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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 Noethers first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noethers 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.
Noethers calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy
The main result of this note is a characterization of the Poisson commutativity of Hamilton functions in terms of their principal action functions.
We give an elementary proof of Noethers first Theorem while stressing the magical fact that the global quasi-symmetry only needs to hold for one fixed integration region. We provide sufficient conditions for gauging a global quasi-symmetry.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. I
This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed