Calculus of variations and optimal control; optimization Basic notions of the calculus of variations Proposition

Beltrami identity

Builds on Euler–Lagrange equation Integral functional with a Lagrangian density Partial derivative

Statement

Let $a < b$ , and let $L \colon \R \times \R \to \R$ , written $L(z, p)$ , be a function of class $C^2$ that does not depend on the independent variable $x$ . Write $L_z$ and $L_p$ for its partial derivatives. Let $y \in C^2([a, b])$ solve the Euler–Lagrange equation $\frac{d}{dx} L_p\big(y(x), y'(x)\big) = L_z\big(y(x), y'(x)\big) \quad \text{on } [a, b].$ Then there is a constant $C$ with $L\big(y(x), y'(x)\big) - y'(x) \, L_p\big(y(x), y'(x)\big) = C \quad \text{for all } x \in [a, b].$

Remarks

The proof is one differentiation. The derivative of $L(y, y') - y' L_p(y, y')$ is $L_z y' + L_p y'' - y'' L_p - y' \frac{d}{dx} L_p = y' \big(L_z - \frac{d}{dx} L_p\big)$ , which is $0$ by the Euler–Lagrange equation.

Example: for the minimal surface of revolution one has $L(z, p) = z \sqrt{1 + p^2}$ . Then $L - p L_p = z / \sqrt{1 + p^2}$ , so $y / \sqrt{1 + y'^2} = C$ . For $C > 0$ the solutions are the catenaries $y(x) = C \cosh\big((x - d)/C\big)$ .

A partial converse holds: if the quantity is constant and $y'(x) \ne 0$ , then the Euler–Lagrange equation holds at $x$ . So constant solutions $y' = 0$ of the identity must be checked separately.

Going deeper

For a student

In mechanics, with $x$ as time and $L$ the Lagrangian, the quantity $y' L_p - L$ is the energy, the Hamiltonian. The identity is then conservation of energy for a time-independent Lagrangian, a special case of Noether's theorem for time translations. It is used in the brachistochrone problem and the catenary problem.

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Euclidean space object
Euler–Lagrange equation concept
Function concept
Integral functional with a Lagrangian density concept
Function of several variables concept
Partial derivative concept
Beltrami identity statement

Where this leads

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Tag	`F7EKTEESQ5`
URI	`m2://F7EKTEESQ5`
Kind	statement, proposition
Writing	W0 stub
MSC	49K05, 70H03
Status	This copy of the site has no build data. Run `m2 status F7EKTEESQ5`.
About	Euler–Lagrange equation
Uses	Euler–Lagrange equation, Integral functional with a Lagrangian density, Partial derivative
Same as	Wikidata Q3303275 Wikipedia Beltrami identity

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