Mathematics · Advanced Topics

Variational Methods

Variational principles — minimizing or extremizing functionals — underlie nearly all of physics. Hamilton's principle (classical mechanics), Fermat's principle (optics), the variational principle in quantum mechanics, and the Einstein-Hilbert action all express physics as the search for an extremal path.

Prerequisites

Lagrangian mechanics (Ch. LA-Mech) \cdot Differential equations (Ch. DE) \cdot Calculus (Ch. 22) \cdot

Linear algebra (Ch. LA)

Learning Goals

$Derive the Euler-Lagrange equation from the stationary-action condition \deltaJ = 0 and apply i$ t to standard physical functionals.
Use the Beltrami identity to reduce problems with no explicit x-dependence, and apply it to the brachistochrone and catenary.
$Compute a functional derivative \deltaF/\delta\rho and state the Hohenberg-Kohn theorem underlying dens$ ity functional theory.
Apply the Rayleigh-Ritz method to bound the ground-state energy of the hydrogen atom using a Gaussian trial wavefunction.
$Interpret Feynman's path integral as a sum over all paths weighted by e^{iS/\hbar} and relate$ $the classical limit \hbar\to0 to the stationary-phase approximation$

VM.1 The Euler-Lagrange Equation

A functionalmaps functions to numbers: J[y] = ∫(a to b) F(x, y, y') dx. The calculus of variations finds functions y(x) that make J stationary.

Theorem VM.1 — Euler-Lagrange Equation

A necessary condition for y(x) to extremize J[y] = \int(a to b) F(x, y, y') dx (with y fixed

at endpoints) is:

\partialF/\partialy - d/dx (\partialF/\partialy') = 0

Derivation: let y \to y + \varepsilon\eta where \eta(a) = \eta(b) = 0. Then \deltaJ = \varepsilon \int[\partialF/\partialy \eta + \partialF/\partialy' \eta'] dx =

\varepsilon \int[\partialF/\partialy - d/dx(\partialF/\partialy')] \eta dx = 0 for all \eta \to E-L equation. Extensions: multiple function

y_{i}(x) \to one E-L equation per y_{i}. Higher derivatives F(y, y', y'', \cdots) \to Ostrogradsky eq

ation.

First integrals: if F has no explicit x-dependence: H = y' ∂F/∂y' − F = const (Beltrami identity — Hamiltonian!). If F has no explicit y-dependence: ∂F/∂y' = const (conserved momentum).

Example VM.1 — Brachistochrone Problem

$Find the curve of fastest descent between two points A = (0,0) and B = (x_{1}, y_{1}) under grav$ ity (the brachistochrone).

Time functional:

dt = ds/v where v = \sqrt(2gy) (energy conservation from rest). ds = \sqrt(1+y'^{2}) dx. T = \int(0 to x

_{1}) \sqrt((1+y'^{2})/(2gy)) dx. Minimize T

F has no x:

Use Beltrami: F - y' \partialF/\partialy' = const. F = \sqrt((1+y'^{2})/(2gy)). \partialF/\partialy' = y'/\sqrt(2gy(1+y'^{2})). Belt

rami: 1/\sqrt(2gy(1+y'^{2})) = 1/\sqrt(2g c^{2}) (constant

Simplify:

y(1+y'^{2}) = c^{2} = const. Parametric solution: x = c^{2}(\theta - sin\theta)/2, y = c^{2}(1 - cos\theta)/2. This i

s a cycloid — traced by a point on the rim of a rolling circle of radius R = c^{2}/2

Result:The fastest path is NOT the straight line, NOT the circular arc, but the cycloid. Johann Bernoulli (1696) — first solved by Newton, Leibniz, l'Hôpital, and Bernoulli using different methods. The brachistochrone is also the tautochrone (period independent of starting point).

VM.2 Functional Derivatives

For a functional F[ρ] = ∫ f(r, ρ(r), ∇ρ(r)) d³r, the functional derivative is:

\deltaF/\delta\rho(r) = \partialf/\partial\rho - \nabla\cdot(\partialf/\partial(\nabla\rho)) \qquad (functional derivative)

(VM.1)

This is the continuum analog of a partial derivative. The condition δF/δρ = 0 gives the Euler-Lagrange equation for field theories.

Density functional theory (DFT): Hohenberg-Kohn theorem (1964) states that the ground-state energy of an N-electron system is a functional E[ρ] of the electron density ρ(r) alone. The Kohn-Sham equations minimize E[ρ] variationally — replacing the interacting problem with an effective non-interacting problem. DFT is the workhorse of computational quantum chemistry (Nobel 1998 to Kohn).

