III.94 Variational Methods

Lawrence C. Evans

The calculus of variations is both a theory in itself and a toolbox of techniques for studying certain kinds of (often extremely nonlinear) ordinary and partial differential equations. These equations, which arise when we seek critical points of appropriate “energy” functionals, are usually far more tractable than other nonlinear problems.

1 Critical Points

Let us begin with a simple observation from first-year calculus, where we learn that if f = f (t) is a smooth function defined on the real line and if f has a local minimum (or maximum) at a point t₀, then (df / dt) (t₀) = 0.

The calculus of variations vastly extends this insight. The basic object to be considered is a functional F, which is applied not to real numbers but to functions, or rather to certain admissible classes of functions. That is, F takes functions u to real numbers F (u). If u₀ is a minimizer of F (that is, F (u₀) ≤ F(u) for all admissible functions u), then we can expect that “the derivative of F at u₀ is zero.” Of course, this idea has to be made precise, which one might expect to be tricky since the space of admissible functions is infinite dimensional. But in practice these so-called variational methods end up using just standard calculus, and they provide deep insights into the nature of minimizing functions u₀.

2 One-Dimensional Problems

The simplest situation to which variational techniques apply involves functions of a single variable. Let us see why minimizers of appropriate functionals in this setting must automatically satisfy certain ordinary differential equations.

2.1 Shortest Distance

As a warmup problem, we shall show that the shortest path between two points in the plane is a line segment. Of course, this is obvious, but the methods we develop can be applied to much more interesting situations.

Suppose, then, that we are given two points P and Q in the plane. We take as our class of admissible functions all smooth, real-valued functions u, defined on some interval I = [a, b], such that u (a) = P and u (b) = Q. The length of this curve is

where u = u (x) and a prime denotes differentiation with respect to x. Now suppose that some particular curve u₀ minimizes the length. We want to deduce that the graph of u₀ is a line segment, which we will do by “setting the derivative of F to zero” at the minimizer u₀.

To make sense of this idea, select any other smooth function w that is defined on our interval I and that vanishes at its endpoints. For each t define f (t) to be F[u₀ + tw]. Since the graph of the function u₀ + tw connects the given endpoints, and since u₀gives the minimum length, it follows that the function f, which is just an ordinary function from to , has a minimum at t = 0. Therefore, (df / dt) (0) = 0. But we can explicitly compute (df/dt) (0) by differentiating under the integral sign and then integrating by parts. This gives

This identity holds for all functions w with the properties specified above, and consequently

everywhere on the interval I.

To summarize the discussion so far: if the graph of u₀ minimizes the distance between the given endpoints, then identically equals zero, and therefore the shortest path is a line segment. This conclusion may not seem too exciting, but even this simple case has an interesting feature. The calculus of variations automatically focuses our attention on the expression

which turns out to be the curvature of the graph of u. The graph of the minimizer u₀ has zero curvature everywhere.

2.2 Generalization: The Euler-Lagrange Equations

It turns out that the technique we used for the previous example is extremely powerful and can be vastly generalized.

One useful extension is to replace the length functional (1) by a more general functional of the form

where L = L (υ, z, x) is a given function, sometimes called the Lagrangian. Then F [u] can be interpreted as the “energy” (or “action”) of a given function u defined on the interval I.

Suppose next that a particular curve u₀ is a minimizer of F, subject to certain fixed boundary conditions. We want to extract information about the behavior of u₀, and to do so we proceed as in our first example. We select a smooth function w as above, define f (t) = F[u₀ + tw], observe that f has a minimum at t = 0, and consequently deduce that (df/dt) (0) = 0. As in the previous calculation, we then explicitly compute this derivative:

Here, L_υ and L_z stand for the partial derivatives L/υ and L/z, evaluated at . This expression equals zero for all functions w satisfying the given conditions. Therefore,

everywhere on the interval I. This nonlinear ordinary differential equation for the function u₀ is called the Euler-Lagrange equation. The key point is that any minimizer of our functional F must be a solution of this differential equation, which often contains important geometrical or physical information.

For example, take L(υ,z,x) = mυ² - W (z), which we interpret as the difference between the kinetic energy and the potential energy W of a particle of mass m moving along the real line. The Euler-Lagrange equation (4) is then

m = -W′(u₀),

which is Newton’s second law of motion. The calculus of variations provides us with an elegant derivation of this fundamental law of physics.

