- Source: 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
→
R
{\displaystyle Q\to \mathbb {R} }
over the time axis
R
{\displaystyle \mathbb {R} }
coordinated by
(
t
,
q
i
)
{\displaystyle (t,q^{i})}
.
This bundle is trivial, but its different trivializations
Q
=
R
×
M
{\displaystyle 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
Γ
{\displaystyle \Gamma }
on
Q
→
R
{\displaystyle Q\to \mathbb {R} }
which takes a form
Γ
i
=
0
{\displaystyle \Gamma ^{i}=0}
with respect to this trivialization. The corresponding covariant differential
(
q
t
i
−
Γ
i
)
∂
i
{\displaystyle (q_{t}^{i}-\Gamma ^{i})\partial _{i}}
determines the relative velocity with respect to a reference frame
Γ
{\displaystyle \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
{\displaystyle X=\mathbb {R} }
. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold
J
1
Q
{\displaystyle J^{1}Q}
of
Q
→
R
{\displaystyle Q\to \mathbb {R} }
provided with the coordinates
(
t
,
q
i
,
q
t
i
)
{\displaystyle (t,q^{i},q_{t}^{i})}
. Its momentum phase space is the vertical cotangent bundle
V
Q
{\displaystyle VQ}
of
Q
→
R
{\displaystyle Q\to \mathbb {R} }
coordinated by
(
t
,
q
i
,
p
i
)
{\displaystyle (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
d
q
i
−
H
(
t
,
q
i
,
p
i
)
d
t
{\displaystyle 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
T
Q
{\displaystyle TQ}
of
Q
{\displaystyle Q}
coordinated by
(
t
,
q
i
,
p
,
p
i
)
{\displaystyle (t,q^{i},p,p_{i})}
and provided with the canonical symplectic form; its Hamiltonian is
p
−
H
{\displaystyle 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) 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).