1 Foundations
Variational calculus studies quantities defined by integrals or other expressions that depend on an entire function rather than on a single variable. The main objective is to determine which functions make these quantities stationary, usually meaning that small perturbations do not change the value to first order. This framework provides a systematic way to treat optimization problems in continuous settings.
1.1 Functionals
A functional is a rule that assigns a number to a function. Typical examples involve integrals whose integrands depend on the function, its derivatives, and the independent variable. Unlike ordinary functions, functionals compare whole curves, surfaces, or fields, and they often arise when measuring length, area, energy, or time.
1.2 Admissible functions
Admissible functions are the candidate functions allowed in a variational problem. They must satisfy the prescribed regularity, boundary values, and any additional restrictions imposed by the problem. Choosing the admissible class carefully is essential, since the existence and form of extrema depend on the space of functions under consideration.
1.3 Variations of a function
A variation is a small perturbation of a function within the admissible class. One usually studies a one-parameter family of nearby functions and examines how the functional changes as the parameter varies. This idea leads to first-variation formulas that identify conditions for stationarity.
1.4 Stationary values and extrema
A stationary value occurs when the first-order change of a functional vanishes under all allowed variations. Such a point may represent a minimum, a maximum, or a saddle point. In many applications, the stationary condition is more fundamental than strict extremality, since it yields the governing equations even when a true minimum is not guaranteed.
2 Classical variational problems
Classical variational problems helped shape the development of the field by showing how geometric and physical questions can be reformulated as optimization problems for functionals. These examples illustrate the power of the variational viewpoint and often lead to elegant differential equations.
2.1 The brachistochrone problem
The brachistochrone problem asks for the curve of fastest descent between two points under gravity. Its solution is a cycloid, not a straight line, demonstrating that the quickest path need not be the shortest. The problem became a landmark example in the early history of the calculus of variations.
2.2 The isoperimetric problem
The isoperimetric problem seeks the figure with the largest area among all closed curves of fixed perimeter. In the plane, the circle is the optimal shape. Variational methods show how a geometric constraint can be incorporated into an optimization principle.
2.3 Geodesics
Geodesics are curves that locally minimize distance on a surface or more general geometric space. On a plane they are straight lines, while on curved surfaces they follow the intrinsic geometry. In differential geometry, geodesics are central examples of variationally defined paths.
2.4 Minimal surfaces
Minimal surfaces arise by minimizing surface area subject to boundary conditions. Soap films spanning wire frames provide a familiar physical model. The resulting equations describe surfaces whose mean curvature vanishes, and they connect the calculus of variations with geometry and physics.
3 Euler–Lagrange theory
Euler–Lagrange theory provides the standard differential equation associated with many variational problems. It translates the condition of stationarity into a local equation that must be satisfied by any extremal function. This result is one of the core tools of the subject.
3.1 Derivation of the Euler–Lagrange equation
The Euler–Lagrange equation is obtained by computing the first variation of a functional and requiring that it vanish for all permissible perturbations. By integrating by parts, derivatives acting on the variation are transferred to the coefficient functions. The resulting condition yields a differential equation for the unknown function.
3.2 Necessary conditions for extrema
A function that minimizes or maximizes a functional must satisfy certain necessary conditions, the most basic being the Euler–Lagrange equation. Additional criteria may be needed to distinguish minima from saddle points or maxima. Second-variation tests and related arguments are often used for this purpose.
3.3 Natural boundary conditions
When endpoints or boundary values are not fixed, extra conditions emerge from the variation process. These are called natural boundary conditions and complement the Euler–Lagrange equation. They describe how an extremal interacts with a free boundary or an unconstrained endpoint.
3.4 Multiple dependent variables
Many problems involve several unknown functions at once, such as vector-valued curves or fields. In that setting, the Euler–Lagrange equation becomes a system of coupled equations, one for each dependent variable. This generalization is common in mechanics, field theory, and geometric analysis.
4 Constraints and generalized formulations
Real variational problems often include restrictions on the admissible functions. Constraints may be geometric, algebraic, or differential, and they require modified methods to describe stationary points correctly. These generalized formulations extend the basic theory to more realistic models.
4.1 Holonomic constraints
Holonomic constraints restrict the variables through equations involving only the functions and independent variables, not their velocities or derivatives. Such constraints can often be built into the variational formulation directly. They reduce the set of admissible functions to a lower-dimensional family.
4.2 Nonholonomic constraints
Nonholonomic constraints involve derivatives and are not always integrable into simple position conditions. They appear in systems with rolling or directional restrictions. Their treatment in variational calculus is subtler, because not every constrained displacement is generated by an ordinary function-space variation.
4.3 Lagrange multipliers in variational problems
Lagrange multipliers allow constraints to be incorporated by adding auxiliary terms to the functional. The multiplier enforces the restriction while preserving the variational structure. This method is widely used because it converts a constrained problem into an unconstrained one on an enlarged space.
