Characterization of Hamiltonian symmetries and their first integrals

•We give criteria when symmetries for a Hamiltonian system correspond to their integrals.•We show conditions that need to be imposed on the Hamiltonian symmetry.•We show that the classical and Noether based approaches are in fact equivalent.•It is shown that when symmetries provide first integrals they form a Lie algebra.•We prove that the Hamilton first integral is invariant under the Hamilton symmetry.

We provide explicit criteria when the Hamiltonian symmetries for a finite dimensional canonical Hamiltonian system correspond to their first integrals. There are two approaches used for the construction of the first integrals once the symmetry is known. In the standard classical approach the first integrals are obtained up to a distinguished function of time t. In the second, which is recent, the integrals are given by a formula which involves the determination of the divergence terms. In both methods utilized, the first integrals are not determined uniquely. Firstly we show what conditions need to be imposed on the Hamiltonian symmetry in order that it constructively and uniquely yields a first integral. Secondly we provide the extra condition on the first integral for the first approach and the integrability conditions on the divergence term for the second. As a consequence, we show that both methods are in fact equivalent. Furthermore, it is shown that when the Hamiltonian symmetries provide first integrals they form a Lie algebra. Moreover, we prove that the Hamilton first integral is invariant under the Hamilton action symmetry. Several examples taken from the literature are given to illustrate our results and conditions.