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On the Existence and Formulation of Variational Principles for Nonlinear Differential Equations
Author(s) -
Atherton R. W.,
Homsy G. M.
Publication year - 1975
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm197554131
Subject(s) - mathematics , nonlinear system , partial differential equation , variational principle , independent equation , simultaneous equations , mathematical analysis , differential equation , physics , quantum mechanics
The inverse problem of the calculus of variations, i.e., the existence and formulation of variational principles for systems of nonlinear partial differential equations, is treated by application of a theorem due to Vainberg. Three types of principles are discussed: potential, alternate potential, and composite principles. Potential principles are those for which the equations, as written, admit a variational formulation. Alternate potential principles are those for which the equations admit a variational formulation only after a differential transformation of variables. Composite principles are those in which, in addition to the original variables, a set of adjoint variables is defined. A composite principle may always be formulated. Consistency relations for the existence of a potential are formulated for general systems of nonlinear partial differential equations of any order. These conditions are quite stringent and severely restrict the class of operators which admit a variational formulation. The results of the development are used to treat many well‐known equations in nonlinear wave mechanics. These include the Burgers, Korteweg‐deVries, and sine‐Gordon equations and evolution and model equations for interfacial waves. The Navier‐Stokes equations and the Boussinesq equations for an infinite Prandtl number fluid are briefly treated; these latter equations do not possess a potential principle.