Construction of canonical difference schemes for Hamiltonian formalism via generating functions. (English) Zbl 0681.70020

Summary: This paper discusses the relationship between canonical maps and generating functions and gives the general Hamilton-Jacobi theory for time-independent Hamiltonian systems. Based on this theory, the general method - the generating function method - of the construction of difference schemes for Hamiltonian systems is considered. The transition of such difference schemes from one time-step to the next is canonical. So they are called the canonical difference schemes. The well known Euler centered scheme is a canonical difference scheme. Its higher order canonical generalizations and other families of canonical difference schemes are given. The construction method proposed in the paper is also applicable to time-dependent Hamiltonian systems.


70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
70H20 Hamilton-Jacobi equations in mechanics