It is the first theorem of this kind and the former results on this problem in the literature all assume that multiple collisions do not occur. It leads us to generalize the definition of billiard trajectory correspondingly, called almost billiard trajectory for distinguishing, and raise a conjecture on the number of collisions of almost billiard trajectories.At last, with the help of the concept of setvalued map and using J. Franks' generalization of the Poincaré-Birkhoff fixed point theorem, we generalize the Birkhoff's theorem on period billiard trajectories over a strictly convex domain to the case over a convex domain. And we raise the problem of studying the existence of almost period billiard trajectories over nonconvex domains and over domains have boundaries with singularities using the concept of upper semicontinuous setvalued map, partially motivated by the fact that, up to the time, one does not know yet whether there exists a period billiard trajectory in any obtuse triangle.