Extension of the principle of least action with focus on dissipative equations

Abstract

In this paper, we extend the \emph{principle of least action} and show that a \emph{Lagrange density} always exists for the usual linear pde or linear fractional problems $\oA\,u=f$ in physics, if the usual causality conditions $u|_{t<0}=0$ and $f|_{t<0}=0$ are assumed. (The approach is actually applicable to uniquely solvable linear operator equations for which an adjoint exist.) The set of Lagrange densities together with the zero vector form a non-trivial vector space and for each different set of variables, e.g. $\{u_t,f\}$, $\{u_{xt},f\}$ or $\{u_t,u_x,u_y,u_z,f\}$, there exists a Lagrange density that implies a Lagrange equation, which is equivalent to the considered problem. The usual Lagrange density is such that it implies the 'original equation'. But there are pde's for which the standard theory does not imply a Lagrange density. We show that for each of these equations a (covariant) Lagrange density exists that leads to an equivalent \emph{higher order pde} (if it is formulated with the above causality conditions). For each of these equations, there exists a Lagrange density that implies a Lagrange equation that equals the original equation, but this Lagrange density contains at least one \emph{linear integral operator}. A new point of view is that each of these equivalent Lagrange densities for a given set of variables implies a (usually different) \emph{generalized Hamiltonian density}, where the respective 'Hamiltonian' is conserved if $\oA$ and $f$ are appropriate. The standard Lagrange density implies an Hamiltonian that (frequently) models the energy. Morever, each conserved Hamiltonian implies countable many higher order Hamiltonians that are conserved (if the solution of the considered problem is sufficiently smooth.)