Noether symmetries and integrability in time dependent Hamiltonian mechanics
We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincare Cartan form, i.e. as symmetries of characteristic line bundles of nondegenerate 1 forms. In the case when the Poincare Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular the Kepler problem. Finally, we prove a variant of the theorem on complete (non commutative) integrability in terms of Noether symmetries of time dependent Hamiltonian systems.