Calculus of Variations - Extrema

The calculus of variations is concerned with the maxima or minima of functionals, which are collectively called extrema. A functional depends on a function, somewhat analogous to the way a function can depend on a numerical variable, and thus a functional has been described as a function of a function. Functionals have extrema with respect to the elements f of a given function space defined over a given domain. A functional J is said to have an extremum at the function f₀ if ΔJ = J - J has the same sign for all f in an arbitrarily small neighborhood of f₀ . The function f₀ is called an extremal function or extremal. The extremum J is called a maximum if ΔJ ≤ 0 everywhere in an arbitrarily small neighborhood of f₀, and a minimum if ΔJ ≥ 0 there. Weak and strong extrema will be discussed after the norms that are necessary for that discussion are defined in the following.

If f is an element of the function space C(a,b) of all continuous functions that are defined on a closed interval, the norm || f ||₀ defined on C(a,b) is the maximum absolute value of f (x) for a ≤ x ≤ b,

Similarly, if f is an element of the function space D₁(a,b) of all functions of C(a,b) that have continuous first derivatives, the norm || f ||₁ defined on D₁(a,b) is the sum of the maximum absolute value of f (x) and the maximum absolute value of its first derivative f ′(x), for a ≤ x ≤ b,

A functional J is said to have a weak extremum at the function f₀ if there exists some δ > 0 such that, J - J has the same sign for all functions f ∈ D₁(a,b) with || f - f₀ ||₁ < δ. Similarly, a functional J is said to have a strong extremum at the function f₀ if there exists some δ > 0 such that, J - J has the same sign for all functions f ∈ C (a,b) with || f - f₀ ||₀ < δ.

Both strong and weak extrema are for a space of continuous functions but weak extrema have the additional requirement that the first derivatives of the functions in the space be continuous. A strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler-Lagrange equation.