1 Calculus of Variations
1.1 Calculus of Variations
The calculus of variations is a field of mathematical analysis that uses variations (small changes in functions and functionals) to mind maxima and minima of functionals (mappings from a set of functions to the real numbers). Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler-Lagrange equation of the calculus of variations.
The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. Recall the following definition of a [[functional]].
A functional is a mapping \(\mathscr{F}:Y^X\to Z\) where \[ Y^X\ni f(x)\mapsto \mathscr{F}[f(x)]\in Z \] where \(Y^X\) denotes the function space of mappings from \(X\) to \(Y\) and \(Z\) is a scalar field.
A functional \(\mathscr{F}\) is said to have an extremum at the function \(f\) if \[ \Delta \mathscr{F} = \mathscr{F}[y]-\mathscr{F}[f] \] has the same sign for all \(y\) in an arbitrarily small neighborhood of \(f\). Here the function \(f\) is called an extremal function or an extremal. The extremum \(\mathscr{F}[f]\) is called a local maximum if \(\Delta \mathscr{F}\leq 0\) everywhere in an arbitrarily small neighborhood of \(f\), and a local minimum if \(\Delta \mathscr{F}\geq 0\) there,
For a function space of continuous functions, extrema of corresponding functionals are called strong extrema and weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.