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Lagrangian; Euler-Lagrange form; dynamical form; Helmholtz-type form; Helmholtz form; Helmholtz conditions
Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ($n=1$), and obtain a generalization of Helmholtz conditions to this case.
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