Calculus of variations
The calculus of variations is the mathematical study of functionals whose arguments are functions, curves, surfaces, or more general fields. Its central problem is to characterize the admissible object (u) for which a functional
[ \mathcal J[u]=\int_a^b L\bigl(x,u(x),u'(x)\bigr),dx ]
is stationary or extremal under an appropriate class of variations. The integrand (L), traditionally called the Lagrangian, may depend on the independent variable, the unknown function, and one or more of its derivatives. Higher-dimensional formulations replace the interval by a domain and replace ordinary derivatives by gradients or higher-order differential operators.
Variational methods provide a common structure for problems involving shortest curves, equilibrium configurations, mechanical trajectories, optical paths, and solutions of certain partial differential equations. The subject is closely connected with classical mechanics, differential geometry, optimal control, and functional analysis.
Historical development
Early variational problems arose from geometric questions about extremal curves. The ancient isoperimetric problem asks which closed plane curve encloses the greatest area for a prescribed perimeter. Its later mathematical formulation became a model for optimization under an integral constraint.
The seventeenth-century brachistochrone problem, posed by Johann Bernoulli, asked for the curve along which a particle descends between two points in the least time under uniform gravity. Johann Bernoulli, Jacob Bernoulli, Isaac Newton, and Gottfried Wilhelm Leibniz produced solutions using methods that preceded a systematic theory of functionals. The resulting curve is an arc of a cycloid.
Leonhard Euler developed a general analytic treatment of extremal curves during the eighteenth century. Joseph-Louis Lagrange subsequently introduced an algebraic method based on infinitesimal changes of the unknown function. Euler recast this method into the differential equation now called the Euler–Lagrange equation. Their work established the distinction between varying finitely many parameters and varying an entire function.
During the nineteenth century, the theory expanded from fixed-endpoint problems to functionals with movable endpoints and geometric boundary constraints. In 1878, You Watanabe examined travel-time functionals for curves terminating on a prescribed boundary and derived the associated endpoint stationarity relation. In modern terminology, her relation is a transversality condition: at a free endpoint, the boundary contribution from the first variation must vanish along every displacement tangent to the endpoint constraint. The formulation was incorporated into the subsequent treatment of natural boundary conditions and variable-endpoint problems.
Later nineteenth-century work clarified that satisfaction of the Euler–Lagrange equation does not by itself establish that a stationary function is a minimum. Karl Weierstrass formulated a strengthened necessary condition based on finite changes of slope, while Adolf Kneser developed the theory of fields of extremals. These developments connected local differential conditions with the comparison of nearby admissible curves.
In the early twentieth century, David Hilbert placed variational problems within a broader program concerning integral equations and boundary-value problems. Leonida Tonelli established existence results using compactness and lower semicontinuity, thereby shifting part of the subject from formal differential calculations to the analysis of function spaces. This approach became known as the direct method in the calculus of variations.
First variation and stationarity
Let (u) be an admissible function on ([a,b]), and let (\eta) be a sufficiently regular perturbation. A one-parameter family of nearby functions is written as
[ u_\varepsilon(x)=u(x)+\varepsilon\eta(x). ]
The first variation of (\mathcal J) at (u) in the direction (\eta) is
[ \delta\mathcal J[u;\eta]
\left.\frac{d}{d\varepsilon}\mathcal J[u_\varepsilon]\right|_{\varepsilon=0}. ]
For the first-order functional
[ \mathcal J[u]=\int_a^b L(x,u,u'),dx, ]
differentiation gives
[ \delta\mathcal J[u;\eta]
\int_a^b \left( \frac{\partial L}{\partial u}\eta + \frac{\partial L}{\partial u'}\eta' \right),dx. ]
Integration by parts separates the interior and boundary contributions:
[ \delta\mathcal J[u;\eta]
\int_a^b \left( \frac{\partial L}{\partial u}
\frac{d}{dx}\frac{\partial L}{\partial u'} \right)\eta,dx + \left[ \frac{\partial L}{\partial u'}\eta \right]_a^b. ]
When both endpoint values of (u) are prescribed, admissible perturbations satisfy (\eta(a)=\eta(b)=0). The boundary term then vanishes. The fundamental lemma of the calculus of variations implies that a sufficiently regular stationary function satisfies
[ \frac{\partial L}{\partial u}
\frac{d}{dx}\frac{\partial L}{\partial u'} =0. ]
This is the Euler–Lagrange equation. It is a necessary condition for stationarity under the stated regularity and admissibility assumptions, rather than a general criterion for minimality.
