Variational principles in continuum mechanics

Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇ X + λ∇ μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.