The Calculus of Variations

Elementary calculus locates the points at which a function of one variable is largest or smallest: differentiate, set the derivative to zero, solve. The calculus of variations poses the same question one level up. The unknown is no longer a number but a whole function, and the quantity to be made stationary is not a function value but an integral whose integrand depends on that function and its derivative. The output of the method is a differential equation: many of the equations solved elsewhere in these notes — the cycloid of the brachistochrone, the catenary, the equations of motion of a mechanical oscillator — arise first as the condition that some integral be as small as possible.¹

Problems that depend on a whole curve

Fix two points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ in the plane and consider the family of functions

whose graphs join $P$ to $Q$. Three classical questions single out one member of this family by a minimizing property:

• Shortest path. Which curve joining $P$ and $Q$ has the least arc length? The length of $y(x)$ is ${\int }_{x_1}^{x_2}\sqrt{1+(y^{\prime }{)}^2}\mathbf{d}x$.
• Minimal surface of revolution. Which curve, revolved about the $x$-axis, sweeps out the surface of least area? The area is ${\int }_{x_1}^{x_2}2\pi y\sqrt{1+(y^{\prime }{)}^2}\mathbf{d}x$.
• Brachistochrone. Down which frictionless wire from $P$ to a lower point $Q$ does a bead slide in the least time? With speed $v=\sqrt{2gy}$ from energy conservation, the descent time is ${\int }_{x_1}^{x_2}\sqrt{1+(y^{\prime }{)}^2}/\sqrt{2gy}\mathbf{d}x$.

Each is a special case of one general problem: among all admissible functions $y(x)$ joining $P$ and $Q$, find the one that gives a stationary value to the integral

The integrand $f$ carries the geometry or physics of the problem; the three questions above correspond to $f=\sqrt{1+(y^{\prime }{)}^2}$, $f=2\pi y\sqrt{1+(y^{\prime }{)}^2}$, and $f=\sqrt{1+(y^{\prime }{)}^2}/\sqrt{2gy}$.

Throughout, $f$ is assumed to have continuous second-order partial derivatives in its three arguments $x$, $y$, $y^{\prime }$, which are treated as independent when the partials $\partial f/\partial y$ and $\partial f/\partial y^{\prime }$ are formed.

Euler's equation for an extremal

Suppose an admissible function $y(x)$ makes $I$ stationary. To turn that supposition into an equation, compare $y$ with nearby admissible functions. Choose any variation $\eta (x)$ with continuous second derivative that vanishes at the endpoints,

and form the one-parameter family $\bar{y}(x)=y(x)+\alpha \eta (x)$. Every member joins $P$ to $Q$, and the whole family collapses onto the minimizing curve $y$ at $\alpha =0$. The deviation of a neighbor from $y$ is $\alpha \eta (x)$.

Substituting $\bar{y}$ and ${\bar{y}}^{\prime }=y^{\prime }+\alpha {\eta }^{\prime }$ into the integral turns $I$ into an ordinary function of the single number $\alpha$,

Since $y$ minimizes the integral, $I(\alpha )$ has a minimum at $\alpha =0$, so $I^{\prime }(0)=0$. Differentiating under the integral sign and using the chain rule,

and setting $\alpha =0$ gives

The derivative ${\eta }^{\prime }$ can be removed by integrating the second term by parts. Because $\eta$ vanishes at both endpoints the boundary term drops, leaving

This holds for every admissible variation $\eta$. If the bracketed factor were nonzero — say positive — at some interior point, continuity would keep it positive on a small subinterval, and choosing an $\eta$ that is positive there and zero elsewhere would make the integral positive, a contradiction. The bracket must therefore vanish identically. That conclusion is the fundamental lemma of the calculus of variations, and it yields Euler's equation.²

Euler's equation is a necessary condition, not a sufficient one. Exactly as $g^{\prime }(x_0)=0$ in ordinary calculus can signal a maximum, a minimum, or an inflection, a solution of Euler's equation only makes the integral stationary. Such a solution is called a stationary function or, when the boundary conditions are left off, an extremal; whether it actually minimizes must be settled by other means, usually the geometry or physics of the problem. Expanding the total derivative shows the equation is second order in general,

so its extremals form a two-parameter family, and the two constants are fixed by the boundary conditions. A second-order nonlinear equation of this kind is rarely solvable in closed form, but three recurring special forms of $f$ collapse it to something integrable.

