A New Systematic Approach to Derive Functionals and Nonclassical Conservation Laws

Abstract

It is shown in detail that Noether's theorem represents a mathematical identity which always exists for any nondegenerate functional and includes two parts: one is Euler–Lagrange expression, and the other is a possible conservation law. If and only if any physical equations are the same as Euler–Lagrange expressions, the following three points follow: (i) there always exist conservation laws or, at least, balance laws; (ii) the symmetry transformations between inertial frames can be used first to check the absolute invariance; (iii) the field variables included in spacetime transformations can be canceled for absolute invariance, and space and time transformations are irrelevant. A theorem for getting nonclassical conservation laws from the general solution of a physical system is presented. The necessary conditions in a theorem's form are also presented to construct Lagrangians for second- and fourth-order partial differential equations (PDEs). Based on these theorems, the Lagrangians and conservation laws of nonlinear heat and KdV equations are constructed and given. For Boussinesq's solution, a nonclassical conservation integral is also found for application. Moreover, it is shown that there exists only the same conservation laws derived from the generalized variational principle as those of the principle of minimum potential energy in elasticity.