Calculus of Variations · aiwiki.page

The calculus of variations is a branch of mathematical analysis concerned with finding functions, curves, or fields that minimize, maximize, or make stationary a functional—a quantity that assigns a number to an entire function. Unlike elementary calculus, where the unknown is usually a number or a finite collection of numbers, variational problems generally have infinitely many degrees of freedom. They provide a common framework for shortest paths, equilibrium configurations, and laws of motion. (courses.maths.ox.ac.uk)

Functionals and admissible functions

A functional can depend on the values of a function throughout a domain. A standard example is the integral functional

J[y]=\int_a^b F\bigl(x,y(x),y'(x)\bigr)\,dx,

where $F$ is a prescribed integrand and $y'$ is the derivative of the unknown function.

The problem must also specify an admissible class: functions with appropriate smoothness or integrability, boundary values, and any additional constraints. For example, one may require $y(a)=A$ and $y(b)=B$ . A global minimizer has the smallest functional value throughout this class; a local minimizer need only be optimal among sufficiently nearby admissible functions. The meaning of “nearby” depends on the chosen norm or topology. (math.cmu.edu)

First variation and the Euler–Lagrange equation

To examine a candidate $y$ , introduce a perturbed function

y_\varepsilon=y+\varepsilon\eta,

where $\eta$ is an admissible variation. Fixed endpoint values require $\eta(a)=\eta(b)=0$ . The first variation is

\delta J[y;\eta] =\left.\frac{d}{d\varepsilon}J[y+\varepsilon\eta]\right|_{\varepsilon=0}.

For a sufficiently smooth integrand and candidate,

\delta J[y;\eta] =\int_a^b\left(F_y\eta+F_{y'}\eta'\right)\,dx.

Here $F_y$ and $F_{y'}$ are partial derivatives with respect to the integrand’s second and third arguments. Integration by parts gives

\delta J[y;\eta] =\left[F_{y'}\eta\right]_a^b+ \int_a^b\left(F_y-\frac{d}{dx}F_{y'}\right)\eta\,dx.

The boundary term vanishes for fixed endpoints. Requiring the remaining integral to vanish for every smooth endpoint-preserving variation yields the Euler–Lagrange equation

\boxed{F_y-\frac{d}{dx}F_{y'}=0}.

This converts a stationarity condition on a functional into a differential equation. It does not, by itself, establish that the candidate is a minimum or maximum. (homepages.ucl.ac.uk)

Boundary conditions, constraints, and minimality

If an endpoint value is free, its variation need not vanish. For the preceding first-order functional, a free value at a fixed endpoint produces the natural boundary condition $F_{y'}=0$ there. Allowing the endpoint’s position to move leads to transversality conditions. Integral constraints can often be treated by introducing a Lagrange multiplier and making an augmented functional stationary, subject to suitable constraint qualifications. (math.cmu.edu)

The second variation examines the next-order change:

\delta^2J[y;\eta] =\left.\frac{d^2}{d\varepsilon^2} J[y+\varepsilon\eta]\right|_{\varepsilon=0}.

A smooth local minimizer must have nonnegative second variation in every admissible direction. Nonnegativity alone is not generally sufficient: stronger estimates controlling the size of variations, together with appropriate regularity assumptions, can establish local minimality. (math.cmu.edu)

Classical examples

Several geometric and mechanical problems illustrate the distinction between optimizing a curve and optimizing a point:

Shortest paths: minimizing arc length between two points produces straight segments in Euclidean space. On curved surfaces, the corresponding stationary curves are geodesics.
Brachistochrone: the brachistochrone problem asks for the curve along which a particle released from rest descends between prescribed points in the least time, under uniform gravity and without friction. Its solution is an arc of a cycloid, rather than generally a straight line.
Minimal surfaces: soap-film idealizations lead to minimizing surface area subject to boundary constraints. The resulting stationary surfaces are called minimal surfaces. (courses.maths.ox.ac.uk)

These examples have different objective functionals—length, travel time, and area—even though each optimizes a geometric object.

Several variables and partial differential equations

For a scalar field $u$ on a domain $\Omega\subset\mathbb R^n$ , consider

J[u]=\int_\Omega F(x,u,\nabla u)\,dx,

where $\nabla u$ is the gradient. Stationarity under variations supported inside the domain gives

F_u-\sum_{i=1}^{n} \frac{\partial}{\partial x_i} \left(F_{u_{x_i}}\right)=0.

