My new article Noether's first theorem in Hamiltonian mechanics in arXiv: 1510.03760
Non-autonomous non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noether's first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noether's first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian mechanics. The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian mechanics, any integral of motion is a symmetry current. In particular, an energy function relative to a reference frame is a symmetry current along a connection on a configuration bundle which is this reference frame. An example of the global Kepler problem is analyzed in detail.
Geometry of fibre bundles over R; Lagrangian mechanics. Integrals of motion; Noether’s first theorem: Energy conservation laws; Noether’s third theorem: Gauge symmetries; Non-autonomous Hamiltonian mechanics; Hamiltonian conservation laws: Noether’s inverse first theorem; Completely integrable Hamiltonian systems; Global Kepler problem