General Dynamical Equations for Free Particles and Their Galilean Invariance

Dynamical equations describing evolution of state functions in space-time of a given metric are important components of physical theories of particles. A method based on a group of the metric is used to obtain an infinite set of general dynamical equations for a scalar and analytical function representing free and spinless particles. It is shown that this set of equations is the same for any group of the metric that consists of an invariant Abelian subgroup of translations in time and space. For Galilean space-time, such group is the extended Galilei group. Using this group, it is proved that the infinite set of equations has only one subset of Galilean invariant dynamical equations, and that the equations of this subset are Schrödinger-like equations.