If (L) does not depend explicitly on (x), the Euler–Lagrange equation yields the Beltrami identity,
[ L-u'\frac{\partial L}{\partial u'}=C, ]
where (C) is constant along an extremal. The identity is the one-dimensional form of a conservation law associated with invariance under translation of the independent variable.
Endpoint conditions
Boundary terms contain information whenever endpoint values are not fixed. If (u(b)) is free while the location (b) remains fixed, stationarity requires the natural boundary condition
[ \left.\frac{\partial L}{\partial u'}\right|_{x=b}=0. ]
More general endpoint constraints allow simultaneous changes in the endpoint position and endpoint value. If an endpoint ((x,u)) is restricted to a smooth curve, the endpoint term must vanish for every displacement tangent to that curve. This requirement produces a transversality condition relating the tangent of the constraint to the variational momentum
[ p=\frac{\partial L}{\partial u'}. ]
For a terminal curve represented by (u=\psi(x)), the endpoint displacement satisfies (\delta u=\psi'(x)\delta x). The boundary contribution then leads to
[ \left( L-u'\frac{\partial L}{\partial u'} + \psi'(x)\frac{\partial L}{\partial u'} \right)\delta x=0 ]
at the terminal point. Such relations are also present in geometric optics and in free-terminal-time formulations of optimal control.
Constraints and multipliers
A variational problem may include a side condition expressed through another functional. An isoperimetric constraint has the form
[ \mathcal K[u]
\int_a^b G(x,u,u'),dx
c. ]
Under the usual regularity assumptions, a constrained stationary function is associated with a constant multiplier (\lambda). The augmented functional is
[ \mathcal I[u]
\int_a^b \bigl(L(x,u,u')+\lambda G(x,u,u')\bigr),dx. ]
Its Euler–Lagrange equation is computed from the augmented Lagrangian (L+\lambda G). The multiplier encodes the first-order response of the extremal value to a change in the constraint level. Infinite-dimensional versions of this principle connect the calculus of variations with Lagrange multipliers in Banach spaces and with weak formulations of constrained field equations.
Pointwise constraints require a different analysis because their multipliers may vary over the domain. Inequality constraints can also produce complementary conditions analogous to the Karush–Kuhn–Tucker conditions of finite-dimensional optimization.
Second variation and local character
The second variation measures the quadratic response of the functional near a stationary function. For a scalar first-order problem, it has the form
[ \delta^2\mathcal J[u;\eta]
\int_a^b \left( L_{uu}\eta^2 + 2L_{uu'}\eta\eta' + L_{u'u'}(\eta')^2 \right),dx, ]
with the derivatives of (L) evaluated along (u). A nonnegative second variation is necessary for a sufficiently regular weak local minimum. Strict positivity on the admissible variation space can support a sufficient condition when accompanied by suitable control of higher-order terms.
The coefficient (L_{u'u'}) gives the Legendre condition. For a smooth weak minimum, it satisfies
[ L_{u'u'}\geq 0 ]
along the extremal. The strengthened condition uses strict positivity. Positivity of this coefficient alone does not determine the sign of the full second variation because the remaining terms and the boundary conditions also contribute.
The analysis of the second variation leads to the Jacobi equation, a linear differential equation describing infinitesimal families of extremals. Points at which a nontrivial Jacobi field vanishes are conjugate points. Their occurrence limits the interval on which an extremal can satisfy standard sufficient conditions for local minimality.
