**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Calculus of variations

Summary

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. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (216)

Related people (35)

Related units (6)

Related concepts (31)

Related lectures (183)

Related courses (31)

Brachistochrone curve

In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the 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. The problem was posed by Johann Bernoulli in 1696. The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies.

Lagrangian mechanics

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space and a smooth function within that space called a Lagrangian. For many systems, where and are the kinetic and potential energy of the system, respectively.

Fermat's principle

Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig. 1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature.

Calculus of Variations: Introduction

Introduces calculus of variations with examples like the Brachistochrone problem in an applied context.

Calculus of Variations

Covers topics in Calculus of Variations, including regularity results and implicit function theorems.

Derivatives and Continuity in Mathematics

Covers derivatives, continuity, Rolle's Theorem, function examples, and extrema.

Jiri Vanicek, Roya Moghaddasi Fereidani

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller’s original thawed Gaussian approximation, it is symplectic, conse ...

2023Using a variational method, we prove the existence of heteroclinic solutions for a 6-dimensional system of ordinary differential equations. We derive this system from the classical Benard-Rayleigh problem near the convective instability threshold. The cons ...

Molecular quantum dynamics simulations are essential for understanding many fundamental phenomena in physics and chemistry. They often require solving the time-dependent Schrödinger equation for molecular nuclei, which is challenging even for medium-sized ...

MATH-101(de): Analysis I (German)

Es werden die Grundlagen der Analysis sowie der Differential- und Integralrechnung von Funktionen einer reellen Veränderlichen erarbeitet.

MATH-515: Topics in calculus of variations

Introduction to classical Calculus of Variations and a selection of modern techniques.

MATH-437: Calculus of variations

Introduction to classical Calculus of Variations and a selection of modern techniques. We focus on inegral functionals defined on Sobolev spaces.