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Homotopy operators for the variational bicomplex, representations of the Euler-Lagrange complex, and the Helmholtz-Sonin conditions. (English) Zbl 1177.49056

Summary: We give formulæ for two distinct local homotopy operators for the horizontal differential in the variational bicomplex. We deduce two different representations of the classes of forms in the Euler-Lagrange complex, and hence two different versions of the Helmholtz-Sonin equations for the local variationality of a source form. We give explicit relationships between these two versions of the equations.

MSC:

49N45 Inverse problems in optimal control
58A20 Jets in global analysis
58E30 Variational principles in infinite-dimensional spaces
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