Physics: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 [math]\displaystyle{ Q\to \mathbb R }[/math] over the time axis [math]\displaystyle{ \mathbb R }[/math] coordinated by [math]\displaystyle{ (t,q^i) }[/math]. This bundle is trivial, but its different trivializations [math]\displaystyle{ 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 [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ Q\to\mathbb R }[/math] which takes a form [math]\displaystyle{ \Gamma^i =0 }[/math] with respect to this trivialization. The corresponding covariant differential [math]\displaystyle{ (q^i_t-\Gamma^i)\partial_i }[/math] determines the relative velocity with respect to a reference frame [math]\displaystyle{ \Gamma }[/math].

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]\displaystyle{ X=\mathbb R }[/math]. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold [math]\displaystyle{ J^1Q }[/math] of [math]\displaystyle{ Q\to \mathbb R }[/math] provided with the coordinates [math]\displaystyle{ (t,q^i,q^i_t) }[/math]. Its momentum phase space is the vertical cotangent bundle [math]\displaystyle{ VQ }[/math] of [math]\displaystyle{ Q\to \mathbb R }[/math] coordinated by [math]\displaystyle{ (t,q^i,p_i) }[/math] and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form [math]\displaystyle{ 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]\displaystyle{ TQ }[/math] of [math]\displaystyle{ Q }[/math] coordinated by [math]\displaystyle{ (t,q^i,p,p_i) }[/math] and provided with the canonical symplectic form; its Hamiltonian is [math]\displaystyle{ p-H }[/math].

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) 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).