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

Applications of Noether conservation theorem to Hamiltonian systems