Calculus of Variations and the Euler-Lagrange Equation

The calculus of variations addresses the problem of finding functions that extremize functionals -- mappings from function spaces to real numbers. This framework underlies classical mechanics (Lagrangian formulation), optics (Fermat's principle), and geometric optimization (geodesics, minimal surfaces).

Functionals and Their Variations

Definition7.1Functional and First Variation

A functional $J[y]$ maps a function $y:[a,b]\to\mathbb{R}$ to a real number. The canonical form is $J[y] = \int_a^b L(x, y(x), y'(x))\,dx$ where $L$ is the Lagrangian density. The first variation of $J$ at $y$ in the direction $\eta$ (with $\eta(a) = \eta(b) = 0$ ) is $\delta J[y;\eta] = \lim_{\varepsilon\to 0}\frac{d}{d\varepsilon}J[y+\varepsilon\eta] = \int_a^b \left[\frac{\partial L}{\partial y}\eta + \frac{\partial L}{\partial y'}\eta'\right]dx$ . A function $y^*$ is a stationary point (extremal) if $\delta J[y^*;\eta] = 0$ for all admissible $\eta$ .

ExampleThe Brachistochrone Problem

Find the curve $y(x)$ from $(0,0)$ to $(x_1,y_1)$ along which a bead slides fastest under gravity. The travel time is $T[y] = \int_0^{x_1}\frac{\sqrt{1+y'^2}}{\sqrt{2gy}}\,dx$ . Here $L = \sqrt{(1+y'^2)/(2gy)}$ . The Euler-Lagrange equation yields a cycloid: $x(\theta) = \frac{R}{2}(\theta - \sin\theta)$ , $y(\theta) = \frac{R}{2}(1-\cos\theta)$ . This was posed by Johann Bernoulli (1696) and solved by Newton, Leibniz, L'Hopital, and the Bernoulli brothers, catalyzing the development of the calculus of variations.

The Euler-Lagrange Equation

Definition7.2Euler-Lagrange Equation

Integration by parts on the first variation gives $\delta J = \int_a^b\left[\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'}\right]\eta\,dx = 0$ . By the fundamental lemma of the calculus of variations (if $\int_a^b f(x)\eta(x)\,dx = 0$ for all smooth $\eta$ with $\eta(a)=\eta(b)=0$ , then $f\equiv 0$ ), the stationary condition becomes the Euler-Lagrange equation: $\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} = 0$ . For multiple dependent variables $y_i$ : $\frac{\partial L}{\partial y_i} - \frac{d}{dx}\frac{\partial L}{\partial y_i'} = 0$ for each $i$ .

ExampleGeodesics on Surfaces

The arc length functional on a surface with metric $ds^2 = g_{ij}dx^i dx^j$ is $\ell[\gamma] = \int \sqrt{g_{ij}\dot{x}^i\dot{x}^j}\,dt$ . The Euler-Lagrange equations (with arclength parametrization) yield the geodesic equation: $\ddot{x}^k + \Gamma^k_{ij}\dot{x}^i\dot{x}^j = 0$ where $\Gamma^k_{ij}$ are Christoffel symbols. On the sphere $S^2$ , geodesics are great circles; on the hyperbolic plane, they are semicircles orthogonal to the boundary.

Special Cases and Conservation Laws

Definition7.3First Integrals of the Euler-Lagrange Equation

(Beltrami identity) If $L$ does not depend explicitly on $x$ : $L - y'\frac{\partial L}{\partial y'} = C$ (a first integral reducing the order). This is the Hamiltonian $H = p\dot{q} - L$ in mechanics. (Cyclic coordinates) If $L$ does not depend on $y$ : $\frac{\partial L}{\partial y'} = C$ (the conjugate momentum is conserved). These first integrals often reduce the Euler-Lagrange ODE from second order to first order, making explicit solutions possible.

RemarkHigher Derivatives and Multiple Integrals

For functionals involving higher derivatives $J[y] = \int L(x,y,y',y'')\,dx$ , the Euler-Lagrange equation becomes $\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} + \frac{d^2}{dx^2}\frac{\partial L}{\partial y''} = 0$ (fourth-order ODE). For multiple integrals $J[u] = \iint L(x,y,u,u_x,u_y)\,dx\,dy$ , the equation is $\frac{\partial L}{\partial u} - \frac{\partial}{\partial x}\frac{\partial L}{\partial u_x} - \frac{\partial}{\partial y}\frac{\partial L}{\partial u_y} = 0$ . This encompasses minimal surface equations, plate bending, and field theories.