Symplectic Symmetries of Hamiltonian Systems

The goal of this paper is to describe some interesting phenomena which occur in Hamiltonian systems with symplectic (locally Hamiltonian) symmetries. 1. The phase space of an abstract Hamiltonian system is a symplectic manifold (M,ω2), dimM = 2n, with the symplectic structure ω2 [1, 2]. Let = : T ∗M 7→ TM be the fibre map induced by the symplectic structure: ω(ξ,=α) = α(ξ) := (α, ξ), ∀α ∈ T ∗ xM, ∀ξ ∈ TxM. Denote by =dH the Hamiltonian vector field generated by the smooth function H :M 7→ R.