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)
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
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
what is a Functional? back
(comment down below your doubts)