Non-autonomous mechanics
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
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., ISBN 981-02-3603-4.
- Giachetta, G., Mangiarotti, L., arXiv:0911.0411).