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arXiv · 1210.0802

Presymplectic current and the inverse problem of the calculus of variations

Igor Khavkine
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Abstract

The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon and Lawson and generalizes an older result of Henneaux from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.

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Igor Khavkine. 2012-10-02. Presymplectic current and the inverse problem of the calculus of variations. https://doi.org/10.1063/1.4828666

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