Ads
related to: calculus of variations example
Search results
Results From The WOW.Com Content Network
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
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 ...
In the calculus of variations and classical mechanics, the Euler–Lagrange equations [1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
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.
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
1 Example. 2 See also. ... In applied mathematics and the calculus of variations, the first variation of a functional J(y) ...
This was a major unsolved problem in the Calculus of Variations, until Šverák gave an counterexample in 1993 for the case and . [11] The case d = 2 {\displaystyle d=2} or m = 2 {\displaystyle m=2} is still an open problem, known as Morrey's conjecture.