Fibre bundle formulation of time-dependent mechanics

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 R. 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. 

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=RxM where R is the time axis. Its phase space RxT*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->R endowed with bundle coordinates (t,q), where t is the standard Cartesian coordinate on the time axis R with transition functions t'=t+const. Its velocity space is the first order jet manifold JQ 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.

Note that relativistic mechanics is adequately formulated as particular classical string theory of one-dimensional submanifolds.