Three solvable cases

The structure of $f$ determines which terms of Euler's equation remain.

The first integral in Case C follows from a total-derivative computation. When $x$ is absent from $f$, the combination $y^{\prime }{f}_{y^{\prime }}-f$ has zero total derivative along an extremal, because

and both the bracket (by Euler's equation) and $\partial f/\partial x$ (by assumption) vanish. The surviving first integral $y^{\prime }{f}_{y^{\prime }}-f=c_1$ is the Beltrami identity, and it turns a second-order problem into a first-order one.

The same problem posed on a curved surface, where the shortest joining curve is called a geodesic, is the seed of differential geometry.

Whether the two constants can be chosen to thread both endpoints is delicate: for some placements of $Q$ there are two catenaries, for others one, and for others none at all, in which case the smallest surface degenerates to two disks joined by a thread (the Goldschmidt solution).

The straight wire is the shortest path but not the fastest: a curve that plunges steeply at first trades extra length for early speed, giving a shorter descent time.

The cycloid also carries the tautochrone property discovered by Huygens: a bead released from any point of an inverted cycloidal arch reaches the bottom in the same time. Bernoulli exulted that the tautochrone of Huygens and his own brachistochrone were the same curve.

Several unknown functions

When the integrand depends on two functions $y(x)$ and $z(x)$,

the same argument with two independent variations ${\eta }_1$, ${\eta }_2$ produces a system of Euler equations, one per unknown,

This extension carries the method into dynamics, where a configuration needs several coordinates.

Isoperimetric problems and Lagrange multipliers

A different class of problem fixes a second integral while making the first stationary. The prototype is the isoperimetric problem the Greeks posed: among all closed plane curves of a given length, which encloses the greatest area? The answer is a circle, and the general pattern — extremize one integral subject to a constraint that a second integral equal a prescribed value $c$ — takes its name from it.

The tool is the same Lagrange multiplier that handles constrained extrema in ordinary calculus. To find where $f(x,y)$ is stationary subject to $g(x,y)=0$, one forms $F=f+\lambda g$ and treats its stationary values as unconstrained; the multiplier $\lambda$ removes the constraint at the cost of one extra unknown, and it keeps the symmetry of $x$ and $y$ that eliminating a variable would destroy.⁴

The derivation mirrors the unconstrained one but disturbs $y$ with a two-parameter family $\bar{y}=y+{\alpha }_1{\eta }_1+{\alpha }_2{\eta }_2$, because a single variation would in general break the side condition $J=c$. The parameters ${\alpha }_1,{\alpha }_2$ are then linked by the constraint — the setting Lagrange multipliers were built for.

Hamilton's principle

The method applies to mechanics, where it recasts Newton's law as the condition that a certain integral be stationary. Consider a particle of mass $m$ moving under a conservative force $\mathbf{F}$ derived from a potential energy $V$, so that $\mathbf{F}=-\nabla V$. With kinetic energy $T=\frac{1}{2}m{v}^2$, define the action over the motion from time $t_1$ to $t_2$ as

The integrand $L=T-V$ is the Lagrangian. For a single particle in rectangular coordinates,

and requiring the action to be stationary means the Euler equations must hold in each coordinate. Carrying out $\frac{\mathbf{d}}{\mathbf{d}t}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0$ and its partners gives

that is, $m\ddot{\mathbf{r}}=\mathbf{F}$ — Newton's second law.⁵

The equivalence runs both ways: assuming Newton's law yields Hamilton's principle, and assuming the principle yields Newton's law, so the vectorial and variational formulations of mechanics are the same physics in two languages. The variational form has one decisive advantage — it is written in energies alone, with no reference to any coordinate system.

For a system with constraints one drops rectangular coordinates in favor of generalized coordinates $q_1,\dots ,q_m$, one per degree of freedom. Hamilton's principle then reads off Lagrange's equations of motion,

a system of second-order equations whose solution is the motion. Conservation of energy $T+V=E$ falls out of them when $L$ has no explicit time dependence, using Euler's theorem on the homogeneous quadratic $T$. Simmons records Planck's verdict that among all the laws of physics the principle of least action comes nearest to the ideal of condensing every natural phenomenon into a single statement; it extends unchanged from one particle to systems, rigid bodies, and continuous media, and reappears in electromagnetism, relativity, and quantum theory.