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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 (), and obtain a generalization of Helmholtz conditions to this case.
Automorphisms of the family of all Sturm-Liouville equations are investigated. The classical Darboux transformation arises as a particular case of a general result.
An approach to the theory of linear differential forms in a radial subset of an (arbitrary) real linear space without a Banach structure is proposed. Only intrinsic (partially linear) topologies on are (implicitly) involved in the definitions and statements. Then a mapping , with , real linear spaces and a radial subset of , is considered. After showing a representation theorem of those bilinear forms on for which
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