Unified Treatment of Birkhoff Coordinates in Billiard Systems of Particles Moving under an Influence of a Potential Including Particle-Particle Interaction : General Physics

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From a unified point of view, we discuss Birkhoff coordinates of Hamiltonian systems with an external potential including particle-particle interaction owing to N invariant quantities defined in a tangent vector space in autonomous N-dimensional dynamical systems with continuous time. We first consider 2D and 3D billiard systems for a single particle moving under an influence of an arbitrary potential to derive Birkhoff coordinates, and discuss a statistical average of a time between two consecutive collisions against a boundary and a dynamically defined pressure (not a thermodynamical pressure) by assuming that an ergodic measure is realized as a micro-canonical ensemble and moreover almost all trajectories repeat bouncings on the boundary without any trapping by the external potential. We then discuss Birkhoff coordinates for 2D systems for two particles interacting with each other. We also consider 2D and 3D magnetic billiard systems with arbitrary scalar and vector potentials. As illustrative examples, we consider 2D and 3D gravitational billiards, a two particle system with a linear repulsive force to obtain statistical averages of the various quantities defined on the boundary.
社団法人日本物理学会の論文
2001-05-15