Jacobi’s last multiplier and Lagrangians for multidimensional systems

Abstract

We demonstrate that the formalism for the calculation of Jacobi’s 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 [Proceedings of the Benares Mathematical Society 2, 53 (1940)] dating from almost 70 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.