2.3 Systems

We can generalize further, by taking

where now we are taking vector-valued functions u that map the interval I into ^m. If u₀ is a minimizer in some appropriate class of functions, then one can compute the Euler-Lagrange equation using ideas similar to those discussed above. We obtain the equations

one for each k. Here L_υ^k and L_z^k represent the partial derivatives of L with respect to the k th variables of u′ and u, evaluated at . These equations form a system of coupled ordinary differential equations for the components of u₀ = .

For a geometric example, put

so that F[u] is the length of the curve u in the RIEMAWNIAW METRIC [I.3 §6.10] determined by the g_ij. When u₀ is a curve of constant unit speed, the Euler-Lagrange system of equations (6) can be rewritten, after some work, to read.

for certain expressions called Christoffel symbols, that can be computed in terms of the g_ij. Solutions of this system of ordinary differential equations are called geodesics. Thus, we have deduced that length-minimizing curves are geodesics.

A physical example is L (υ,z,x) = m|υ|² – W (z), for which the Euler-Lagrange equation is

This is Newton’s second law of motion for a particle in ^m moving under the influence of the potential energy W.

3 Higher-Dimensional Problems

Variational methods also apply to expressions involving functions of several variables, in which case the resulting Euler-Lagrange equations are partial differential equations (PDEs).

3.1 Least Area

A first example extends our earlier examination of shortest curves. For this problem we are given a region U in the plane, with boundary U, and a real-valued function g defined on the boundary. We then look at a class of admissible real-valued functions u, defined on U, with the condition that u should equal g on the boundary. We can think of the graph of u as a two- dimensional curved surface with a boundary equal to the graph of g. The area of this surface is

Let us assume that a particular function u₀ minimizes the area among all other surfaces with the given boundary. What can we deduce about the geometric behavior of this so-called minimal surface?

Yet again we proceed by writing f (t) = F[u₀ + tw], differentiating with respect to t, and so on. After some calculation we eventually discover that

within the region U, where “div” denotes the divergence operator. This nonlinear PDE is the minimal surface equation. The left-hand side turns out to be a formula for (twice) the mean curvature of the graph of u₀. Consequently, we have shown that a minimal surface has zero mean curvature everywhere.

Minimal surfaces are sometimes regarded physically as the surfaces formed by soap films when they are stretched between a fixed wire frame that traces out the boundary specified by the function g.

3.2 Generalization: The Euler-Lagrange Equations

It is now straightforward, and sometimes very profitable, to replace the area functional (7) by the general expression

in which we now take U to be a region in ⁿ. Assuming that u₀ is a minimizer, subject to given boundary conditions, we deduce the Euler-Lagrange equation

This is a nonlinear PDE that a minimizer must satisfy. A given PDE is called variational if it has this form.

If, for example, we take L(υ,z,x) = |υ|² + G(z), the corresponding Euler-Lagrange equation is the nonlinear Poisson equation

Δu = g (u),

where g = G′ and is the LAPLACLAN [I.3 §5.4] of u. We have shown that this important PDE is variational. This is a valuable insight, since we can then find solutions by constructing minimizers (or other critical points) of the functional F[u] =

4 Further Issues in the Calculus of Variations

Our examples have shown pretty convincingly how useful our simple method, called computing the first variation, can be when applied to the right geometrical and physical problems. And indeed, variational principles and methods appear in several branches of both mathematics and physics. Many of the objects that mathematicians consider most important have an underlying variational principle of some kind. The list is impressive and, besides the examples we have discussed, includes Hamilton’s equations, the Yang-Mills and Selberg-Witten equations, various nonlinear wave equations, Gibbs states in statistical physics, and dynamic programming equations from optimal control theory.

Many issues remain. For example, if f = f (t) has a local minimum at a point t₀, then we know not only that (df / dt)(t₀) = 0, but also that (d²f/dt²)(t₀) ≥ 0. The attentive reader will correctly guess that a generalization of this observation, called computing the second variation, is important for the calculus of variations. It provides an insight into appropriate convexity conditions that are needed to ensure that critical points are in fact stable minimizers. Even more fundamental is the question of the existence of minimizers or other critical points. Here mathematicians have devoted great ingenuity to designing appropriate function spaces within which “generalized” solutions can be found. But these weak solutions need not be smooth, and so the further question of their regularity and/or possible singularities must then be addressed.

However, these are all highly technical mathematical issues, far beyond the scope of this article. We end our discussion here, in the hope that our excessive demands upon the reader’s attention have been minimized.