The National Natural Science Foundation，No. 11272076，2013.1-2016.12
In this project, the high performance and structure-preserving methods are studied for the nonlinear Hamiltonian dynamic systems with no constrains, holonomic constrains and non-holonomic constrains. For the above three kinds of dynamic systems, based on the dual variational principle and generating functions, a unified theory for constructing structure-preserving numerical algorithms will be established and the numerical algorithms with arbitrary order of precision and with symplectic-preserving and symmetric-preserving properties are constructed. For Hamiltonian dynamic systems with holonomic constrains, the structure-preserving algorithms which can satisfy precisely the displacement and velocity constrains are studied, which can overcome the numerical difficulty result from violating constrains. For Hamiltonian dynamic systems with non-holonomic constrains, how to satisfy the non-holonomic constrains is a key problem for numerical integration. In this project, the structure-preserving algorithms which can satisfy precisely non-holonomic constrains are studied, which can resolve the difficult problem in constructing numerical method for dynamic sysmtems with non-holonomic constrains. Moreover, in this project, the symmetric property is used to analyze the algebraic structure of the matrix exponential corresponding to the periodic linear Hamiltonian dynamic systems and an accurate, efficient and structure-preserving algorithm for computing the dynamic responses of periodic linear Hamiltonian dynamic systems is established. This method can overcome the difficulties of huge computational costs and memory requirements, and gives an effective way for analyzing the periodic linear Hamiltonian dynamic systems.