Non-autonomous mechanics explained

Q\to R over the time axis R coordinated by (t,q ^{i)} .

This bundle is trivial, but its different trivializations

Q = R x 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 R which takes a form \Gamma ^{i} =0 with respect to this trivialization. The corresponding covariant differential (q ^{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 = R . Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold J ^{1Q} of Q\to R provided with the coordinates (t,q ^{i,q} ^{i} ^{t)} . Its momentum phase space is the vertical cotangent bundle VQ 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 - H(t,q} _{idq} ^{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 .

See also

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

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) .
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) .