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,qⁱ). This bundle is trivial, but its different trivializations $Q=\mathbb R\times M$ correspond to the choice of different non-relativistic reference frames.

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 J¹Q 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,qⁱ,p_i) and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form p_idqⁱ − H(t,qⁱ,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,qⁱ,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 9810236034.
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 9814313726 (arXiv: 0911.0411).

See also

• Analytical mechanics
• Hamiltonian mechanics
• Symplectic manifold
• Covariant Hamiltonian field theory