How relativity determines the Hamiltonian description of an object in classical mechanics

Abstract

For a single object, the Poisson bracket relations for the Galilei group or the Poincaré group and the bracket relations that describe how the position should be changed by space translations, rotations, and either Galilei transformations or Lorentz transformations, imply that if the canonical coordinates are the position coordinates, the generators of the relativity group can be put in their familiar forms by a canonical transformation of the canonical momentum, which is a gauge transformation. In particular, the form of the Hamiltonian as a function of the canonical momentum, and the implied relation of the canonical momentum to the velocity and interpretation of the canonical momentum and Hamiltonian as physical quantities, are made more specific and unique by relativity; if Galilei or Lorentz transformations were not used, the Hamiltonian could be any function of the canonical momentum magnitude. Conversely, the same two sets of bracket relations imply that if the generators have their familiar forms, the canonical coordinates must be the position coordinates.