Some theorems related to the Jacobi variational principle of analytical dynamics

TL;DR

This paper explores the mathematical relationship between classical dynamics and geodesic variational principles in stationary spacetimes, using Routh's and Jacobi's methods to establish a commuting diagram of mappings.

Contribution

It introduces a novel framework connecting classical mechanics and general relativity through explicit mappings based on classical variational procedures.

Findings

Established a commuting diagram linking classical dynamics and geodesic variational principles.

Reviewed and extended Routh's and Jacobi's procedures for this context.

Provided mathematical foundations for analyzing stationary spacetimes in general relativity.

Abstract

It is shown that there exists a commuting diagram of mappings between dynamics of classical systems on one side and variational principles for geodesic lines in stationary spacetimes of general relativity on the other. The construction of the mappings is based on classical Routh's and Jacobi's procedures and on corresponding inverse procedures which are reviewed in the paper.