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117 relations: Adrien-Marie Legendre, Alfred Clebsch, Arc length, Archimedes, Augustin-Louis Cauchy, Beltrami identity, Brachistochrone curve, Calculus, Carl Friedrich Gauss, Carl Gustav Jacob Jacobi, Catenary, Chaplygin problem, Constant (mathematics), Convenient vector space, Converse (logic), David Hilbert, De Donder–Weyl theory, Derivative, Dimitri Bertsekas, Direct method in the calculus of variations, Dirichlet's principle, Domain of a function, Dynamic programming, Ekeland's variational principle, Elliptic partial differential equation, Emmy Noether, Encyclopedia of Mathematics, Euler–Lagrange equation, Fermat Prize, Fermat's principle, First variation, First-order partial differential equation, Function (mathematics), Function space, Functional (mathematics), Functional analysis, Functional derivative, Fundamental lemma of calculus of variations, Geodesic, Gottfried Wilhelm Leibniz, Guillaume de l'Hôpital, Hamilton's principle, Hamilton–Jacobi equation, Hamiltonian mechanics, Hamiltonian optics, Henri Lebesgue, Hilbert's problems, Hilbert's twentieth problem, Hilbert's twenty-third problem, Hu Washizu principle, ..., Infinite-dimensional optimization, Infinitesimal, Integral, Integration by parts, Inverse problem for Lagrangian mechanics, Isaac Newton, Isoperimetric inequality, Jacob Bernoulli, Jacques Hadamard, Johann Bernoulli, John Hewitt Jellett, Joseph-Louis Lagrange, Karl Weierstrass, Kinetic energy, Laplace's equation, Legendre transformation, Leonhard Euler, Leonida Tonelli, Lev Pontryagin, Louisiana State University, Luke's variational principle, Map (mathematics), Marston Morse, Mathematical analysis, MathWorld, Maxima and minima, Mechanics, Mikhail Lavrentyev, Mikhail Ostrogradsky, Minimal surface, Morse theory, Mountain pass theorem, Necessity and sufficiency, Nehari manifold, Newton's minimal resistance problem, Noether's theorem, Obstacle problem, Optimal control, Ordinary differential equation, Oskar Bolza, Otto Hesse, Perturbation theory, Peter Gustav Lejeune Dirichlet, Pierre Frédéric Sarrus, PlanetMath, Plateau's problem, Potential energy, Principle of least action, R. Tyrrell Rockafellar, Rayleigh–Ritz method, Real number, Richard Courant, Richard E. Bellman, Sign (mathematics), Siméon Denis Poisson, Snell's law, Stampacchia Medal, Strauch, Topology, Total derivative, Variational Bayesian methods, Variational bicomplex, Variational principle, Vincenzo Brunacci, Wave equation, Young measure, YouTube. Expand index (67 more) »
Adrien-Marie Legendre
Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.
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Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory.
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Arc length
Determining the length of an irregular arc segment is also called rectification of a curve.
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Archimedes
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
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Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
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Beltrami identity
The Beltrami identity, named after Eugenio Beltrami, is a simplified and less general version of the Euler–Lagrange equation in the calculus of variations.
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Brachistochrone curve
In mathematics and physics, a brachistochrone curve, or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.
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Calculus
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
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Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
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Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
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Catenary
In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.
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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.
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Constant (mathematics)
In mathematics, the adjective constant means non-varying.
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Convenient vector space
In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.
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Converse (logic)
In logic, the converse of a categorical or implicational statement is the result of reversing its two parts.
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David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
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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 time coordinates on equal footing.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Dimitri Bertsekas
Dimitri Panteli Bertsekas (b. 1942, Athens, Δημήτρης Παντελής Μπερτσεκάς) is an applied mathematician, electrical engineer, and computer scientist, and a professor at the department of Electrical Engineering and Computer Science in School of Engineering at the Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts.
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Direct method in the calculus of variations
In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900.
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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.
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Domain of a function
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method.
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Ekeland's variational principle
In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems.
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Elliptic partial differential equation
Second order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic.
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Emmy Noether
Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.
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Encyclopedia of Mathematics
The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
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Euler–Lagrange equation
In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.
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Fermat Prize
The Fermat prize of mathematical research bi-annually rewards research works in fields where the contributions of Pierre de Fermat have been decisive.
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Fermat's principle
In optics, Fermat's principle or the principle of least time, named after French mathematician Pierre de Fermat, is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time.
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First variation
In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional \delta J(y) mapping the function h to where y and h are functions, and ε is a scalar.
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First-order partial differential equation
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Function space
In mathematics, a function space is a set of functions between two fixed sets.
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Functional (mathematics)
In mathematics, the term functional (as a noun) has at least two meanings.
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Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
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Functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional to a change in a function on which the functional depends.