VM.3 Rayleigh-Ritz Method

Approximate the solution by a finite-dimensional trial function: y(x) ≈ Σᵢ cᵢ φᵢ(x). The functional becomes a function of the cᵢ. Minimize: ∂J/∂cᵢ = 0 → linear system for the coefficients.

For the variational principle in quantum mechanics — approximate ψ by trial function ψ_trial(α₁, α₂, ...):

E_{trial} = ⟨\psi_trial|H|\psi_trial⟩/⟨\psi_trial|\psi_trial⟩ \ge E_{0} \qquad (variational bound)

(VM.2)

E_trial is always an upper bound on the ground state energy. The best trial function minimizes E_trial. Gaussian basis sets in quantum chemistry (Pople 6-31G* etc.) use the Rayleigh-Ritz method to compute electronic structure. The finite element method (FEM) for PDEs is another application: divide space into elements, minimize energy.

VM.4 Isoperimetric Problems and Constraints

Constrained optimization with a functional constraint G[y] = const uses aLagrange multiplier: extremize J[y] − λ G[y]. The E-L equation becomes: ∂(F − λG)/∂y − d/dx ∂(F − λG)/∂y' = 0.

Isoperimetric problem: maximize the enclosed area for a fixed perimeter → solution is a circle (proven by Euler). Equivalent: for a soap bubble, surface tension minimizes area for fixed volume → spherical shape.

Plateau's problem: find the minimal surface with a given boundary. E-L equation: H = 0 (mean curvature = 0). Solutions: catenoid (rotation of a catenary), helicoid (helix surface), Schwarz surfaces (triply periodic). Soap films realize these automatically.

VM.5 Path Integrals as Functional Integrals

Feynman's path integral quantizes a system by summing over all paths:

K(x_{f}, t_{f}; x_{i}, t_{i}) = \int Dx(t) e^(iS[x]/(\hbar)) \qquad (Feynman path integral)

(VM.3)

where S[x] = ∫(t_i to t_f) L(x, ẋ) dt is the action. The classical path (δS = 0 → E-L equations) dominates in the limit ℏ → 0 (stationary phase). Quantum fluctuations around the classical path give O(ℏ) corrections — the WKB approximation, loop expansion in QFT.

The stationary phase approximation: ∫ e^(iS[x]/ℏ) Dx ≈ e^(iS_cl/ℏ) × (det(−δ²S/δx²))^(−1/2) — functional determinant. This connects semiclassical mechanics (WKB) to one-loop quantum corrections.

Definition VM.1 — Common Traps

Stationary does not always mean minimum: actions can be saddle points.
Boundary conditions are part of the variation: fixed endpoints change which terms vanish.
Generalized coordinates need not be Cartesian: choose coordinates that match constraints.
Constraints introduce multipliers or reduced coordinates: ignoring them gives unphysical variations.

Exercises — VM.1–VM.5 Variational Methods

1.Find the shape of a hanging chain (catenary) by minimi

zing the potential energy subject to the constraint of fixed length. Derive the result y =

Straightforward

Apply the variational principle to the hydrogen atom using trial function \psi = e^{-\alphar}. Sho

w that minimizing ⟨H⟩ over \alpha gives the exact ground state energy (eV

Intermediate

3.Find the minimal surface of revolution (catenoid) as a solution to the Euler-Lagrange equation. What is the Goldschmidt discontinuity?

Intermediate

4.Compute the one-loop path

integral for the harmonic oscillator in Euclidean time to find the partition function Z =

tional determinant gives the correct energy levels.

Challenging

Key Takeaways

$Euler-Lagrange: \partialF/\partialy - d/dx(\partialF/\partialy') = 0. Derived from \deltaJ = 0 for all variations$ .
$Beltrami identity: if F has no x, then F - y'\partialF/\partialy' = const (energy conservation analog$ .
$Rayleigh-Ritz: approximate by trial functions \to upper bound on ground state energy$ .
$Functional derivative: \deltaF/\delta\rho = \partialf/\partial\rho - \nabla\cdot(\partialf/\partial\nabla\rho). DFT minimizes E[\rho] over density$ .
$Brachistochrone \to cycloid. Catenary \to cosh. Minimal surface \to catenoid. All from E-L$ .
$Path integral: K = \intDx e^{iS/\hbar}. Classical path dominates (\hbar\to0). Quantum corrections from$ fluctuations.