Galilean relativism with coupled parameters in hyperbolic functions

In the current paper, Galilean relativism with coupled parameters is discussed, which, in the special case of non-relativistic conditions, transitions into the well-known Galilean relativity. The extension of the principles of Galilean relativity is considered, which includes the application of proper time and proper coordinates, connected through a hyperbolic function of rapidity. Relativistic Galilean coordinates have been obtained. The results show that the proper relativistic Galilean coordinates are invariant under Lorentz transformations with respect to the proper Galilean interval. An extension of the Jacobi theorem, formulated by Einstein for dynamic functions with coupled parameters, is presented in the form of the Jacobi–Milekhin theorem. Invariants with respect to the proper coordinates have been derived from the Jacobi equation. The motion of a relativistic particle is demonstrated through the Galilean and Lorentz coordinates in a one-dimensional laser pulse with linear polarization.