nForum - Search Results Feed (Tag: variational-calculus) 2020-01-20T09:00:17-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher De Donder-Weyl-Hamilton equation https://nforum.ncatlab.org/discussion/5342/ 2013-10-21T11:07:46-04:00 2018-10-28T11:02:29-04:00 Urs https://nforum.ncatlab.org/account/4/ I have finally added some actual content to de Donder-Weyl-Hamilton equation. In particular I have spelled out the “non-relativistic” form (&iota; v n&ctdot;&iota; v ...

I have finally added some actual content to de Donder-Weyl-Hamilton equation.

In particular I have spelled out the “non-relativistic” form

$(\iota_{v_n} \cdots \iota_{v_1}) \omega = \mathbf{d}(H + e)$

and the “relativistic” form

$(\iota_{v_n} \cdots \iota_{v_1} )\Omega = 0$

and discussed the relation.

(I think I even sorted out all the signs correctly ;-)

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diffiety https://nforum.ncatlab.org/discussion/8191/ 2017-12-11T05:59:34-05:00 2017-12-11T05:59:34-05:00 Urs https://nforum.ncatlab.org/account/4/ I gave diffiety more of an Idea-section

I gave diffiety more of an Idea-section

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evolutionary derivative https://nforum.ncatlab.org/discussion/8181/ 2017-11-30T07:52:54-05:00 2017-11-30T07:52:54-05:00 Urs https://nforum.ncatlab.org/account/4/ created evolutionary derivative (what Olver calls the “Fréchet derivative of tuples of differential functions”) with basic definitions and properties

created evolutionary derivative (what Olver calls the “Fréchet derivative of tuples of differential functions”) with basic definitions and properties

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Functorial equations - are there equations whose solution is functor (variational calculus for categ https://nforum.ncatlab.org/discussion/6001/ 2014-06-01T14:50:39-04:00 2014-06-02T13:23:15-04:00 tomr2 https://nforum.ncatlab.org/account/1243/ Just by simple analogy:Are there functorial Euler-Lagrange equations - equations whose solution is functor (variational calculus for categories)?Are there optimization methods (topology on ... Just by simple analogy:

Are there functorial Euler-Lagrange equations - equations whose solution is functor (variational calculus for categories)?

Are there optimization methods (topology on categories) whose solution is one distinct object of category or class of objects from category?

This is question from stackoverflow, sorry for double post. If answers to these question exists, then there are several possible applications of this to the other brances of mathematics and sciences, but for the time being I will keep those prospects to myself (in such times and social systems we are living now). ]]>
variational calculus https://nforum.ncatlab.org/discussion/4398/ 2012-10-16T22:46:11-04:00 2012-10-17T00:40:17-04:00 Urs https://nforum.ncatlab.org/account/4/ at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for S&colon;[&Sigma;,X] &PartialD;&Sigma;&rightarrow;&Ropf; S \colon ...

at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for

$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}$

a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces

$\mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$

The notation can still be optimized. But I am running out of energy now. Has been a long day.

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