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

Saturday, 21 April 2012

Lagrangian dynamics of higher-dimensional submanifolds

Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamiltonian theory on a fibre bundle Q-> R over the time axis R, where R is provided with the Cartesian coordinate t possessing the transition functions t'=t+const.  A velocity space of non-relativistic mechanics is the first order jet manifold JQ of sections of Q-> R. Lagrangians of non-relativistic mechanics are defined as densities on JQ. This formulation is extended to time-reparametrized non-relativistic mechanics subject to time-dependent transformations which are bundle automorphisms of Q-> R.

Thus, one can think of non-relativistic mechanics as being particular classical field theory on fibre bundles over X=R. However, an essential difference between non-relativistic mechanics and field theory on fibre bundles Y->X, dim X>1, lies in the fact that connections on Q-> R always are flat. Therefore, they fail to be dynamic variables, but characterize non-relativistic reference frames.

In comparison with non-relativistic mechanics, relativistic mechanics admits transformations of the time depending on other variables, e.g., the Lorentz transformations in Special Relativity on a Minkowski space Q. Therefore, a configuration space Q of relativistic mechanics has no preferable fibration Q-> R, and its velocity space is the first order jet manifold J[1]Q of one-dimensional submanifolds of a configuration space Q. Fibres of the jet bundle J[1]Q-> Q are projective spaces, and one can think of them as being spaces of the three-velocities of a relativistic system. The four-velocities of a relativistic system are represented by elements of the tangent bundle TQ of a configuration space Q.

One can provide a generalization of the above mentioned formulation of relativistic mechanics to the case of submanifolds of arbitrary dimension n. For instance, if n=2, this is the case of classical string theory.

G.Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics arXiv: 1112.0216

No comments:

Post a Comment