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Fundamental lemma of calculus of variations
In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point.
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Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
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Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
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Guillaume de l'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital (1661 – 2 February 1704) was a French mathematician.
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Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action (see that article for historical formulations).
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Hamilton–Jacobi equation
In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation.
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Hamiltonian mechanics
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
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Hamiltonian optics
Hamiltonian opticsH.
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Henri Lebesgue
Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration—summing the area between an axis and the curve of a function defined for that axis.
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Hilbert's problems
Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
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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.
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Hilbert's twenty-third problem
Hilbert's twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert.
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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 C is the elastic stiffness tensor.
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Infinite-dimensional optimization
In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body.
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Infinitesimal
In mathematics, infinitesimals are things so small that there is no way to measure them.
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.
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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 Lagrangian function.
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Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.
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Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.
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Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.
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Jacques Hadamard
Jacques Salomon Hadamard ForMemRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
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Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.
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John Hewitt Jellett
John Hewitt Jellett (25 December 1817 – 19 February 1888) was an Irish mathematician whose career was spent at Trinity College Dublin (TCD), where he rose to the rank of Provost.
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Joseph-Louis Lagrange
Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.
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Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".
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Kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
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Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.
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Legendre transformation
In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable.
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Leonhard Euler
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
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Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations.
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Lev Pontryagin
Lev Semyonovich Pontryagin (Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.
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Louisiana State University
The Louisiana State University (officially Louisiana State University and Agricultural and Mechanical College, commonly referred to as LSU) is a public coeducational university located in Baton Rouge, Louisiana.
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Luke's variational principle
In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity.
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Map (mathematics)
In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.
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Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory.
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Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
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MathWorld
MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.
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Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).
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Mechanics
Mechanics (Greek μηχανική) is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.
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Mikhail Lavrentyev
Mikhail Alekseevich Lavrentyev or Lavrentiev (Михаи́л Алексе́евич Лавре́нтьев) (November 19, 1900 – October 15, 1980) was a Soviet mathematician and hydrodynamicist.
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Mikhail Ostrogradsky
Mikhail Vasilyevich Ostrogradsky (transcribed also Ostrogradskiy, Ostrogradskiĭ) (Михаил Васильевич Остроградский, Михайло Васильович Остроградський, September 24, 1801 – January 1, 1862) was a Ukrainian mathematician, mechanician and physicist in the Russian Empire.
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Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area.
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Morse theory
"Morse function" redirects here.
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Mountain pass theorem
The mountain pass theorem is an existence theorem from the calculus of variations.
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Necessity and sufficiency
In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.
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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.
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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 direction of the axis of revolution, named after Isaac Newton, who studied the problem in 1685 and published it in 1687 in his Principia Mathematica.
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Noether's theorem
Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
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Obstacle problem
The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems.
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Optimal control
Optimal control theory deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved.
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Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.
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Oskar Bolza
Oskar Bolza (12 May 1857 – 5 July 1942) was a German mathematician, and student of Felix Klein.
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Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician.
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Perturbation theory
Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.
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Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
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Pierre Frédéric Sarrus
Pierre Frédéric Sarrus (10 March 1798, Saint-Affrique – 20 November 1861) was a French mathematician.
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PlanetMath
PlanetMath is a free, collaborative, online mathematics encyclopedia.
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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.
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Potential energy
In physics, potential energy is the energy possessed by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
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Principle of least action
The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system.
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R. Tyrrell Rockafellar
Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics.
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Rayleigh–Ritz method
The Rayleigh-Ritz method is a method of finding approximations to eigenvalue equations that cannot be solved easily (or at all) analytically.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician.
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Richard E. Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and important contributions in other fields of mathematics.
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Sign (mathematics)
In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.
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Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.
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Snell's law
Snell's law (also known as Snell–Descartes law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.
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Stampacchia Medal
The Stampacchia Gold Medal is an international prize awarded every three years by the Italian Mathematical Union (Unione Matematica Italiana - UMI) together with the Ettore Majorana Foundation (Erice), in recognition of outstanding contributions to the field of Calculus of Variations and related applications.
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Strauch
Strauch, a German word meaning bush or shrub, is a surname.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Total derivative
In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.
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Variational Bayesian methods
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.
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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.
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Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which extremize the value of quantities that depend upon those functions.
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Vincenzo Brunacci
Vincenzo Brunacci (3 March 1768 – 16 June 1818) was an Italian mathematician born in Florence.
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Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.
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Young measure
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions.
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YouTube
YouTube is an American video-sharing website headquartered in San Bruno, California.
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Calculus of Variations, Calculus of variation, Minimum principle, Strong extrema, Variation calculus, Variational Calculus, Variational Method, Variational calculus, Variational method, Variational method (disambiguation), Variational methods.
References
[1] https://en.wikipedia.org/wiki/Calculus_of_variations
