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'''Non-autonomous mechanics''' describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose [[Lagrangian]]s and [[Hamiltonian mechanics|Hamiltonian]]s depend on the time. The configuration space of non-autonomous mechanics is a [[fiber bundle]] <math>Q\to \mathbb R</math> over the time axis <math>\mathbb R</math> coordinated by <math>(t,q^i)</math>.

This bundle is trivial, but its different trivializations <math>Q=\mathbb R\times M</math> correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a [[connection (mathematics)|connection]]
<math>\Gamma</math> on <math>Q\to\mathbb R</math> which takes a form <math>\Gamma^i =0</math> with respect to this trivialization. The corresponding covariant differential <math>(q^i_t-\Gamma^i)\partial_i</math>
determines the relative velocity with respect to a reference frame <math>\Gamma</math>.

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 <math>X=\mathbb R</math>. Accordingly, the velocity phase space of non-autonomous mechanics is the [[jet bundle|jet manifold]] <math>J^1Q</math> of <math>Q\to \mathbb R</math> provided with the coordinates <math>(t,q^i,q^i_t)</math>. Its momentum phase space is the vertical cotangent bundle <math>VQ</math> of <math>Q\to \mathbb R</math> coordinated by <math>(t,q^i,p_i)</math> and endowed with the canonical [[Poisson manifold|Poisson structure]]. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form <math>p_idq^i-H(t,q^i,p_i)dt</math>.

One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle <math>TQ</math> of <math>Q</math> coordinated by <math>(t,q^i,p,p_i)</math> and provided with the canonical [[symplectic manifold|symplectic form]]; its [[Hamiltonian mechanics|Hamiltonian]] is <math>p-H</math>.

== 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., [[Gennadi Sardanashvily|Sardanashvily, G.]], Gauge Mechanics (World Scientific, 1998) ISBN 981-02-3603-4.
* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ([http://xxx.lanl.gov/abs/0911.0411 arXiv: 0911.0411]).

==See also==
* [[Analytical mechanics]]
* [[Non-autonomous system (mathematics)]]
* [[Hamiltonian mechanics]]
* [[Symplectic manifold]]
* [[Covariant Hamiltonian field theory]]
* [[Free motion equation]]
* [[Relativistic system (mathematics)]]

[[Category:Theoretical physics]]
[[Category:Classical mechanics]]
[[Category:Hamiltonian mechanics]]
[[Category:Symplectic geometry]]

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