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
Associated With
Source Calculus of Variations (Wikipedia)
Includes
Source Wikipedia: Constantin Carathéodory
Source Herglotz's variational principle (Wikipedia)
Source Johann Bernoulli, Biography (MacTutor History of Mathematics)
Source Johann Radon (Wikipedia)
Source Siméon Denis Poisson (Wikipedia)
Source Wikipedia: Tibor Radó
Sources
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.
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
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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