4.4 Constrained extrema
A constrained extremum is a stationary point found within the admissible set defined by the restrictions. Such extrema may satisfy modified Euler–Lagrange equations together with constraint equations. The resulting conditions often describe balance between the objective functional and the imposed limitations.
5 Higher-order variational problems
Some functionals depend on higher derivatives of the unknown function, not just on the function and its first derivative. These problems occur in beam theory, elasticity, and geometric models where curvature or bending is important. They require extensions of the standard variational machinery.
5.1 Functionals involving higher derivatives
Higher-order functionals depend on second, third, or even higher derivatives. Because these derivatives measure more refined geometric or physical features, they are useful in models involving stiffness, smoothness, or curvature control. Their analysis is more involved due to the increased number of boundary terms.
5.2 Higher-order Euler–Lagrange equations
The Euler–Lagrange equation generalizes to higher-order problems by repeated integration by parts. The resulting differential equations contain derivatives of the unknown function up to twice the order appearing in the functional. These equations characterize stationary points in the higher-order setting.
5.3 Boundary conditions for higher-order problems
Higher-order functionals require additional boundary data to determine a well-posed problem. Depending on which derivatives are fixed, the variation process yields corresponding natural boundary conditions. These conditions ensure consistency between the functional and the admissible class.
6 Variational methods in mechanics
Variational ideas play a fundamental role in classical mechanics. Physical trajectories can often be characterized as stationary points of an action functional, which unifies the description of motion across many systems. This approach is both conceptually elegant and practically powerful.
6.1 Principle of least action
The principle of least action states that the actual motion of a system makes the action stationary. Despite its name, the action is not always minimized; it may instead be a saddle point. The principle provides a compact route from physical assumptions to differential equations of motion.
6.2 Lagrangian mechanics
Lagrangian mechanics expresses dynamics in terms of kinetic and potential energy through a Lagrangian function. The equations of motion arise from applying variational principles to the action integral. This formulation is especially useful for systems with constraints or generalized coordinates.
6.3 Hamilton’s principle
Hamilton’s principle asserts that the path taken by a system between two times is stationary with respect to variations that keep the endpoints fixed. It serves as a central statement connecting mechanics and variational calculus. Many classical equations of motion can be derived from it directly.
6.4 Conservation laws and symmetries
Symmetries in a variational system often lead to conserved quantities. For example, invariance under time translation is associated with energy conservation, while spatial symmetries relate to momentum. This relationship reveals a deep structural link between geometry and dynamics.
7 Weak and generalized solutions
Not all variational problems have smooth solutions. In many important settings, one must work with weaker notions of differentiability and broader function spaces. This modern framework allows existence theory to handle more realistic and less regular problems.
7.1 Weak derivatives
Weak derivatives extend the classical derivative concept to functions that may not be differentiable in the usual sense everywhere. They are defined through integration against test functions. This idea permits variational equations to be interpreted meaningfully even when solutions lack smoothness.
7.2 Sobolev spaces
Sobolev spaces collect functions whose weak derivatives satisfy integrability conditions. They provide the natural setting for many variational problems because they balance flexibility with enough structure for analysis. These spaces are central in partial differential equations and the direct method.
7.3 Direct methods in the calculus of variations
The direct method seeks minimizers by showing that a functional is coercive and lower semicontinuous on an appropriate space. Instead of solving an Euler–Lagrange equation first, one studies minimizing sequences and compactness. This strategy is especially effective for proving existence results.
7.4 Existence of minimizers
An existence theorem establishes that at least one function actually attains the infimum of the functional. Such results often depend on compactness, weak convergence, and lower semicontinuity. They are essential because a formally derived stationary condition does not by itself guarantee that a solution exists.
8 Advanced topics
Advanced developments broaden the reach of variational calculus into modern analysis, geometry, and control theory. These topics reveal the depth of the subject and its continued influence across mathematics and applied sciences.
8.1 Noether’s theorem
Noether’s theorem states that every continuous symmetry of an action corresponds to a conservation law. This principle provides a systematic explanation for the origin of conserved quantities in physics. It is one of the most influential results connecting symmetry with dynamics.
8.2 Calculus of variations in several variables
When the unknown is a function of several independent variables, the theory applies to functionals defined over regions rather than intervals. This setting includes surfaces, fields, and multidimensional media. The resulting equations are often partial differential equations.
8.3 Optimal control
Optimal control seeks control functions that guide a dynamical system while minimizing a cost functional. It extends variational ideas to time-dependent systems with inputs and constraints. The theory is used in engineering, economics, and automated decision-making.
8.4 Convex variational problems
Convex variational problems have functionals with convex structure, which often simplifies analysis and ensures uniqueness of minimizers. Convexity also supports powerful existence and stability results. These problems occupy an important place in optimization and modern analysis.