Discrete Variational Calculations For Special Nonholonomic Systems

In mechanical systems,the holonomic system has a better symplectic geometry structure,which meets the requirements of the self-accommodating condition,and from the traditional structure-preserving algorithm,the holonomic system can sustain the stability for a long time.However,a different situation arises for nonholonomic systems,where the nonholonomic constraint makes the symplectic structure of the system destroyed,and our previous traditional structure-preserving algorithms can no longer be used for this system to solve dynamics-related problems.Therefore,the development of the theory of discrete variational methods for nonholonomic systems and their applications has become one of the hot issues of current research.If the kinematic problems of nonholonomic systems are related to holonomic systems,it is necessary to make the requirement that the generalized forces under nonholonomic systems satisfy the self-accommodating condition.For the nonholonomic system whose generalized force satisfies the self-accommodating condition,the symplectic structure of this system can also be maintained by the method of variational integration from the discrete variational principle.In this paper,the discrete variational method is used to calculate the generalized force satisfying the self-accommodating condition under special nonholonomic systems,which is discussed in depth in the following parts.First,from the viewpoint of variational self-accompaniment,the special nonholonomic system related content is introduced,that is,the generalized force function under the nonholonomic system should have the generalized potential.The special Chaplygin equation and the Routh equation are explored according to the nature of variational self-accommodating.The Chaplygin and Routh equations under the nonholonomic system are written in the form with the generalized force to determine whether the generalized force satisfies the self-accommodating condition.Secondly,according to the action of the discrete Hamiltonian principle and,from the viewpoint of Lagrangian mechanics and Hamiltonian mechanics,considering the standard symplectic method,the form of the variational product numerator is written in terms of the midpoint rule and the Verlet method,and the discrete dynamical equations for the special nonholonomic system in discrete form are given in terms of the variational product numerator.Finally,specific numerical calculations are performed corresponding to the special nonholonomic examples.Example 1 compares the results with the traditional 2nd order symplectic method,and Example 2 analyzes the results with the analytical solution of the equations in terms of absolute errors.The conclusions obtained above can verify the feasibility and superiority of the discrete variational method.Thus,it is reasonable and effective to use the discrete variational method for numerical simulation of some special incomplete systems,provided that the self-accommodating conditions are satisfied.