Euler-Lagrange Equation

Euler-Lagrange Equation 

Before we move on to the euler-lagrange equation first let's look at some definitions and notations that we will use.


Stationary Point : A point x₀ at which the derivative of a function f (x) vanishes, that is,

                         f ' (x₀) = 0

is called a stationary point.
A stationary point may be a minimum or maximum.

For example : let
                               f (x) = x²
  it's derivative is 
                              f ' (x) = 2 x

clearly at x₀ = 0 we have
                                   f ' (0) = 0

Hence x₀ = 0 is a stationary point of f (x) = x²


Notations : we will denote a functional by  L [y] , where y is a function of x.

y' is the partial differentiation of y(x) with respect to x.

f_x  , f_y   and f_y'  are partial derivatives of functional f with respect to x, y and y' respectively.
        

              Euler-Lagrange Equatioin 

A necessary condition for 

to have a stationary point is that f(x, y, y') satisfies the following equation
                    

which is the Euler-Lagrange Equation.

Other form of Euler-Lagrange Equation is 

On solving euler-lagrange equation we will get a function y(x) which makes the given functional stationary. In other words, maximum or minimum of a functional happens on the function y(x).



Let us take a look at an example : Consider the functional 
                 

Here 
       f(x, y, y') = y'² - y²
then
       f_y  = - 2 y  and f_y'  = 2 y'

using euler-lagrange equation we get

Hence the given functional achieves it's extremum only on the curve  y = sin x .  



what is a Functional? back     

(comment down below your doubts)                                                

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]