# 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 $Q\to \mathbb R$ over the time axis $\mathbb R$ coordinated by $(t,q^i)$.

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

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 $X=\mathbb R$. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold $J^1Q$ of $Q\to \mathbb R$ provided with the coordinates $(t,q^i,q^i_t)$. Its momentum phase space is the vertical cotangent bundle $VQ$ of $Q\to \mathbb R$ 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 $p_idq^i-H(t,q^i,p_i)dt$.

One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle $TQ$ 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).