Abstract
In the present paper, the well-known Noether's identity, which represents the connection between symmetries and first integrals of Euler–Lagrange equations, is rewritten in terms of the Hamiltonian function. This approach, based on the Hamiltonian identity, provides a simple and clear way to find first integrals of canonical Hamiltonian equations without integration. A discrete analog of the Hamiltonian identity is developed. It leads to a connection between symmetries and first integrals of difference Hamiltonian equations that can be used to conserve the structural properties of Hamiltonian equations under discretizaton. The results are illustrated by a number of examples for both continuous and difference Hamiltonian equations.
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