Structure-Preserving Algorithm For Nonholonomic Mechanics

Nonholonomic mechanics are more widely and general in actual mechanical calculation. The existence of external force and dissipative, which makes calculation steps become very complicated and integral would take longer time. As the consequence the error would be increased more accordingly. In the case of that, traditional algorithm is not suit any more. It is obviously to be urgent to seek a new kind of algorithm which can keep the accuracy of problem.Nonholonomic constraints of the non-complete systems, would destruct the mechanical in phase and symplectic structure, that makes symplectic-preserving of complete system is no longer applicable.So in order to get more accurate numerical results, it is indeed to find a new structure-preserving algorithm in the non-complete numerical calculation. Compared with the complete theoretical system of Hamilton, in the background of Birkhoff System, we called augmented space,which include time variable and Hamiltonian. When geometric variational function in the augment space,we would get the common symplectic structure of augment space as well as a Euler-Lagrange equation with a set of external force condition and a set of energy flow formula.What’s more, standard symplectic manifold formula can still be got despite of the independent dimension. In this case, the energy(momentum)formula in the independent variable direction can be viewed as the constraints conditions of independent variable. Thus the augment space would keep a structure which combined with constraints conditions and symplectic manifold formula of standard phase space.The first chapter is introductory section which briefly discusses the nonholomic mechanics, the origin of symplectic-preserving algorithm as well as the development history of and current situation of the symplectic algorithm in the nonholonomic mechanical systems application status and research results.The second chapter discusses the Birkhoffian symplectic algorithm based on discrete variational principle. First, introduce some prior knowledge of Pfaff variational, and then get the discrete equation according to the discrete variational principle. Moreover, it can verify the Birkhoff-two form is closed. At last use an example of quantum system to verify the advantages of reliability and numerical algorithms.The third chapter focuses on the structure-preserving algorithm of variable mass nonholonomic system. First give brief introduction to variable mass nonholonomic system and some basic theory, Birkhoffian Finally we discusses the feasible illustrates of Birkhoff Sim algorithm to Tzenoff equation from the discrete variational principle. The fourth chapter on contents of the full text is a brief summary and outlook about the structure-preserving algorithm in this issue still need further clarification on other aspects of nonholonomic mechanical system.