  The Noether theorem connecting symmetries and conservation laws can be
applied directly in a Hamiltonian framework without using any intermediate
Lagrangian formulation. This requires a careful discussion about the invariance
of the boundary conditions under a canonical transformation and this paper
proposes to address this issue. Then, the unified treatment of Hamiltonian
systems offered by Noether's approach is illustrated on several examples,
including classical field theory and quantum dynamics.
