The Application Of Symplectic Algorithm In Nonholonomic Mechanical Systems

With gradually geometrization of analytical mechanics, the nonholonomic mechanics made a far-reaching development. Currently the research field of Nonholonomic mechanical systems which we are mainly concerned is the geometric dynamics constraint systems and the way to maintain its inherent structural characteristics algorithm for preserving structure. Various numerical calculations have played an increasingly important role in the validation of the nonholonomic mechanics theoretical results and its practical application. Numerical calculation not only is a strong supplement from the the nonholonomic constraint system geometric dynamics practicality, but also is a bridge between theory and practices.Under a Nonholonomic mechanical system, the Hamilton symplectic geometry structure of the phase space was the destruction by the nonholonomic constraints, the corresponding Hamilton symplectic algorithm is failure of a Nonholonomic mechanical system. We need to find a new Algorithm of preserving structure as a numerical calculation of the non-holonomic mechanical systems, and get a more accurate and undistorted numerical results. In the Birkhoff theoretical framework, the non-complete restraint system has a Birkhoff structure. With a Birkhoff symplectic tensor, the vector field on constraints manifolds can be tructured of a Lie algebra. Thus we expected Birkhoff symplectic geometry theory is a certain reference value in a accurate numerical calculation of the non-holonomic mechanical systems. The Birkhoff symplectic algorithm open up new areas of Nonholonomic mechanical system and get some profound result of the theoretical and practical application.The first chapter is conspectus, discusses briefly the definition of symplectic geometry, the status of nonholonomic mechanical system and the structure of symplectic geometry, the development history and current situation of symplectic algorithm, The practical applications status and research results of the symplectic algorithm in nonholonomic mechanical systems.The second chapter discusses the transformation of Boltzmann-Hamel equations in nonholonomic mechanical systems into the Birkhoffian systems. This section introduce briefly the form of the Boltzmann-Hamel equation and related prior knowledges, and then transforlate the geometric structure of the Boltzmann-Hamel equations into the Birkhoffian structure. The third chapter discusses the way that makes the difference scheme to the Boltzmann-Hamel equation of Birkhoffian structure. And we compared the results of the algorithm with the classical R-K method, the advantage of Birkhoff symplectic Scheme is more accurate in a long-term following. Then this chapter discusses the relaxation of the constraints conditions in the transformation of Boltzmann-Hamel equations in nonholonomic mechanical systems into the Birkhoffian systems.Finally, we summarizes the full text content and prospect the subject of application of symplectic algorithm in the nonholonomic mechanical systems will have an in-depth development.