The calculus of variations is a field of mathematical analysis that uses variations, small changes in functions and functionals, to find the maxima and minima of functionals. Its central tool is the Euler-Lagrange equation, used to identify the functions that make a given functional attain an optimal value.
Facts
Central QuestionThe calculus of variations asks, among all functions satisfying given conditions, which one maximizes or minimizes a functional, a quantity such as length, area or energy that depends on the whole shape of a function rather than on a single point. 1 Key DebateBernhard Riemann argued that a smooth minimizing function for the Dirichlet principle was guaranteed to exist because it corresponded to a physical membrane settling into a state of minimal potential energy, but Karl Weierstrass later showed this reasoning was unsound by giving a variational problem with no solution at all. 1 Classification
Pure or Applied Connections
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Source Calculus of Variations (Wikipedia)
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Source Alexis Fontaine des Bertins (Wikipedia)
Source Wikipedia: Constantin Carathéodory
Source Guido Stampacchia (MacTutor)
Source Hans Lewy (MacTutor)
Source Herglotz's variational principle (Wikipedia)
Source Jesse Douglas (Wikipedia)
Source Johann Bernoulli, Biography (MacTutor History of Mathematics)
Source Johann Radon (Wikipedia)
Source MacTutor History of Mathematics, University of St Andrews: Martin Ohm (biography)
Source Mikhail Ostrogradsky (Wikipedia)
Source Oskar Bolza (Wikipedia)
Source Paul du Bois-Reymond (Wikipedia)
Source MacTutor History of Mathematics, University of St Andrews: Robert Woodhouse (biography)
Source Siméon Denis Poisson (Wikipedia)
Source Wikipedia: Tibor Radó
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1. Calculus of Variations (Wikipedia)
Wikimedia FoundationLead section
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
History section, Dirichlet principle passage
Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet.
In Group: Optimization, Control and Operations, lead paragraph
The calculus of variations (or variational calculus) is a field of mathematical analysis
View the Source Herglotz's variational principle (Wikipedia)
Includes: Herglotz's Variational Principle, LeadQuote, Includes: Herglotz's Variational Principle, Lead
Herglotz's variational principle, named after German mathematician and physicist Gustav Herglotz, is an extension of the Hamilton's principle
Jesse Douglas (Wikipedia)
Includes: Jesse Douglas, Life and careerQuote, Includes: Jesse Douglas, Life and career
is part of the calculus of variations
Mikhail Ostrogradsky (Wikipedia)
Includes: Mikhail Ostrogradsky, WorkQuote, Includes: Mikhail Ostrogradsky, Work
He worked mainly in the mathematical fields of calculus of variations
Alexis Fontaine des Bertins (Wikipedia)
Includes: Alexis Fontaine des Bertins, Article textQuote, Includes: Alexis Fontaine des Bertins, Article text
brachistochrone curve
MacTutor History of Mathematics, University of St Andrews: Robert Woodhouse (biography)
MacTutor History of Mathematics, University of St AndrewsIncludes: Robert Woodhouse, Biography, worksQuote, Includes: Robert Woodhouse, Biography, works
A Treatise on Isoperimetrical Problems and the Calculus of Variations
MacTutor History of Mathematics, University of St Andrews: Martin Ohm (biography)
MacTutor History of Mathematics, University of St AndrewsIncludes: Martin Ohm, Biography, treatise on maxima and minimaQuote, Includes: Martin Ohm, Biography, treatise on maxima and minima
The portion of the book extending over pages 87 - 127 is called Calculus of Variations.
Paul du Bois-Reymond (Wikipedia)
Includes: Paul du Bois-Reymond, LeadQuote, Includes: Paul du Bois-Reymond, Lead
He is also associated with the fundamental lemma of calculus of variations
Oskar Bolza (Wikipedia)
Includes: Oskar Bolza, Lead, research fieldQuote, Includes: Oskar Bolza, Lead, research field
known for his research in the calculus of variations
Hans Lewy (MacTutor)
Includes: Hans Lewy, Biography, obituaryQuote, Includes: Hans Lewy, Biography, obituary
fundamental papers on partial differential equations and the calculus of variations
Guido Stampacchia (MacTutor)
Includes: Guido Stampacchia, Biography, Pisa and Naples yearsQuote, Includes: Guido Stampacchia, Biography, Pisa and Naples years
a passionate specialist in calculus of variations
Johann Radon (Wikipedia)
Includes: Johann Radon, Lead paragraph [in-branch]Quote, Includes: Johann Radon, Lead paragraph [in-branch]
calculus of variations
Siméon Denis Poisson (Wikipedia)
Includes: Siméon Denis Poisson, Lead paragraph [in-branch 4]Quote, Includes: Siméon Denis Poisson, Lead paragraph [in-branch 4]
the calculus of variations
Wikipedia: Tibor Radó
Includes: Tibor Radó, BiographyQuote, Includes: Tibor Radó, Biography
Radó published "On the Problem of Plateau" in which he gave a solution to Plateau's problem
Wikipedia: Constantin Carathéodory
Includes: Constantin Carathéodory, LeadQuote, Includes: Constantin Carathéodory, Lead
He made significant contributions to real and complex analysis, the calculus of variations, and measure theory.
Johann Bernoulli, Biography (MacTutor History of Mathematics)
Includes: Johann Bernoulli, SummaryQuote, Includes: Johann Bernoulli, Summary
Johann Bernoulli was a Swiss mathematician who studied reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by series and the brachistochrone.
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