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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 Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle $Q \to ℝ$ over the time axis $ℝ$ coordinated by $(t, q^{i})$ .

This bundle is trivial, but its different trivializations $Q = ℝ \times M$ correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection $Γ$ on $Q \to ℝ$ which takes a form $Γ^{i} = 0$ with respect to this trivialization. The corresponding covariant differential $(q_{t}^{i} - Γ^{i}) \partial_{i}$ determines the relative velocity with respect to a reference frame $Γ$ .

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 $X = ℝ$ . Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold $J^{1} Q$ of $Q \to ℝ$ provided with the coordinates $(t, q^{i}, q_{t}^{i})$ . Its momentum phase space is the vertical cotangent bundle $V Q$ of $Q \to ℝ$ coordinated by $(t, q^{i}, p_{i})$ and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form $p_{i} d q^{i} - H (t, q^{i}, p_{i}) d t$ .

One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle $T Q$ of $Q$ coordinated by $(t, q^{i}, p, p_{i})$ and provided with the canonical symplectic form; its Hamiltonian is $p - H$ .

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) ISBN 981-02-3603-4.
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 (arXiv: 0911.0411).

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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>.
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+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]]
+{{physics-stub}}
+{{applied-math-stub}}

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