Euler-Lagrange equation

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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.

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

only if it satisfies

{\frac  {d}{dt}}{\frac  {dL}{dq'_{k}}}-{\frac  {dL}{dq_{k}}}=0 \forall k=1,2,...n

This formulation is particularly useful in physics when L is taken to be be the Lagrangian.

Another multi-dimensional generalization comes from considering a function on n variables. If \Omega is some surface, then

J=\int _{{\Omega }}L(f,x_{1},...,x_{n},f_{{x_{1}}},...,f_{{x_{n}}})d\Omega

is extremized only if f satisfies the partial differential equation

L_{f}-\sum _{{i=0}}^{{n}}{\frac  {d}{dx_{i}}}{\frac  {dL}{dz_{{x_{i}}}}}=0

When n=2 and L is the energy functional, this leads to the soap-film minimal surface problem.

External links

MathWorld's article on Euler-Lagrange