The null set of the Euler-Lagrange operator

Summary

The most general element of the null set of the Euler-Lagrange operator is shown to be a polynomial in the derivatives of the independent variables of degree less than or equal to the minimum of the number of independent and dependent variables. The coefficients of such polynomials are solutions to an exhibited system of linear algebraic and first-order partial differential equations. It is then shown that these polynomials may be represented in terms of the divergence of a vector-ordered polynomial, provided similar linear algebraic and differential conditions are satisfied by the coefficients. Thus, an arbitrary divergence is not variationally deletable.

Although the calculus of variations often provides elegant and fruitful means of attacking a wide range of problems, it carries with its use certain intrinsic difficulties. The origin of these difficulties lies in the fact that there is a continuum of variational statements which leads to one and the same system of Euler-Lagrange equations. This in turn implies that any solution of the EulerLagrange equations satisfies a continuum of laws of balance. The mathematical reason for the non-uniqueness of the variational statements and the laws of balance is the well known fact that there exist Lagrangian functions for which the Euler-Lagrange equations are identically satisfied. For the case of one independent variable, such Lagrangian functions are known to be total derivatives. When several independent variables are involved, the form that Lagrangian functions must have in order to be members of the null class of the Euler-Lagrange operator is more complicated. While examining a related problem in the theory of liquid crystals, Ericksen [1] obtained results which indicated that the characterization of the null set given in Theorem 3.2 of [2] was in error. A reexamination of the theorem in question showed that this was indeed the case. This note is presented to correct the error in Theorem 3.2 and its implications.