The God has created a man in order that he creates that the God fails to do

Saturday, 5 May 2012

Classical mechanics and field theory admit comprehensive geometric formulation

Classical non-relativistic mechanics and classical field theory are adequately formulated in geometric terms of fibre bundles Y->X where dim X>1 in field theory and X=R in mechanics.

Configuration space of a classical non-relativistic mechanics is a fibre bundle Y->R over the time axis R. Its velocity space is a first order jet manifold JY of Y. Its phase space is the vertical tangent bundle VY of Y. Connections on a fibre bundle Y->R characterize non-relativistic reference frames.

Geometric formulation of classical field theory is based on a representation of classical fields by sections of fibre bundles Y->X where dim X>1. Their Lagrangians are densities on finite order jet manifolds of Y.  Connections on Y->X also are fields, e.g., gauge fields which are sections of the first order jet bundle JY->Y. In a very general setting, in order to include odd fields, e.g., fermions  and ghosts, field theory is formulated on a graded manifold whose body is a fibre bundle Y->X.

The formulation of relativistic mechanics generalizes that of non-relativistic mechanics. It is phrased in terms of one-dimensional submanifolds of a configuration space. In a case of two-dimensional submanifolds, we come to classical string theory.


G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
G.Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886
G.Sardanashvily, Advanced mechanics. Mathematical introduction arXiv: 0911.0411

No comments:

Post a Comment