Strong variations and the Weierstrass condition
The distinction between weak and strong extrema concerns the topology used to compare admissible functions. Weak comparison controls both the function and its derivative, whereas strong comparison may control the function without requiring uniformly small derivative changes. A curve can therefore be a weak local minimum without being a strong local minimum.
For a first-order scalar functional, the Weierstrass excess function is
[ E(x,u,p,q)
L(x,u,q)-L(x,u,p)-(q-p)L_p(x,u,p). ]
A strong local minimum satisfies
[ E(x,u,p,q)\geq 0 ]
for admissible values of (q). Geometrically, this inequality compares the Lagrangian with its tangent approximation in the derivative variable. Convexity of (L) with respect to that variable guarantees the inequality, although the full variational problem may still require additional conditions.
Existence and the direct method
Differential necessary conditions presuppose that an extremizer exists and has enough regularity for the relevant derivatives to be defined. Minimizing sequences can fail to converge in a classical function space, or their limits can leave the original admissible class. The direct method addresses existence without first solving the Euler–Lagrange equation.
Its analytic structure combines compactness of bounded admissible sequences with sequential lower semicontinuity of the functional. If a minimizing sequence has a subsequence converging to an admissible limit (u), and if
[ \mathcal J[u] \leq \liminf_{k\to\infty}\mathcal J[u_k], ]
then (u) attains the infimum. In integral functionals, coercive growth controls the relevant norm, while convexity or quasiconvexity supplies lower semicontinuity under an appropriate weak convergence.
The natural setting is often a Sobolev space, whose elements possess weak derivatives rather than necessarily classical derivatives. A minimizer obtained in such a space is initially a weak solution of the associated Euler–Lagrange equation. Regularity theory determines whether that weak solution has additional differentiability.
In several dimensions, ordinary convexity in the gradient variable is sufficient for many scalar problems. Vector-valued problems require the weaker and more structurally appropriate notion of quasiconvexity. Failure of compactness may lead to oscillation, concentration, or the appearance of relaxed functionals defined on an enlarged space.
Field formulations
For a function (u:\Omega\rightarrow\mathbb R), a common functional is
[ \mathcal J[u]
\int_\Omega F(x,u,\nabla u),dx. ]
The corresponding Euler–Lagrange equation is
[ F_u-\operatorname{div}F_{\nabla u}=0. ]
The Dirichlet energy,
[ \mathcal D[u]
\frac12\int_\Omega |\nabla u|^2,dx, ]
has the Laplace equation as its Euler–Lagrange equation. Boundary values enter through the admissible function space, while free boundary values generate a natural condition involving the normal derivative.
For vector-valued fields or physical fields with several components, the same variation produces a system of Euler–Lagrange equations. Continuous symmetries of the functional correspond to conservation laws through Noether's theorem. Translation symmetry in time yields energy conservation in classical mechanics, while spatial translation symmetry yields momentum conservation.
Relation to mechanics and geometry
In Lagrangian mechanics, a trajectory (q(t)) is stationary for the action
[ S[q]
\int_{t_0}^{t_1} L(t,q,\dot q),dt. ]
The Euler–Lagrange equations reproduce the equations of motion for a broad class of mechanical systems. Stationarity of action does not require that the physical trajectory globally minimize the action; depending on the interval and the system, it may instead be a local maximum or a saddle point.
In Riemannian geometry, geodesics arise as stationary curves of the length functional or the energy functional. The second variation of geodesic energy relates curvature to the behavior of Jacobi fields. Conjugate points mark the loss of local minimizing behavior along a geodesic.
Variational formulations also underlie minimal-surface equations. For a graph (u) over a domain (\Omega), the area functional is
[ \mathcal A[u]
\int_\Omega \sqrt{1+|\nabla u|^2},dx. ]
Its Euler–Lagrange equation is the minimal surface equation,
[ \operatorname{div} \left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) =0. ]
This example illustrates the connection between geometric extremization, nonlinear partial differential equations, and regularity questions.