Friday, February 22, 2019

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.


fx  , fy   and fy'  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
       fy  = - 2 y  and fy'  = 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 " 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,...,xn)

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)] =  abf(x)dx

A slightly more general form is 

                    Fw[f] = aw(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]