On the strangeness of relativistic mechanics

From the mathematical viewpoint, relativistic mechanics fails to be a step between non-relativistic mechanics and classical field theory. Classical field theory is formulated in terms of fibre bundles Y->X (Archive). Non-relativistic mechanics can be treated as a particular field theory in terms of fibre bundles over the time axis X=R (Archive). Relativistic mechanics is formulated in terms of one-dimensional submanifolds of a its configuration space Q (Archive). In a sense, this is a generalization of non-relativistic mechanics because sections of a fibre bundle Q->R are particular one-dimensional submanifolds of Q. However, this is a generalization towards string theory, but not field theory. Indeed, one can develop theory whose dynamic variables are submanifolds and, if they are two-dimensional submanifolds, we are in the case of classical string theory.

From the physical viewpoint, we do not observe classical relativistic masses of velocities more than 0.0001 of the light one.

References:

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)