2020/12/10 by Richard Kowar, Kowar, Richard
Engineering · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Elasticity and Material Modeling #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2012.05521
openalex publication_date 2020/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we extend the \principle of least action and show that a\n\Lagrange density always exists for the usual linear pde or linear\nfractional problems oA ,u=f in physics, if the usual causality conditions\nu|t<0=0 and f|t<0=0 are assumed. (The approach is actually applicable\nto uniquely solvable linear operator equations for which an adjoint exist.) The\nset of Lagrange densities together with the zero vector form a non-trivial\nvector space and for each different set of variables, e.g. ut,f ,\n uxt,f or ut,ux,uy,uz,f , there exists a Lagrange density that\nimplies a Lagrange equation, which is equivalent to the considered problem. The\nusual Lagrange density is such that it implies the 'original equation'. But\nthere are pde's for which the standard theory does not imply a Lagrange\ndensity. We show that for each of these equations a (covariant) Lagrange\ndensity exists that leads to an equivalent \higher order pde (if it is\nformulated with the above causality conditions). For each of these equations,\nthere exists a Lagrange density that implies a Lagrange equation that equals\nthe original equation, but this Lagrange density contains at least one\n\linear integral operator. A new point of view is that each of these\nequivalent Lagrange densities for a given set of variables implies a (usually\ndifferent) \generalized Hamiltonian density, where the respective\n'Hamiltonian' is conserved if oA and f are appropriate. The standard\nLagrange density implies an Hamiltonian that (frequently) models the energy.\nMorever, each conserved Hamiltonian implies countable many higher order\nHamiltonians that are conserved (if the solution of the considered problem is\nsufficiently smooth.)\n