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Calculus of Variations

Computation, Optimization and Control

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 Question
The 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 Debate
Bernhard 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
Applied Mathematics 1
Connections

Associated With

Analysis, Branches of Mathematics
Source Calculus of Variations (Wikipedia)

Includes

Johann Radon, Mathematicians
Source Johann Radon (Wikipedia)
Simeon Denis Poisson, Mathematicians
Source Siméon Denis Poisson (Wikipedia)
Sources
1. Calculus of Variations (Wikipedia)
Wikimedia Foundation
  • Lead 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.
  • Associated With: Analysis, Wikipedia lead paragraph, Calculus of variations
    The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals.
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Johann Radon (Wikipedia)
Includes: Johann Radon, Lead paragraph [in-branch]
Quote, Includes: Johann Radon, Lead paragraph [in-branch]
calculus of variations
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Siméon Denis Poisson (Wikipedia)
Includes: Simeon Denis Poisson, Lead paragraph [in-branch 4]
Quote, Includes: Simeon Denis Poisson, Lead paragraph [in-branch 4]
the calculus of variations
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