Study And Application Of Elastic Collision System And Billiard Dynamic System

A dynamic system is a system that evolves over time.The dynamics of the Hamiltonian dynamics system follows the principle of variation.The study on the periodic trajectory of the Hamiltonian system is divided into two directions:geometry and dynamics.In this article,we respectively introduced the dynamic method and theoretical basis of studying collision problems and the Birkhoff billiard theory in geometric methods.The one-dimensional elastic collision system and the three-body collision system on a two-dimensional ring are typical Hamiltonian systems.Therefore,not only can the general dynamics method be used to study it,but also the dynamic system can be converted into a billiard system for research by mapping.In this article,we map the dynamic system into a billiard system by linearly transforming the coordinates of the dynamic system.Then,in the billiard system,the problem of the number of collisions in the one-dimensional three-body elastic collision motion was studied,and a very interesting result related to π was obtained.And we used analytical and numerical methods to verify the results.Then we used the same method to map the three-body collision system on the two-dimensional ring into a triangular billiard system.We found that this method of mapping the elastic collision process into a billiard system is very helpful for understanding the movement of the system.At the end of the article,we introduced the application of the elastic collision system and their corresponding billiard system as the basic physical model for one-dimensional heat conduction problems and the ergodicity of the triangular billiard.