The Research Of Finite Element Method Of Henon-Heiles System

Hamiltonian system is the mechanical system which is used to describe dissipationless physical process and phenomenon,and it has universality which was applied widely in the fields of physics,mechanics,engineering, pure and ap-plied mathematics,etc. The typical Hamiltonian systems have two most impor-tant characteristics:(1) energy is conserved. (2)symplectic structures,meaning corresponding symplectic flow is to maintain the same area.Using traditional numerical methods, such as multistep method,RK method to simulate the Hamilton system, it would destroy the symplectic structure, the numerical simulation will fail after a long time calculation, the problem even beyond recognition.And was first in 1984, Feng Kang put forward symplectic algorithm based on symplectic geometry,it can maintain symplectic structure after after a long time calculation,numerical simulate is also valid.After it, some formats are arised at home and abroad,for example symplectic Runge-kutta format, partitioned-symplectic (PSRK), symplectic algorithm theory is gradually mature.However, any discrete algorithms, in general, can not maintain energy conservative and symplectic simultaneously.(Ge-Masden theorem). In the past two decades,Hamiltonian system algorithms research focused mostly concen-trated in its symplectic structures, These algorithms can be very good to keep symplectic properties,but the energy properties of the study involved fewer .However, in many areas, that energy conservative is more important, while the use of finite element method to highlight the conservation of Hamilton systems, so it is meaningful to study finite element method.This article focuses on the finite element method of Henon-Heiles system which is a nonlinear classical chaotic Hamiltonian systems.we choose some traditional representative methods such as:RK method, symplectic difference method, symplectic RK method and continuous finite element method to com-pare.The principal content of this paper is as follows: (1) The numerical results shows that any order finite element method to calculate Henon-Heiles system always conserve energy, energy error is time-independent. Calculating a long time has good stability and high precision.(2)first proposed the vision from the three-dimensional energy surface to research the calculation trajectories of Henon-Heiles system, compared to the use of traditional two-dimensional Poincare phase plane, it is more intuitive, and comparing to the traditional numerical algorithms,can prove that finite is more important in energy conservation.(3)to prove Henon-Heiles system, when H＜1/6 of its movement is nor-mal, when H> 1／6, the movement of non-formal, that generate chaos.