basic_tools:variational_calculus

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basic_tools:variational_calculus [2018/05/05 15:41] jakobadmin |
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====== Variational Calculus ====== | ====== Variational Calculus ====== | ||

+ | <tabbox Why it is interesting?> | ||

Variational calculus is the alternative to the usual calculus methods when we want to find functions that minimize something. As an analogy, usually when we search for the extrema of a function, we differentiate the function, set the derivative of the function to zero and find the point that yields the extrema. Similar results can be computed by using variational calculus.In variational calculus we find extrema of functionals which are functions of functions with respect some function (instead of variable). This is extremely important for the [[formalisms:lagrangian_formalism|Lagrangian formalism]]. | Variational calculus is the alternative to the usual calculus methods when we want to find functions that minimize something. As an analogy, usually when we search for the extrema of a function, we differentiate the function, set the derivative of the function to zero and find the point that yields the extrema. Similar results can be computed by using variational calculus.In variational calculus we find extrema of functionals which are functions of functions with respect some function (instead of variable). This is extremely important for the [[formalisms:lagrangian_formalism|Lagrangian formalism]]. | ||

+ | <tabbox Concrete> | ||

- | | + | **What we know from Calculus** |

- | === What we know from Calculus === | + | |

On [[basic_tools:calculus|calculus]], when we want to find the extremum of a function, we use the derivative: | On [[basic_tools:calculus|calculus]], when we want to find the extremum of a function, we use the derivative: | ||

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- | === Concept of Variational Calculus === | + | **Concept of Variational Calculus** |

Instead of a function $f$ that takes numbers to numbers, the object of interest is a [[basic_tools:variational_calculus:functional|functional]], a function of functions of sorts. To find the stationary functions of the functionals, we need to change a bit the differentiation process, and we use the //[[basic_tools:variational_calculus:functional_derivatives|functional derivative]]// or variational derivative, and we equate the result to zero. | Instead of a function $f$ that takes numbers to numbers, the object of interest is a [[basic_tools:variational_calculus:functional|functional]], a function of functions of sorts. To find the stationary functions of the functionals, we need to change a bit the differentiation process, and we use the //[[basic_tools:variational_calculus:functional_derivatives|functional derivative]]// or variational derivative, and we equate the result to zero. | ||

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- | === References === | + | **References** |

* //Calculus of Variations//, Gelfand and Fomin | * //Calculus of Variations//, Gelfand and Fomin | ||

* Calculus of Variations by MacCluer. | * Calculus of Variations by MacCluer. | ||

+ | <tabbox Abstract> | ||

+ | |||

+ | See [[https://www.ams.org/journals/notices/201903/rnoti-p303.pdf|Karen Uhlenbeck and the Calculus of Variations]] by Simon Donaldson | ||

<tabbox Quotes> | <tabbox Quotes> |

basic_tools/variational_calculus.txt · Last modified: 2019/03/04 09:39 by 129.13.36.189

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