Analogy between geodesic equation and the GCHS on Riemannian manifolds

Abstract

Enlightened by the similar equation form between the GCHS \footnote{GCHS: Generalized Covariant Hamilton System\\GSPB:Generalized structural Poisson bracket} defined by the GSPB and the geodesic equation expressed by geospin variable, we find a deep connection between the geospin matrix and S-dynamics. In this contrastive way, we actually proves that the GCHS is a compatible theory suitable for the curved spacetime as primitively stated. By contrast, geospin matrix in Riemannian geometry has the same physical nature as S-dynamics in GCHS. We obtain a fact that geodesic equation can be naturally derived by the GCHS in terms of the velocity field. We strictly prove that the geometrio $\hat{S}{{\left( {{x}_{k}},{{p}_{i}},H \right)}^{T}}={{\left( {{b}_{k}},{{A}_{i}},w \right)}^{T}}$ holds by using structural operator $\hat{S}$ directly induced by structural derivative ${A}_{i}$ in terms of position ${x}_{k}$, momentum ${p}_{i}$ and Hamiltonian $H$ respectively. It evidently proves that the GCHS on the Riemannian manifold is certainly determined by the Christoffel symbols. As an application, we consider the GCHS on Riemannian geometry.