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Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [ l ] is defined as the linear part of the change in the functional, and the second variation [ m ] is defined as the quadratic part.
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
Category: Calculus of variations. 34 languages. ... Download as PDF; Printable version; In other projects Wikimedia Commons; Wikidata item; Appearance. move to ...
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations.
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
This is a list of variational topics in from mathematics and physics.See calculus of variations for a general introduction.. Action (physics) Averaged Lagrangian; Brachistochrone curve
In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf , the coefficient of ...