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 $(q_{t}^{i}-\Gamma ^{i})\partial _{i}$ 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 $J^{1}Q$ of $Q\to {\mathbb R}$ provided with the coordinates $(t,q^{i},q_{t}^{i})$. 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 $p_{i}dq^{i}-H(t,q^{i},p_{i})dt$.

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).

See also

• Analytical mechanics
• Non-autonomous system (mathematics)
• Hamiltonian mechanics
• Symplectic manifold
• Covariant Hamiltonian field theory
• Free motion equation
• Relativistic system (mathematics)