Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems

TL;DR

This paper extends Jacobi's last multiplier formalism from single to multidimensional systems, connecting historical results and illustrating with coupled oscillators to show multiple Lagrangians can be derived.

Contribution

It generalizes the Jacobi last multiplier method to systems with multiple degrees of freedom, building on historical work and demonstrating its application to coupled oscillators.

Findings

Extended Jacobi's last multiplier to multidimensional systems.

Connected historical results from Whittaker and Rao.

Showed multiple Lagrangians can be obtained for coupled oscillators.

Abstract

We demonstrate that the formalism for the calculation of the Jacobi last multiplier for a one-degree-of-freedom system extends naturally to systems of more than one degree of freedom thereby extending results of Whittaker dating from more than a century ago and Rao ({\it Proc Ben Math Soc} {\bf 2} (1940) 53-59) dating from almost seventy years ago. We illustrate the theory with an application taken from the theory of coupled oscillators. We indicate how many Lagrangians can be obtained for such a system.