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.
Contents
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
