Invariant Variational Principles (Enhanced Edition‪)‬

Quite generally, the calculus of variations deals with the problem of
determining the extreme values (maxima or minima) of certain variable
quantities called functionals. Bya functional, we mean a rule which associates
a real number to each function in some given class of so-called admissible
functions. More precisely, let A be a set offunctionsx, y, ... ;then afunctional
J defined on A is a mapping J: A -+ R1 which associates to each x E A a
real number J(x). Inthe most general case, the set of admissible functions may,
in fact, be any set of admissible objects, for example, vectors, tensors, or
other geometric objects. Afundamental problem of the calculus of variations
may then be stated as follows: given a functional J and a well-defined set of
admissible objects A, determine which objects in A afford a minimum value
to J. Here, we may interpret the word minimum in either the absolute sense,
i.e.,a minimum in the whole set A, or in the local sense, i.e.,a relative'minimum
ifA is equipped with a device to measure closeness of its objects. For the most
part, the classical calculus of variations restricts itself to functionals defined
by integrals, either single integrals or multiple integrals, and to the determination
of both necessary and sufficient conditions for a functional to be
extremal.