Calculus of Variations: Optimizing Over Functions

J[y] = integral from a to b of F(x, y(x), y'(x)) dx

• x is the independent variable on [a, b]
• y(x) is the unknown function (the curve we seek)
• y'(x) = dy/dx is the derivative of y
• F(x, y, p) is the Lagrangian density, assumed smooth

A = (y in C^1[a,b] : y(a) = alpha, y(b) = beta)

This class requires y to be continuously differentiable on [a, b] and to satisfy fixed endpoint conditions. Natural (Neumann) boundary conditions relax the endpoint requirements and lead to additional transversality conditions.

y_epsilon(x) = y*(x) + epsilon * eta(x)

Then phi(epsilon) = J[y_epsilon] is a function of the real parameter epsilon, and y* is an extremum only if:

phi'(0) = 0 for all admissible eta

First Variation:

delta J[y; eta] = integral of (F_y - d/dx F_y') * eta dx = 0

where F_y = partial F / partial y and F_y' = partial F / partial y'. Since eta is arbitrary, the fundamental lemma of the calculus of variations forces the integrand to vanish.

F - y' * F_y' = C (constant)

This reduces the Euler-Lagrange ODE from second order to first order — a major simplification. The Beltrami identity applies directly to the brachistochrone and geodesic problems.

F_y'(a, y(a), y'(a)) = 0 (if left endpoint is free)

F_y'(b, y(b), y'(b)) = 0 (if right endpoint is free)

More generally, if an endpoint lies on a curve g(x, y) = 0, the transversality condition imposes an orthogonality-type relation between the extremal and the endpoint curve.

Setup:

By energy conservation: v = sqrt(2gy) (taking y downward). Time = integral of ds/v = integral of sqrt((1 + y'^2) / (2gy)) dx

Solution via Beltrami Identity:

F = sqrt((1 + y'^2) / y) is independent of x, so F - y' F_y' = C. This simplifies to: y(1 + y'^2) = 1/C^2 = 2R (constant).

Parametric Solution (Cycloid):

x = R(theta - sin theta), y = R(1 - cos theta)

Plane Geodesic:

Minimize J[y] = integral of sqrt(1 + y'^2) dx. Euler-Lagrange: d/dx(y' / sqrt(1 + y'^2)) = 0, so y' = const. Solution: y = mx + b (straight lines, as expected).

Sphere Geodesic:

On the unit sphere with metric ds^2 = d(theta)^2 + sin^2(theta) d(phi)^2, the geodesic equations from the Euler-Lagrange system yield great circles: intersections of the sphere with planes through the origin.

d^2 x^k/ds^2 + Gamma^k_ij (dx^i/ds)(dx^j/ds) = 0

where Gamma^k_ij are Christoffel symbols of the metric. In general relativity, freely falling particles follow geodesics of spacetime — this is the variational content of Einstein's equivalence principle.

Plateau's Problem:

Given a closed curve in space, find the surface of least area spanning it. This was solved rigorously by Jesse Douglas and Tibor Radó in 1930-31, earning Douglas the first Fields Medal.

Catenoid (Surface of Revolution):

Minimize 2 pi * integral of y * sqrt(1 + y'^2) dx. Beltrami identity: y / sqrt(1 + y'^2) = C. Solution: y = C * cosh((x - x_0)/C) (catenary).

Formulation:

Maximize: A[y] = integral of y dx (area under curve) Subject to: L[y] = integral of sqrt(1 + y'^2) dx = L (fixed length)

Solution via Lagrange Multiplier:

Form H[y] = A[y] + lambda * L[y] and apply Euler-Lagrange. The extremals are circles of radius R = L / (2 pi).

Let y* be an extremal (solution of Euler-Lagrange). The second variation is the quadratic form in eta:

delta^2 J[y*; eta] = integral of (F_yy eta^2 + 2 F_yy' eta eta' + F_y'y' eta'^2) dx

For y* to be a weak local minimum, we need delta^2 J ≥ 0 for all admissible eta. This is the necessary condition for a minimum.

F_y'y'(x, y*, y*') ≥ 0 for all x in [a, b]

If F_y'y' is strictly positive everywhere (strengthened Legendre condition), the functional is locally convex in the y' direction, which is a necessary step toward confirming a minimum. The Legendre condition fails for maximization problems.

Jacobi Equation:

d/dx(F_y'y' u') - (F_yy - d/dx F_yy') u = 0

Conjugate Point:

The point x = c is conjugate to a with respect to the extremal y* if the Jacobi equation has a nontrivial solution u with u(a) = 0 and u(c) = 0.

Jacobi Condition for a Minimum:

The interval (a, b] contains no conjugate point to a. If the open interval (a, b) contains a conjugate point, y* is not even a weak local minimum.

Method:

Introduce a constant Lagrange multiplier lambda and extremize the unconstrained functional:

H[y] = J[y] + lambda * K[y] = integral of (F + lambda G) dx

Apply the Euler-Lagrange equation to the combined Lagrangian F + lambda G. The constant lambda is determined by the constraint K[y] = c.

For several dependent variables y_1, ..., y_n with holonomic constraints g_k(x, y_1, ..., y_n) = 0, use Lagrange multiplier functions lambda_k(x) (not constants) and extremize:

integral of (F + sum lambda_k g_k) dx

This yields n + m Euler-Lagrange equations for n unknowns y_i and m multiplier functions lambda_k, along with the m constraint equations.

d/dt(partial L / partial q-dot_i) - partial L / partial q_i = 0

for each generalized coordinate q_i, i = 1, ..., n. These n second-order ODEs completely determine the motion of the system.

