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 over the time axis
coordinated by
.
This bundle is trivial, but its different trivializations
correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a
connection
on
which takes a form
with respect to this trivialization. The corresponding covariant differential
determines the relative velocity with respect to a reference frame
.
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 fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and ...
(in particular
covariant Hamiltonian field theory) on
. Accordingly, the velocity phase space of non-autonomous mechanics is the
jet manifold of
provided with the coordinates
. Its momentum phase space is the vertical cotangent bundle
of
coordinated by
and endowed with the canonical
Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form
.
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle
of
coordinated by
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 R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
; its
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 ...
is
.
See also
*
Analytical mechanics
*
Non-autonomous system (mathematics)
*
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
*
Symplectic manifold
*
Covariant Hamiltonian field theory
*
Free motion equation A free motion equation is a differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, t ...
*
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) ().
Theoretical physics
Classical mechanics
Hamiltonian mechanics
Symplectic geometry
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