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.
References:
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
WikipediA: Relativistic system (mathematics)
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