Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamiltonian theory on a fibre bundle

*Q->*over the time axis**R****, where***R***is provided with the Cartesian coordinate***R**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->*. Lagrangians of non-relativistic mechanics are defined as densities on**R***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=*. However, an essential difference between non-relativistic mechanics and field theory on fibre bundles**R***Y->X*, dim*X>1*, lies in the fact that connections on*Q->*always are flat. Therefore, they fail to be dynamic variables, but characterize non-relativistic reference frames.**R**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->*, and its velocity space is the first order jet manifold**R***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.**Reference:**

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

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