This is generally a partial differential equation. For example,

J[u]=\frac12\int_\Omega |\nabla u|^2\,dx

has Euler–Lagrange equation $\Delta u=0$ , Laplace’s equation. Prescribed boundary data make this a boundary-value problem. Vector-valued fields yield systems of equations, while functionals involving higher derivatives yield corresponding higher-order equations. (webhomes.maths.ed.ac.uk)

Mechanics and symmetry

In classical mechanics, Hamilton’s principle states that a physical trajectory makes the action

S[q]=\int_{t_0}^{t_1}L(q,\dot q,t)\,dt

stationary under endpoint-preserving variations. For a standard conservative system, the Lagrangian is kinetic energy minus potential energy. The resulting Euler–Lagrange equations describe its motion. “Least action” is therefore a traditional name, not a universal assertion that the action is minimized. (courses.maths.ox.ac.uk)

Noether’s theorem connects continuous symmetries of the action with conservation laws. Spatial translation and rotation symmetries give conservation of momentum and angular momentum, respectively. This makes variational formulations useful not only for deriving equations, but also for identifying their conserved quantities. (webhomes.maths.ed.ac.uk)

Existence and the direct method

Solving stationarity equations does not prove that a minimum exists. The direct method instead starts with a minimizing sequence $u_k$ , whose functional values approach the infimum. It seeks a convergent subsequence with an admissible limit $u$ , then establishes

J[u]\leq\liminf_{k\to\infty}J[u_k].

This lower-semicontinuity inequality ensures that the limit attains the infimum.

The main ingredients are coercivity or other bounds on minimizing sequences, suitable compactness, closure of the admissible class, and lower semicontinuity. Sobolev spaces and weak convergence supply natural settings for many such arguments. These methods belong to the interaction between variational calculus and functional analysis. (arxiv.org)

For vector-valued functions in several dimensions, ordinary convexity is often too restrictive. Quasiconvexity, rank-one convexity, and polyconvexity play important roles in existence and regularity theory, particularly in nonlinear elasticity. Existence of a minimizer also does not imply that it is smooth or unique. (maths.ox.ac.uk)

Numerical methods and applications

The Rayleigh–Ritz method replaces an infinite-dimensional admissible class with a finite-dimensional approximation:

u_N=u_0+\sum_{j=1}^{N}c_j\phi_j.

Here $u_0$ supplies prescribed boundary values, and the trial functions $\phi_j$ preserve the corresponding homogeneous conditions. Optimizing the coefficients $c_j$ turns the variational problem into finite-dimensional mathematical optimization. Locally supported trial functions provide a route to the finite element method, widely used in structural mechanics and other field problems. (compute.cba.mit.edu)

Variational methods also appear in optimal control, eigenvalue problems, and image processing. Their application requires attention to the admissible function space, boundary conditions, and convergence of approximations; a formal stationary equation alone does not settle these issues. (doi.org)

Historical development

Johann Bernoulli posed the brachistochrone challenge in 1696. Work by Johann and Jakob Bernoulli helped turn individual curve-optimization problems into a broader mathematical subject. Leonhard Euler’s 1744 treatise developed a general framework and a differential equation for extremizing integral functionals. Joseph-Louis Lagrange subsequently introduced a systematic analytic treatment of variations; his published account of 1760 explained the use of the symbol $\delta$ to distinguish variation from ordinary differentiation. (mathshistory.st-andrews.ac.uk)

The classical emphasis on necessary conditions later expanded into rigorous existence theory. Around 1900, David Hilbert advanced the direct-method approach. Twentieth-century developments in weak convergence and lower semicontinuity, including work by Charles B. Morrey Jr. and Norman G. Meyers, helped establish the modern theory of multidimensional variational problems. (arxiv.org)

References

ASO: Calculus of Variations (2025–26)courses.maths.ox.ac.uk
Calculus of Variations — Riccardo Cristoferi lecture notesmath.cmu.edu
MATH0043 §2: Calculus of Variationshomepages.ucl.ac.uk
Brief notes on the calculus of variationswebhomes.maths.ed.ac.uk
Weak lower semicontinuity of integral functionals and applicationsarxiv.org
Calculus of Variations — Mathematical Institutemaths.ox.ac.uk
Finite Elementscompute.cba.mit.edu
MATH 333 Lecture Notes: FEM—General Rayleigh-Ritz Approachsites.calvin.edu
Brachistochrone problem — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk