Foundations of Modern Physics 7: Non-relativistic time-dependent mechanics

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 on T*M. 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 Om fails to be invariant under these transformations. As a palliative variant, one has developed time-dependent mechanics on a configuration space Q=RxM where R is the time axis. Its phase space RxT*M is provided with the pull-back presymplectic form. 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->R. Its velocity space is the first order jet manifold of sections of Q->R. A phase space is the vertical cotangent bundle 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 R are flat, and they fail to be dynamic variables, but describe reference frames.

References:

WikipediA: Non-autonomous_mechanics

Advanced mechanics. Mathematical introduction, arXiv: 0911.0411

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