SYMMETRIES OF HAMILTONIAN EQUATIONS AND Λ-CONSTANTS OF MOTION

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Lambda-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Lambda-symmetries. We show that in the presence of a Lambda-symmetry for the Hamiltonian equations, one can introduce the notion of "Lambda-constant of motion". The presence of a Lambda-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Lambda-invariant Lagrangian. We illustrate how the Lagrangian Lambda-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Lambda-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler-Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.