Why is the Euler-Lagrange equation only a necessary condition?

It identifies functions where the functional is stationary, the analogue of a critical point, but such a point may be a minimum, a maximum, or neither. Determining which requires examining the second variation or applying convexity or direct-method arguments.

What is a natural boundary condition?

When the endpoints of the competing functions are not fixed, requiring the first variation to vanish forces an extra condition at those endpoints, derived from the boundary terms. These natural boundary conditions emerge automatically from the variational principle rather than being imposed.

Euler-Lagrange Equation

The Euler-Lagrange equation is the differential equation that any function extremizing an integral functional must satisfy, the central necessary condition of the calculus of variations.

Definition

For a functional given by an integral of a Lagrangian depending on a function and its derivative, the Euler-Lagrange equation states that the partial derivative of the Lagrangian with respect to the function equals the derivative with respect to the independent variable of its partial derivative with respect to the function's derivative.

Scope

This topic covers the first variation of a functional and its vanishing condition, the derivation of the Euler-Lagrange equation, the fundamental lemma of the calculus of variations, natural and essential boundary conditions, first integrals such as the Beltrami identity, and generalizations to several functions, higher derivatives, and multiple integrals.

Core questions

What equation must an extremal of a functional satisfy?
How is the condition derived from the first variation?
What boundary conditions accompany the equation?
When do first integrals simplify the resulting equation?

Key theories

First variation and stationarity: Setting the first variation of a functional to zero for all admissible perturbations, together with the fundamental lemma of the calculus of variations, yields the Euler-Lagrange equation.
Natural boundary conditions: When endpoints are free rather than fixed, the vanishing first variation imposes additional natural boundary conditions on the extremal beyond the differential equation itself.
First integrals and the Beltrami identity: When the Lagrangian does not depend explicitly on the independent variable, a conserved quantity, the Beltrami identity, reduces the second-order equation to a first-order one.

Clinical relevance

The Euler-Lagrange equation turns variational principles into solvable differential equations, producing the equations of motion in Lagrangian mechanics, the geodesic equations in geometry, and the governing equations of elasticity, optics, and field theory.

History

Euler derived the equation geometrically in 1744, and Lagrange recast the derivation through his algebraic method of variations around 1755, giving the equation its modern form and name. Noether later connected symmetries of the Lagrangian to conserved quantities through the equation.

Key figures

Leonhard Euler
Joseph-Louis Lagrange
Emmy Noether
Eugenio Beltrami

Seminal works

gelfand1963
courant1953

Frequently asked questions

Why is the Euler-Lagrange equation only a necessary condition?: It identifies functions where the functional is stationary, the analogue of a critical point, but such a point may be a minimum, a maximum, or neither. Determining which requires examining the second variation or applying convexity or direct-method arguments.
What is a natural boundary condition?: When the endpoints of the competing functions are not fixed, requiring the first variation to vanish forces an extra condition at those endpoints, derived from the boundary terms. These natural boundary conditions emerge automatically from the variational principle rather than being imposed.