**G.Sardanashvily**, Fibre bundle formulation of time-dependent mechanics, arXiv:

**1303.1735**

**Abstract**. We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis

**. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization.**

*R***Introduction**

The technique of symplectic
manifolds is well known to provide the adequate Hamiltonian formulation of
autonomous mechanics . Its realistic example is a mechanical system whose configuration
space is a manifold

*M*and whose phase space is the cotangent bundle*T*M*of*M*provided with the canonical symplectic form*W*. Any autonomous Hamiltonian system locally is of this type.
However, this geometric
formulation of autonomous mechanics is not extended to mechanics under
time-dependent transformations because the symplectic form

*W*fails to be invariant under these transformations. As a palliative variant, one has developed time-dependent mechanics on a configuration space*Q=*where**R**xM**is the time axis. Its phase space***R**R**xT*M*is provided with the pull-back of the form*W*. However, this presymplectic form also is broken by time-dependent transformations.
We address non-relativistic
mechanics in a case of arbitrary time-dependent transformations. Its configuration
space is a fibre bundle Q->

**endowed with bundle coordinates***R**(t,q)*, where*t*is the standard Cartesian coordinate on the time axis**with transition functions***R**t'=t+*const. Its velocity space is the first order jet manifold*JQ*of sections of*Q->*. A phase space is the vertical cotangent bundle**R***V*Q*of*Q->*.**R**
This formulation of
non-relativistic mechanics is similar to that of classical field theory on
fibre bundles over a base of dimension

*>*1. A difference between mechanics and field theory
however lies in the fact that connections on bundles over are flat, and they fail to be dynamic variables, but describe reference frames.**R**
Note that relativistic
mechanics is adequately formulated as particular classical string theory of
one-dimensional submanifolds.

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