PlanetMath: Euler-Lagrange differential equation (elementary)

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Euler-Lagrange differential equation (elementary)

(Definition)

Let be a twice differentiable function from $\mathbb{R}$ to $\mathbb{R}$ and let be a twice differentiable function from $\mathbb{R}^3$ to $\mathbb{R}$ . Let $\dot{q}$ denote $\frac{d}{d{t}}{q}$ .

Define the functional as follows:

$\displaystyle I(q) = \int_a^b L (t, q(t), \dot{q}(t)) \, dt$

Suppose we regard the function

and the limits of integratiuon

and

as fixed and allow

to vary. Then we could ask for which functions

(if any) this integral attains an extremal (minimum or maximum) value. (Note: Especially in Physics literature, the function

is known as the Lagrangian.)

Suppose that a differentiable function $q_0 : [a,b] \to \mathbb{R}$ is an extremum of . Then, for every differentiable function $f \colon [-1,+1] \times [a,b] \to \mathbb{R}$ such that , the function $g \colon [-1, +1] \to \mathbb{R}$ , defined as

$\displaystyle g (\lambda) = \int_a^b L \left( t, f(\lambda, t), {\partial f \over \partial t} (\lambda, t) \right) \, dt$

will have an extremum at $\lambda = 0$ . If this function is differentiable, then $dg/d\lambda = 0$ when $\lambda = 0$ .

By studying the condition $dg/d\lambda = 0$ (see the addendum to this entry for details), one sees that, if a function is to be an extremum of the integral , then must satisfy the following equation:

$\displaystyle \frac{\partial}{\partial {q}} L - \frac{d}{d{t}} \left( \frac{\partial}{\partial {\dot{q}}} L \right) = 0.$

(1)

This equation is known as the Euler-Lagrange differential equation or the Euler-Lagrange condition. A few comments on notation might be in order. The notations $\frac{\partial}{\partial {q}} L$ and $\frac{\partial}{\partial {\dot{q}}} L$ denote the partial derivatives of the function

with respect to its second and third arguments, respectively. The notation $\frac{d}{d{t}}$ means that one is to first make the argument a function of

by replacing the second argument with

and the third argument with $\dot{q}(t)$ and secondly, differentiate the resulting function with respect to

. Using the chain rule, the Euler-Lagrange equation can be written as follows:

$\displaystyle \frac{\partial}{\partial {q}} L - {\partial^2 \over \partial t \p... ...l q \partial \dot{q}} L - \ddot{q} {\partial^2 \over \partial \dot{q}^2} L = 0.$

(2)

This equation plays an important role in the calculus of variations. In using this equation, it must be remembered that it is only a necessary condition and, hence, given a solution of this equation, one cannot jump to the conclusion that this solution is a local extremum of the functional . More work is needed to determine whether the solution of the Euler-Lagrange equation is an extrmum of the integral or not.

In the special case $\frac{\partial}{\partial {t}} L = 0$ , the Euler-Lagrange equation can be replaced by the Beltrami identity.

"Euler-Lagrange differential equation (elementary)" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Other names:

Euler-Lagrange condition

Also defines:

Euler-Lagrange differential equation, Lagrangian

Attachments:: derivation of Euler-Lagrange differential equation (elementary) (Derivation) by rspuzio

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Cross-references: Beltrami identity, conclusion, solution, necessary, calculus of variations, chain rule, differentiate, arguments, partial derivatives, equation, differentiable, extremum, integral, fixed, limits, function, functional, differentiable function
There are 11 references to this entry.

This is version 29 of Euler-Lagrange differential equation (elementary), born on 2002-02-18, modified 2008-04-01.
Object id is 2092, canonical name is EulerLagrangeDifferentialEquation.
Accessed 22655 times total.

Classification:

AMS MSC:	47A60 (Operator theory :: General theory of linear operators :: Functional calculus)
	70H03 (Mechanics of particles and systems :: Hamiltonian and Lagrangian mechanics :: Lagrange's equations)

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