How does the calculus of variations differ from ordinary calculus?

Ordinary calculus finds points where a function is largest or smallest, while the calculus of variations finds whole functions, such as curves or surfaces, that extremize an integral. The unknown is a function rather than a number, and the condition for an extremum is a differential equation.

What is the principle of least action?

It is the physical statement that the motion of a system makes a quantity called the action stationary. Applying the calculus of variations to the action yields the equations of motion, so much of classical and quantum physics can be derived from a single variational principle.

Calculus of Variations

The calculus of variations seeks functions that extremize integral functionals, generalizing ordinary maximization and minimization from points to curves and fields.

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Definition

The calculus of variations studies functionals, which assign numbers to functions, and seeks the functions at which a functional is stationary or attains an extreme value, subject to boundary and side conditions.

Scope

This area covers the derivation of the Euler-Lagrange equations as necessary conditions for an extremal, variational problems with constraints and free boundaries, second-variation and convexity conditions for minima, the direct method establishing existence of minimizers, and the connection to Hamiltonian mechanics and optimal control.

Sub-topics

Core questions

Which functions make a given integral functional stationary?
What necessary and sufficient conditions identify a minimizer?
When does a minimizer actually exist?
How do variational principles encode the laws of physics?

Key theories

Euler-Lagrange equations: A function that extremizes an integral functional must satisfy the Euler-Lagrange differential equation, the variational analogue of setting a derivative to zero.
Direct method: Existence of a minimizer is established by taking a minimizing sequence and using compactness and lower semicontinuity, bypassing explicit solution of the Euler-Lagrange equation.
Variational principles in physics: Hamilton's principle of stationary action recasts mechanics and field theory as variational problems, unifying their governing equations through the calculus of variations.

Clinical relevance

Variational methods express fundamental laws across physics through least-action and minimal-energy principles, and they underpin optimal control, geometry of minimal surfaces and geodesics, image processing, and the finite-element method in engineering.

History

The subject began with the brachistochrone problem posed by Johann Bernoulli in 1696. Euler and Lagrange developed the general theory and the Euler-Lagrange equation in the eighteenth century, Hamilton recast mechanics variationally, and Hilbert's twentieth-century direct method and his twenty-third problem revitalized existence theory.

Key figures

Leonhard Euler
Joseph-Louis Lagrange
William Rowan Hamilton
David Hilbert

Seminal works

gelfand1963
courant1953
dacorogna2008

Frequently asked questions

How does the calculus of variations differ from ordinary calculus?: Ordinary calculus finds points where a function is largest or smallest, while the calculus of variations finds whole functions, such as curves or surfaces, that extremize an integral. The unknown is a function rather than a number, and the condition for an extremum is a differential equation.
What is the principle of least action?: It is the physical statement that the motion of a system makes a quantity called the action stationary. Applying the calculus of variations to the action yields the equations of motion, so much of classical and quantum physics can be derived from a single variational principle.