Gauge Mechanics

This book presents in a unified way modern geometric methods in analytical me-chanics, based on the application of jet manifolds and connections. As is well known, the technique of Poisson and symplectic spaces provide the adequate Hamiltonian formulation of conservative mechanics. This formulation, however, cannot be ex-tended to time-dependent mechanics subject to time-dependent transformations. We will formulate non-relativistic time-dependent mechanics as a particular field theory on fibre bundles over a time axis. The geometric approach to field theory is based on the identification of classical fields with sections of fibred manifolds. Jet manifolds provide the adequate mathe-matical language for Lagrangian field theory, while the Hamiltonian one is phrased in terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian field theory leads us in a straightforward manner to Lagrangian time-dependent mechanics. At the same time, the canonical polysymplectic form on a momentum phase space of time-dependent mechanics reduces to the canonical exterior 3-form which plays the role similar to a symplectic form in conservative mechanics. Wit