1997/02/25 by G. Giachetta, Giachetta, G., L. Mangiarotti +3
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #dg-ga #gr-qc #hep-th #math.DG
paper · pdf · doi:10.48550/arxiv.dg-ga/9702020
79 pages, Latex
arxiv created 1997/02/25 · arxiv updated 2009/11/30
The usual formulations of time-dependent mechanics start from a given splitting Y=R× M of the coordinate bundle Y→ R. From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical transformations. Our goal is to formulate time-dependent mechanics in gauge-invariant form, i.e., independently of any reference frame. The main ingredient in this formulation is a connection on the bundle Y→ R which describes an arbitrary reference frame. We emphasize the following peculiarities of this approach to time-dependent mechanics. A phase space does not admit any canonical contact or presymplectic structure which would be preserved under reference frame transformations, whereas the canonical Poisson structure is degenerate. A Hamiltonian fails to be a function on a phase space. In particular, it can not participate in a Poisson bracket so that the evolution equation is not reduced to the Poisson bracket. This fact becomes relevant to the quantization procedure. Hamiltonian and Lagrangian formulations of time-dependent mechanics are not equivalent. A degenerate Lagrangian admits a set of associated Hamiltonians, none of which describes the whole mechanical system given by this Lagrangian.