The Jacobi's principle of stationary full action and its consequences

TL;DR

This paper extends Jacobi's principle of stationary action to non-conservative systems by independently varying time and coordinates, deriving new equations of motion and broadening the principle's applicability.

Contribution

It introduces a novel approach of independently varying time in the Jacobi action, enabling application to non-conservative systems.

Findings

Extended Jacobi principle to non-conservative systems

Derived new equations of motion from the extended principle

Demonstrated broader applicability of stationary action

Abstract

The purpose of this article is to extend the applicability of the stationarity principle of the full Jacobi action to non-conservative natural systems and to derive equations of motion corresponding to this extended principle. To this end, in addition to the well-known variation of the Jacobi action with respect to coordinates, we propose independently variating time. Small variations in coordinates and time depend on the point on the true trajectory with the usual boundary conditions.