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We demonstrate that the formalism for the calculation of the Jacobi lastmultiplier for a one-degree-of-freedom system extends naturally to systems ofmore than one degree of freedom thereby extending results of Whittaker datingfrom 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 anapplication taken from the theory of coupled oscillators. We indicate how manyLagrangians can be obtained for such a system.

Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems