Summary
We show that the Euler-Lagrange operator and the Poincaré-Cartan form arise in a very simple and natural way from the contact structure of the second order jet space, in a purely differential context, without any reference to a variational problem. By the way we obtain an intrinsic expression of the Euler-Lagrange operator.
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Costantini, P.G. On the geometrical structure of Euler-Lagrange equations. Annali di Matematica pura ed applicata 167, 389–402 (1994). https://doi.org/10.1007/BF01760341
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DOI: https://doi.org/10.1007/BF01760341