Abstract
A novel approach to a coordinate-free analysis of the multiplier question in the inverseproblem of the calculus of variations, initiated in a previous publication, is completed in thefollowing sense: under quite general circumstances, the complete set of passivity or integrabilityconditions is computed for systems with arbitrary dimension n. The results are appliedto prove that the problem is always solvable in the case that the Jacobi endomorphism of thesystem is a multiple of the identity. This generalizes to arbitrary n a result derived byDouglas for n=2.
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Sarlet, W., Crampin, M. & Martínez, E. The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations. Acta Applicandae Mathematicae 54, 233–273 (1998). https://doi.org/10.1023/A:1006102121371
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DOI: https://doi.org/10.1023/A:1006102121371