Euler-Lagrange equation

From Exampleproblems

The Euler-Lagrange Equation is the major formula of the Calculus of variations. It provides a way to solve for functions which extremize a given cost functional. It is widely used to solve optimization problems, and in conjunction with the action principle to calculate trajectories. It is analogous to the result from Calculus that a function attains its extreme values when its derivative vanishes.

1 Statement
2 Examples
3 Multidimensional Variations
4 External links

Statement

Formally, given a functional F(x,f(x),f'(x)) with continuous first partial derivatives, any function f which extremizes the cost functional

$J = \int_a^b F(x,f(x),f'(x)) dx$

must also satisfy the ordinary differential equation

$\frac{dF}{df} - \frac{d}{dx} \frac{dF}{df'} = 0$

Examples

A standard example is finding the shortest path between two points in the plane. Assume that the points to be connected are (a,c) and (b,d). The length of a path y=f(x) between these two points is:

$L = \int_{a}^{b} \sqrt{1+\left(\frac{dy}{dx}\right)^2} dx$

The Euler-Lagrange equation yields the differential equation:

$\frac{d}{dx} \sqrt{1+\left(\frac{dy}{dx}\right)^2} = 0 \Rightarrow \frac{dy}{dx} = C$

In other words, a straight line.

Multidimensional Variations

There are also various multi-dimensional versions of the Euler-Lagrange equations. If q is a path in n-dimensional space, then it extremizes the cost functional

$J = \int_{t1}^{t2} L(t, q(t), q'(t)) dt$