Geometry as an aspect of dynamics

Abstract

Contrary to the predominant way of doing physics, we claim that the geometrical structure of a general differentiable space-time manifold can be determined from purely dynamical considerations. Anyn-dimensional manifoldV _a has associated with it a symplectic structure given by the2n numbersp andx of the2n-dimensional cotangent fiber bundle TV_n. Hence, one is led, in a natural way, to the Hamiltonian description of dynamics, constructed in terms of the covariant momentump (a dynamical quantity) and of the contravariant position vectorx (a geometrical quantity). That is, the Hamiltonian description furnishes a natural way of relating dynamics and geometry. Thus, starting from the Hamiltonian state function (for a particle)—taken as the fundamental dynamical entity—we show that general relativistic physics implies a general pseudo-Riemannian geometry, whereas the physics of the special theory of relativity is tied up with Minkowski space-time, and nonrelativistic dynamics is bound up to Newton-Cartan space-time.