Category: Calculus of variations

Envelope theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of th
Inverse problem for Lagrangian mechanics
In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some
Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Plateau's problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who
Bounded variation
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are
Palais–Smale compactness condition
The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of ce
Fundamental lemma of calculus of variations
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.Accordingly, the necessary condi
Action (physics)
In physics, action is a scalar quantity describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle
Noether's second theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given acti
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was
Pseudo-monotone operator
In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in t
First variation
In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional mapping the function h to where y and h are functions, and ε is a sc
Variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the v
Energy principles in structural mechanics
Energy principles in structural mechanics express the relationships between stresses, strains or deformations, displacements, material properties, and external effects in the form of energy or work do
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a
Γ-convergence
In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter
Tonelli's theorem (functional analysis)
In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for function
Transversality (mathematics)
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the id
Hu–Washizu principle
In continuum mechanics, and in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action is stationary, where is the elastic stiffness tenso
Lagrangian system
In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem
Quasiconvexity (calculus of variations)
In the calculus of variations, a subfield of mathematics, quasiconvexity is a generalisation of the notion of convexity. It is used to characterise the integrand of a functional and related to the exi
Mosco convergence
In mathematical analysis, Mosco convergence is a notion of convergence for functionals that is used in nonlinear analysis and set-valued analysis. It is a particular case of Γ-convergence. Mosco conve
Direct method in the calculus of variations
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and
Regularized canonical correlation analysis
Regularized canonical correlation analysis is a way of using ridge regression to solve the singularity problem in the cross-covariance matrices of canonical correlation analysis. By converting and int
Dirichlet energy
In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space H1. The Dirichlet energy is intimately connected to
Calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functionsand functionals, to find maxima and minima of functio
Mountain pass theorem
The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonst
Stampacchia Medal
The Stampacchia Gold Medal is an international prize awarded every three years by the Italian Mathematical Union (Unione Matematica Italiana - UMI {it}) together with the Ettore Majorana Foundation (E
History of variational principles in physics
A variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum (minimum, maximum or saddle point) of a functio
Beltrami identity
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals
Homicidal chauffeur problem
In game theory, the homicidal chauffeur problem is a mathematical pursuit problem which pits a hypothetical runner, who can only move slowly, but is highly maneuverable, against the driver of a motor
Malliavin calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to
Nehari manifold
In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of Zeev Nehari . It is a differentiable manifold associa
Lagrange multipliers on Banach spaces
In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The meth
Signorini problem
The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid
Carathéodory function
In Mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measu
Convenient vector space
In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimens
Chaplygin problem
In mathematics, particularly in the fields of nonlinear dynamics and the calculus of variations, the Chaplygin problem is an isoperimetric problem with a differential constraint. Specifically, the pro
Functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that a
Geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential ge
In applied mathematics, the Ostrogradsky instability is a feature of some solutions of theories having equations of motion with more than two time derivatives (higher-derivative theories). It is sugge
List of variational topics
This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction. * Action (physics) * Averaged Lagrangian * Brachistochrone curve * Calc
Saint-Venant's theorem
In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity i
Variational vector field
In the mathematical fields of the calculus of variations and differential geometry, the variational vector field is a certain type of vector field defined on the tangent bundle of a differentiable man
Maupertuis's principle
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of pa
Variational bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is th
Noether identities
In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whos
Variational inequality
In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematica
Transportation theory (mathematics)
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mat
Minkowski's first inequality for convex bodies
In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequ
Fermat's principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray bet
Minkowski–Steiner formula
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" o
Almgren–Pitts min-max theory
In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the effo
De Donder–Weyl theory
In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and t
Lagrangian (field theory)
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete
Weierstrass–Erdmann condition
The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be
Costate equation
The costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order
Newton's minimal resistance problem
Newton's minimal resistance problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the directi
Brunn–Minkowski theorem
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The origi
Legendre–Clebsch condition
In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum. For the problem of minim
Morse–Palais lemma
In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be ex
Obstacle problem
The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic mem
Hilbert's twentieth problem
Hilbert's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether all boundary value problems can be solved (that is, do var
Path of least resistance
The path of least resistance is the physical or metaphorical pathway that provides the least resistance to forward motion by a given object or entity, among a set of alternative paths. The concept is
Caccioppoli set
In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli s