A new class of invariance transformations in classical Hamiltonian particle dynamics

Abstract

The relation between Hamiltonian dynamics and the representation of relativity groups is explored. The implications of relativitic invariance are first explored for classical particle systems. The relation to representations of the relativity transformations is then considered, in particular showing that the preservation of canonical Poisson bracket relations (the analog to canonical commutation relations) is a strong additional requirement on some types of relativity transformations (like the Lorentz rotation). In this discussion, a system is said to be relativistically invariant if a solution to the second order equation of the position transforms into a solution to an equation of motion of the same functional form. The formalism is developed from a Hamiltonian point of view, and in a form which is analogous to the equivalent quantum theory. For a concrete illustration of the discussion, the representation of dilation for simple harmonic motion is developed.