  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.
