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tautochrone

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tautochrone
tautochrone Math. (ˈtɔːtəkrəʊn) [f. tauto- + Gr. χρόνος time: cf. F. tautochrone (Dict. Trévoux 1771).] That curve upon which a particle moving under the action of gravity (or any given force) will reach the lowest (or some fixed) point in the same time, from whatever point it starts. So tautochroni... Oxford English Dictionary
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Tautochrone curve
The tautochrone problem The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659. For the tautochrone problem, is constant. wikipedia.org
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When is a brachistochrone not a tautochrone? I thought I understood why the cycloid is both a brachistochrone (path of shortest time) and tautochrone (path of equal time for different starting points and the same end ...
Any brachistochrone that contains a minimum is a tautochrone to its minimum (from either side, of course). On the other hand, any tautochrone that contains a cusp is a brachistochrone from that cusp to any other point on it.
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Cycloid
period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone See also Cyclogon Cycloid gear List of periodic functions Tautochrone curve References Further reading An application from physics: Ghatak, A. wikipedia.org
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What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? If possible, show some r...
Here an illustration of the Tautochrone from Wikipedia (by Claudio Rocchini): ! Tautochrone By comparison, this is the problem you are trying to solve with a Brachistochrone (Maxim Razin on Wikipedia): !Bachistochrone
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Semicubical parabola
In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the wikipedia.org
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Brachistochrone curve
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal. wikipedia.org
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limit of integral function As part of an investigation, a student of mine needed to evaluate the limit, $$\lim_{\beta\to 0+}\int_0^\beta\frac{1}{\sqrt{\cos\theta-\cos\beta}}\,d\theta.$$ Mathematica gives the answer as...
if you substitute $\theta = \beta t$, you get \begin{equation} \int_0^1 dt \frac{\beta}{\sqrt{\cos\beta t - \cos\beta}} \end{equation} and you do a Taylor expansion of the integrand in $\beta$ (essentially take the limit $\beta$ goes to $0$). You are left with $\int_0^1 dt\sqrt{2}\frac{1}{\sqrt{1-t^...
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A claim regarding Fourier Series Claim : "A periodic function $f(u)$ satisfying $$\int_{0}^{1}f(u)du=0$$ can generally expanded into a Fourier Series: $$f(u)=\sum_{m=1}^{\infty}[a_m\sin{(2 \pi m u)}+b_m\cos{(2 \pi m u...
If you read Greiner closely, $f(x)$ has period $1$ because that's how it is constructed. $a_0=0$ because $a_0=\int_{0}^{1}f(u)du=0$
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Alexis Fontaine des Bertins
He first got a taste for maths by reading the Géométrie de l'infini of Fontenelle and gave solutions to the problems of the tautochrone curve, the brachistochrone wikipedia.org
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Vincenzo Miotti
Mainly main of card and wood, few of them now survive, though one for demonstrating the tautochrone curve of a cycloid and another for parabolic motion wikipedia.org
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منحنى متساوي الزمن
منحنى متساوي الزمن (tautochrone or isochrone curve من البادئة اليونانية tauto- ، التي تعني «نفس» أو iso- ، «يساوي»، و chrono ، «الوقت») هو المنحنى حيث wikipedia.org
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1754 in science
Lagrange begins to work on the problem of tautochrone. wikipedia.org
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自学弦论，最好先掌握什么知识？
支持一下高赞答主 @Tautochrone ，牛津MMathPhys内容的确挺适合理论物理的初学，它的特点在于课多课短，每个unit是16小时的课程，一些比较基础比较重要的主课是1.5units，其余的都是1个unit，选课没有上限，比较方便学生去探索自己的兴趣，了解不同的领域。 zhihu
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List of variational topics
surface Morse theory Noether's theorem Path integral formulation Plateau's problem Prime geodesic Principle of least action Soap bubble Soap film Tautochrone wikipedia.org
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