Variational principles for nonpotential operators

Fractional variations for dynamical systems: Hamilton and Lagrange approaches

“…In this subsection, we consider the brief review of Helmholtz conditions [36,37] to fix notation and provide a convenient reference. The Helmholtz equations give the necessary and sufficient conditions for equations to be the Euler-Lagrange equations that can be derived from stationary action principle.…”

Conjugate flow action functionals

“…is not sufficient even in the case of homogeneous boundaries 13 . In any case, it is useful to establish formal symmetry conditions for particular classes of field equations.…”

“…This is the classical condition arising from the symmetry requirement of a second-order nonlinear scalar field equation 13,15 .…”

“…The problem of finding an action functional whose Euler-Lagrange equations correspond to a prescribed set of partial differential equations (PDEs) is known as the inverse problem of the calculus of variations and it has attracted the attention of researches for more than one century 5,13 . Perhaps, one the main reasons is that the formulation of a field theory in terms of an action functional is very elegant and, more importantly, it allows us to establish an immediate connection between symmetry principles and conservation laws 23,24,34,51 .…”

“…The path integral of an operator along a trajectory of fields is defined in terms of a bilinear form 26,43 which can be selected in a rather arbitrary way. In this sense, the solution to the inverse problem of the calculus of variations is reduced to look for a bilinear form that makes the given nonlinear operator potential 13 , if any. It was shown by Tonti 44 that there exist not just one but an infinite number of such forms.…”