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)