Physical significance of Nöther’s symmetries and extremum principles

Abstract

In this volume we express the basic aspects of thermo-hydrodynamics in the language of variational calculus, the branch of mathematics that deals with the problem of determining the extreme values of the quantities called functionals. By a functional, we mean a rule that assigns a real number to each (scalar, vector, etc.) function belonging to a given class of admissible functions. One may distinguish such a definition from the definition of the function, the rule that associates a number to each (n-tuples of) number(s) belonging to the given set of numbers. Roughly, the functional is defined on a set of the functions, the function on a set of the numbers. More precisely, let Q will be the set of functions; q, r, s …; then a functional A defined on Q is a mapping A: Q → R¹ which associates to each q ∈ Q a real number A(q). In general, the set of the functions may constitute any set of geometric objects, scalars, vectors, tensors, etc.