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

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

s depend on the time. The configuration space of non-autonomous mechanics is a

fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...

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

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 mathematical physics, covariant classical field theory represents classical field theory, classical fields by Section (fiber bundle), sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of field ( ...

(in particular covariant Hamiltonian field theory) on

X=\mathbb R

. Accordingly, the velocity

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

of non-autonomous mechanics is the jet manifold

J^1Q

Q\to \mathbb R

provided with the coordinates

(t,q^i,q^i_t)

. Its momentum phase space is the vertical cotangent bundle

VQ

Q\to \mathbb R

coordinated by

(t,q^i,p_i)

and endowed with the canonical

Poisson structure In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hami ...

. 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

Q

coordinated by

(t,q^i,p,p_i)

and provided with the canonical

symplectic form In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping \omega : V \times V \to F that is ; Bilinear: ...

; its

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) . * Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) (). Classical mechanics Hamiltonian mechanics Symplectic geometry {{classicalmechanics-stub

See also

References