Lagrange's equation for continua

Abstract

Lagrange’s equation provides the cornerstone for the generalized methods of classical mechanics [l-5]. Despite the unnecessary assumptions of many existing derivations, the excess of specialized terminology regarding kinematic constraints, and the questionable procedure of regarding a rigid body as an “infinite number” of particles, there may appear to be little point to yet another derivation of Lagrange’s equation. However, this view proves erroneous when we consider the dynamics of nonrigid continua. At present there simply does not exist a Lagrangian formulation which applies to nonrigid continua, ignoring certain physically and mathematically questionable extended versions of what is known as Hamilton’s principle [5-71. Thus, the dynamic equations for a given nonrigid continuum are commonly obtained by considering a small element, applying Newton’s law for a particle, and taking limits as the size of the element approaches zero. This elemental procedure generally requires very strong assumptions regarding spatial continuity and differentiability of the velocity and displacement functions, and often there is no apparent physical reason why these assumptions should hold. In deriving the one-dimensional wave equation for the simple string by using the elemental method, we assume the existence of two spatial derivatives of the deflection curve. However, we violate this assumption immediately upon considering a triangular initial deflection curve. Even if we consider only twice-differentiable initial deflection curves, we may still have difficulties. In the multidimensional wave equation, for example, twice-differentiable intial data do not necessarily produce a twice-differentiable solution due to focusing effects [&lo]. A great deal of study has been devoted to a generalized version of the multidimensional wave equation [9, lo]. The generalized solution obtained does not have the strong continuity and differentiability required of the classical solution and, when restricted to an appropriate Sobolev space,