Definition:Euler-Lagrange Equation

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Definition

The Euler–Lagrange equation is an equation satisfied by a function $\mathbf q$ of a real argument $t$, which is a stationary point of the functional:

$\displaystyle S \left({\mathbf q}\right) = \int_a^b L \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) \, \mathrm d t$

where:

$\mathbf q$ is the function to be found:
$\mathbf q : \left[{a \,.\,.\, b}\right] \subset \R \to X : t \mapsto x = \mathbf q \left({t}\right)$

such that:

$\mathbf q$ is differentiable
$\mathbf q \left({a}\right) = \mathbf x_a$
$\mathbf q \left({b}\right) = \mathbf x_b$
$\mathbf q'$ is the derivative of $\mathbf q$:
$\mathbf q' : \left[{a \,.\,.\, b}\right] \to T_{\mathbf q \left({t}\right)} X: t \mapsto v = \mathbf q' \left({t}\right)$
$T_{\mathbf q \left({t}\right)} X$ denotes the tangent space to $X$ at the point $\mathbf q \left({t}\right)$
$L$ is a real-valued function with continuous first partial derivatives:
$L: \left[{a \,.\,.\, b}\right] \times T X \to \R: \left({t, x, v}\right) \mapsto L \left({t, x, v}\right)$

where:

$T X$ is the tangent bundle of $X$ defined by:
$T X = \bigcup_{x \mathop \in X} \left\{ {x}\right\} \times T_x X$

The Euler–Lagrange equation, then, is given by:

$\displaystyle L_x \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) - \dfrac {\mathrm d} {\mathrm d t} L_v \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) = 0$

where:

$L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively.



Also known as

The Euler-Lagrange equation is often taken in the plural: Euler-Lagrange equations, as it actually defines a system of differential equations.

They are sometimes referred to as Lagrange's equations of motion.


Source of Name

This entry was named for Leonhard Paul Euler and Joseph Louis Lagrange.


Historical Note

The Euler-Lagrange equation was developed in the $1750$s by Leonhard Paul Euler and Joseph Louis Lagrange in connection with their studies of the Tautochrone Problem.

Lagrange solved it in $1755$ and sent the solution to Euler.

They further developed Lagrange's method and applied it to mechanics.

This led to the formulation of Lagrangian mechanics.

Their correspondence ultimately led to the calculus of variations.