The Research And Computer Simulation Of Symplectic Structure Representation In Mechanical System

The principle of this paper introduces the differential geometry representation andproperties of general mechanical system and the different mechanical manifold. Andintroduces several mechanical system has a symplectic structure representation andusing symplectic mapping to simulate the variation of mechanical system incomputer.In the first,use differentiable manifold to describe the mechanical system,introduces the basic concepts of differential geometry, such as the mappingrepresentation of the rate of the change of space which named with the tangent bundle,and calculation in different coordinate such as antiderivative, differential, Poissonbrackets.And introduces some special manifolds of mechanical system, such as thecharacteristics of symplectic manifolds, presymplectic manifold, Poisson manifolds.Secondly, if the various coordinate of mechanical system equations consisting ofa matrix is a self adjoint matrix, then it will meet the Lie algebraic properties of thePoisson bracket, symplectic structure of mechinical system can be constructed. Andhave a detailed introduction of the a effective symmetry representation of mechanicalsystem--Birkhoff equation representation, in Birkhoff representation,mechanicalsystem in effective to meet variational self adjoint matrix and Lie algebra properties,so it is able to find the symplectic structure of mechanical system.Thirdly, we introduce the discrete variational methods which simulation the varytime of mechanical system in the computer, Birkhoff equation discrete variationalmethod and the Lagrange discrete variational method is both a kind of symplecticmapping, and comparison of various calculation format such as Verlet format,Runge-Kutta format or various structure method, framework of Lagrange andBirkhoff system to make the effect comparison.