Calculus of variations Calculus of variations In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. The path of an object is the one that yields a stationary value for a quantity called the action. ...more on Wikipedia about "Action (physics)"
Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such functionals can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions: those making the functional attain a maximum or minimum value. Some classical problems on curves were posed in this form: one example is the brachistochrone, the path along which a particle would descend under gravity in the shortest time from a given point A to a point B not directly beneath it. Amongst the curves from A to B one has to minimise the expression representing the time of descent. ...more on Wikipedia about "Calculus of variations"
In mathematics, Dirichlet's principle in potential theory states that the harmonic function on a domain with boundary condition ...more on Wikipedia about "Dirichlet principle"
The envelope theorem is a basic theorem used to solve maximization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity. The statement of the theorem is: ...more on Wikipedia about "Envelope theorem"
The Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. It provides a way to solve for functions which extremize a given cost functional. It is widely used to solve optimization problems, and in conjunction with the action principle to calculate trajectories. It is analogous to the result from Calculus that a function attains its extreme values when its derivative vanishes. ...more on Wikipedia about "Euler-Lagrange equation"
Fermat's principle in optics states: ...more on Wikipedia about "Fermat's principle"
The fundamental lemma of the calculus of variations states that if f is a function in C [a,b], and ...more on Wikipedia about "Fundamental lemma of calculus of variations"
Isoperimetry literally means "having an equal perimeter". In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. ...more on Wikipedia about "Isoperimetry"
This is a list of variational topics, from mathematics and physics, by Wikipedia page. See calculus of variations for a general introduction ...more on Wikipedia about "List of variational topics"
Noether's theorem is a central result in theoretical physics that expresses the one-to-one correspondence between symmetries and conservation laws. This exact equivalence holds for all physical laws based upon the action principle defined over a symplectic space. It is named after the early 20th century mathematician Emmy Noether. ...more on Wikipedia about "Noether's theorem"
Plateau's problem is to show the existence of a minimal surface with a given boundary. It is named after Joseph Plateau, who was interested in soap films, but was raised by Joseph-Louis Lagrange in 1760. The problem is considered part of the calculus of variations. ...more on Wikipedia about "Plateau's problem"
The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that "Nature is thrifty in all its actions". See action (physics). Others who developed the idea included Euler and Leibniz. It should be said that, from the point of view of the calculus of variations, a principle of stationary action is a more accurate formulation. ...more on Wikipedia about "Principle of least action"
(Variational principle) For a hamiltonian H that describes the studied system and any normalizable function Ψ with arguments appropriate for the unknown wave function of the system, we define the functional ...more on Wikipedia about "Variational principle"
In the calculus of variations, Γ-convergence is a notion of convergence for functionals. It was introduced by Ennio de Giorgi. ...more on Wikipedia about "Γ-convergence"
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