Symmetry in Nonlinear Mathematical Physics - 2009

Roman Kozlov (Norwegian School of Economics and Business Administration, Bergen, Norway)

Hamiltonian formalism for discrete equations. Symmetries and first integrals

We consider the relation between symmetries and first integrals for discrete Hamiltonian equations. The results are built on those for canonical Hamiltonian equations. The canonical Hamiltonian equations can be obtained by variational principle from an action functional. We develop an analog of the well-known Noether's identity for canonical Hamiltonian equations and their discrete counterparts (discrete Hamiltonian equations). The approach based on symmetries of the discrete action functional provides a simple and clear way to construct first integrals of discrete Hamiltonian equations just by means of algebraic manipulations. It can be used to conserve structural properties of underlying differential equations under discretization that is useful for numerical implementation. The results are illustrated by a number of examples.