Calculus of Variation

What you will learn :

• What is a Functional?
• Variation of a functional.
• Euler-Lagrange equation.
• Necessary and sufficient condition for extrema.
• Variation methods for boundary value problems in ordinary and partial differential equation.

What is a Functional?

A functional is a function whose inputs are functions and outputs are scalars. Input functions can be of one or several variables. Hence we can say that " a functional is a function from a vector space into the scalar field. "

In simple words, a functional is a function of functions.

The calculus of variation  is concerned with solving extremal problems for a Functional, that is, to say Maximum and Minimum problems for functions whose domain contains functions 

              

                  Y(x)  or  Y(x₁,x₂,...,x_n)

The range of the functional will be the real numbers.


Some examples of are :


1. A definite integral over a continuous function f(x)

                   F[f(x)] =  ∫_a^bf(x)dx

A slightly more general form is 

                    F_w[f] = ∫_a^b  w(x)f(x)dx

that is an integral over the function f with a fixed weight function w(x).

2.  A prescription which associates a function with the value of this function at a  particular point in the interior of a given interval [a, b] 

                          F[f] = f(x)             x ϵ [a, b]


So far, all examples are characterized by the fact that they depend linearly on the function f(x), so that they satisfy the relation 

  F[c₁ f₁ + c₂ f₂] = c₁ F[f₁] + c₂ F[f₂] 

   Variation of a Functional : 

The variation δF of the functional F[f] which results from variation of f by δf 

                       δF := F[f + δf] - F[f]

                                                            Next Euler-Lagrange Equation

  (comment down below your doubts)