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Sunday, 5 April 2015

Foundations of Modern Physics 8: Relativistic mechanics

Non-relativistic mechanics (FMP-7) as like as classical field theory (FMP-3) is adequately formulated in the terms of fiber bundles Q->R over the time axis R and jet manifolds of their sections.

If a configuration space Q of a mechanical system has no preferable fibration Q->R, we obtain a general formulation of relativistic mechanics, including Special Relativity on the Minkowski space Q=R^4. This fomulation involves a more sophisticated technique of jets of one-dimensional submanifolds. In the framework of this formalism, submanifolds of a manifold Q are identified if they are tangent to each other at points of Q with some order. Jets of sections are particular jets of submanifolds when Q->R is a fiber bundle and these submanifolds are its sections. In contrast with jets of sections, jets of submanifolds in relativistic mechanics admit arbitrary transformations of time t’= t(q) including the Lorentz transformations, but not only t’=t+const. in the non-relativistic case.

Note that jets of two-dimensional submanifolds provide a formulation of classical string theory.

A velocity space of relativistic mechanics is the first-order jet manifold J^1Q of one-dimensional submanifolds of the configuration space Q. The jet bundle J^1Q → Q is projective, and one can think of its fibers 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. Lagrangian formalism of relativistic mechanics on the jet bundle J^1Q → Q is developed.


G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010)

G. Sardanashvily, Relativistic mechanics in a general setting, Int. J. Geom. Methods Mod. Phys. 7 (2010) 1307-1319

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