Invariance and first integrals of continuous and discrete Hamiltonian equations

Abstract

The relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations is considered. The observation that canonical Hamiltonian equations can be obtained by a variational principle from an action functional makes it possible to consider invariance properties of a functional in the same way as done in the Lagrangian formalism. The well-known Noether identity is rewritten in terms of the Hamiltonian function and symmetry operators. This approach, which is based on symmetries of the Hamiltonian action, provides a simple and clear way to construct first integrals of Hamiltonian equations without integration. A discrete analog of this identity is developed. It leads to a relation between symmetries and first integrals for discrete Hamiltonian equations that can be used to conserve structural properties of Hamiltonian equations in numerical implementation. The results are illustrated by a number of examples for both continuous and discrete Hamiltonian equations.