Analogy between geodesic equation and the GCHS on Riemannian manifolds

2020/02/25 by Wang, Gen
#37K05 #37K25 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2002.10825

Abstract

Enlightened by the similar equation form between the GCHS \footnoteGCHS: 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 S( x_k,p_i,H )^T=( b_k,A_i,w )^T holds by using structural operator 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.

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