Non-autonomous mechanics
Introduction '''Non-autonomous mechanics''' describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle  over the time axis  coordinated by . This bundle is trivial, but its different trivializations  correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection on  which takes a form  with respect to this trivialization. The corresponding covariant differential determines the relative velocity with respect to a reference frame .As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian field theory) on . Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold  of  provided with the coordinates . Its momentum phase space is the vertical cotangent bundle  of  coordinated by  and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form . One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle  of  coordinated by  and provided with the canonical symplectic form; its Hamiltonian is .
See also * Analytical mechanics* Non-autonomous system (mathematics)* Hamiltonian mechanics* Symplectic manifold* Covariant Hamiltonian field theory* Free motion equation* Relativistic system (mathematics)
References * De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989). * Echeverria Enriquez, A., Munoz Lecanda, M., Roman Roy, N., Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys. '''3''' (1991) 301. * Carinena, J., Fernandez-Nunez, J., Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys., '''41''' (1993) 517. * Mangiarotti, L., Sardanashvily, G., Gauge Mechanics (World Scientific, 1998) . * Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010)  ().