Foundations of anisotropic relativistic mechanics

We lay down the foundations of particle dynamics in mechanical theories that satisfy the relativity principle and whose kinematics can be formulated employing reference frames of the type usually adopted in special relativity. Such mechanics allow for the presence of anisotropy, both conventional (due to non-standard synchronisation protocols) and real (leading to detectable chronogeometrical effects, independent of the choice of synchronisation). We give a general method for finding the fundamental dynamical quantities (Lagrangian, energy and momentum), and write their explicit expression in all the kinematics compatible with the basic requirements. We also write the corresponding dispersion relations and outline a formulation of these theories in terms of a pseudo-Finslerian spacetime geometry. Although the treatment is restricted to the case of one spatial dimension, an extension to three dimensions is almost straightforward.