Example: Particle in Potential V(x):

L = (1/2)m x-dot^2 - V(x)

d/dt(m x-dot) + V'(x) = 0

m x-ddot = -V'(x) = F(x) (Newton's second law)

Conjugate Momentum:

p_i = partial L / partial q-dot_i

Hamiltonian:

H(q, p, t) = sum_i p_i q-dot_i - L(q, q-dot, t)

The Legendre transform converts L (a function of (q, q-dot, t)) to H (a function of (q, p, t)). For natural mechanical systems (T quadratic in q-dot, V independent of q-dot): H = T + V = total energy.

div(q-dot, p-dot) = sum_i (partial^2 H / partial q_i partial p_i - partial^2 H / partial p_i partial q_i) = 0

By the divergence theorem, any volume in phase space is preserved under Hamiltonian flow. This has profound consequences: the system is incompressible in phase space, preventing trajectories from converging to attractors (a Hamiltonian system cannot have asymptotically stable fixed points with a basin of attraction).

Definition:

(f, g) = sum_i (partial f/partial q_i * partial g/partial p_i - partial f/partial p_i * partial g/partial q_i)

Time Evolution:

df/dt = (f, H) + partial f/partial t

A quantity f is conserved if and only if (f, H) = 0 and f has no explicit time dependence. Canonical coordinates satisfy the fundamental Poisson bracket relations: (q_i, p_j) = delta_ij, (q_i, q_j) = 0, (p_i, p_j) = 0.

Each continuous symmetry yields a conserved current j^mu (density and flux) satisfying the continuity equation:

partial_mu j^mu = 0 (Einstein summation)

The conserved charge Q = integral of j^0 d^3x is the spatial integral of the charge density. In electromagnetism, U(1) gauge symmetry yields charge conservation. In the Standard Model, SU(3) x SU(2) x U(1) gauge symmetries yield the conservation laws governing the strong, weak, and electromagnetic forces.

Definition of W^(1,2)(Omega) = H^1(Omega):

H^1(Omega) = (u in L^2(Omega) : weak gradient Du in L^2(Omega))

with inner product: (u, v)_H1 = integral of (u*v + Du * Dv) dx. This is a Hilbert space — complete, with an inner product.

Sobolev Embedding:

In dimension n, H^1(Omega) embeds continuously into L^(2n/(n-2))(Omega) for n ≥ 3 (Sobolev embedding), and compactly into L^2(Omega) by the Rellich-Kondrachov theorem. These compact embeddings are crucial for the direct method.

Tonelli's Theorem:

If F(x, u, p) is convex in p (the gradient variable) and satisfies appropriate growth conditions (coercivity: F ≥ alpha |p|^r - beta for some alpha greater than 0, r greater than 1), then the functional J[u] = integral of F(x, u, Du) dx is weakly lower semicontinuous on W^(1,r)(Omega) and attains its minimum on closed convex subsets.

Quasiconvexity (Morrey):

For vector-valued problems (elasticity, geometric problems), convexity in p must be replaced by quasiconvexity, a weaker condition introduced by Morrey. Quasiconvexity is necessary and (with growth conditions) sufficient for weak lower semicontinuity of the functional.

T[y] = (1/c) integral of n(x, y) * sqrt(1 + y'^2) dx

where n(x, y) is the refractive index. The Euler-Lagrange equation yields the ray equation, which in a homogeneous medium (n = constant) gives straight lines. At an interface between two media, Snell's law follows as a consequence:

n_1 sin(theta_1) = n_2 sin(theta_2)

tau[x] = integral of sqrt(-g_mu nu (dx^mu/dlambda)(dx^nu/dlambda)) dlambda

The Euler-Lagrange equations yield the geodesic equation with Christoffel symbols encoding the curvature of spacetime. In the Schwarzschild metric, geodesics account for the perihelion precession of Mercury and the bending of light by the sun.

Problem Formulation:

Minimize J[u] = integral of f_0(x, u, t) dt + phi(x(T)) subject to: x-dot = f(x, u, t), x(0) = x_0, u(t) in U

Pontryagin's Maximum Principle:

Introduce costate variables p(t) (analogous to momenta) and the control Hamiltonian H(x, p, u) = p * f(x, u, t) - f_0(x, u, t). Optimal control u* must maximize H at every time t:

H(x*, p*, u*) = max over u in U of H(x*, p*, u)

E[u] = integral over Omega of W(Du) dx - integral over partial Omega of g * u dS

where u is the displacement vector, W(F) is the stored energy density (a function of the deformation gradient F = Du), and g is the surface traction. For linear elasticity, W(F) is quadratic in the symmetric strain tensor epsilon = (F + F^T)/2.

The Euler-Lagrange equations yield the Lamé-Navier equations of linear elasticity. For nonlinear elasticity (rubber, biological tissue), quasiconvexity of W is the condition for well-posedness of the variational problem.

Variational Principle for Ground State Energy:

E_0 ≤ (psi, H psi) / (psi, psi) for any trial function psi

The ground state energy E_0 is the minimum of the Rayleigh quotient over all nonzero functions psi in the domain of H. Minimizing over a parametric family of trial functions (Ritz method) gives upper bounds for E_0 and approximate ground states.

The Rayleigh-Ritz method is the foundation of finite element methods and density functional theory (DFT) in computational chemistry.