Period of a pendulum

[latexpage]
Commenting my statement that “we can now return to the imaginary time period of the pendulum” in my last post Elliptic addition theorem, Bjab aked “What about the real time period?”

And indeed, we did not quite finish the case of the real period. Let me recall what it is about in general terms.

He was seventeen and bored listening to the Mass being celebrated in the cathedral of Pisa. Looking for some object to arrest his attention, the young medical student began to focus on a chandelier high above his head, hanging from a long, thin chain, swinging gently to and fro in the spring breeze. How long does it take for the oscillations to repeat themselves, he wondered, timing them with his pulse. To his astonishment, he found that the lamp took as many pulse beats to complete a swing when hardly moving at all as when the wind made it sway more widely. The name of the perceptive young man, destined to make other momentous scientific discoveries, was Galileo Galilei.

Though Galileo discovered the isochronism of the pendulum as a fact of nature, he did not offer an underlying reason for his seminal observation. That explanation had to wait for the great work of Isaac Newton.

(From “Galileo’s Pendulum: From the Rhythm of Time to the Making of Matter“, by Roger. G. Newton

Later it was discovered that the isochronism of the pendulum is not the fact of nature. Nevertheless, quoting from “Thus Spoke Galileo: The great scientist’s ideas and their relevance to the present day” by Andrea Frova and Mariapiera Marenzana:

It is interesting to note that Galileo’s mistake in treating this isochronism as perfect led to some important scientific and technological advances. Without this error, for example, perhaps no one would ever have thought of using the pendulum as a device for measuring time.


After this introduction let us do some simple math. In fact, we have already started to discuss the problem in “Nonlinear pendulum period and Kozyrev’s mirrors“. We have noticed that for the mathematical pendulum with the swinging amplitude $\alpha$ smaller than $\pi,$ we have the formula:
\[T(m)=\frac{4K(1/m)}{\omega},\]
where $m=k^2$ and
\[K(m)=F(\pi/2,m)=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-m\sin^2\theta}}.\]
The “classical period” from the schoolbooks, one that is a good approximation for small oscillations is
\[ T_0=\frac{2\pi}{\omega},\]
where
\[\omega =\sqrt{\frac{g}{l}.\]
In the discussion under that post we have also found the relation between the value of $k$ and the amplitude of the oscillations $\alpha$
\[\frac{1}{k}=\sin\frac{\alpha}{2}.\]
Therefore we can write down the formula telling us how the period of the pendulum depends on the amplitude:
\begin{equation}T(\alpha)=T_0\, C(\alpha),\end{equation}
where
\begin{equation}C(\alpha)=\frac{2K(\sin^2\frac{\alpha}{2})}{\pi}.\end{equation}
Here is the plot of the function $C(\alpha)$

We see that it is almost constant and equal to 1 for $\alpha<\pi/2$. At $\alpha=\pi/2$ we have, in fact: \[C(\pi/2)=1.18034,\] that is about 18% larger than $T_0$. Then the deviation from the constancy of the period becomes larger and larger, and dramatically larger near $\alpha=\pi.$ But, of course, experiments with bobs hanging on strings were impossible to conduct for $\alpha>\pi/2.$

What about experiments with $\alpha<\pi/2$? How their results fit the theory? What should we tell to students asking such questions? A review can be found in the paper "An accurate formula for the period of a simple pendulum oscillating beyond the small-angle regime” by F. M. S. Lima and P. Arun. They propose a simple approximation to the function $C(\alpha),$ and approximation based on the logarithm function:

\[C(\alpha)\approx -\frac{\log \cos \frac{\alpha}{2}}{1-\cos\frac{\alpha}{2}}\]
For comparison I plotted both, where the approximation is in red:

Here is the comparison with experiments, taken from the paper:

Now we are ready to move into imaginary time. In the next post.

Pendulum in imaginary time

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Today’s post needs mathematical imagination and practice. Practice is important. For instance lot of practice is needed for a successful target slinging. Like in this video below

There are many stiles of slinging. For us probably the best is is Apache style, when the stone (representing our mathematical pendulum) is making vertical circles. We will practice the math behind such slings right below. But after we are done with that, we will check what happens with the sling when time is imaginary rather than than real. The problem of the pendulum with imaginary time has been posed by John Baez as an exercise in his notes on elliptic functions. It seems that the solution of this exercise has not been published yet.

But first let us considered the stone (or a bucket of water, if you wish) making vertical circles on a string. It is clear that there are certain cases (velocities) that are “forbidden”, for instance if you do not give the bucket on the string sufficient initial velocity, the water will fall on your head from the bucket.

The same with certain amusement park rides – not all speed are safe:

We have started looking into this problem yesterday in Cosmoplanetary pendulum, and Bjab, in his comments, proposed an answer, but without providing details. I promised to look into it and to post my own solution (in fact yesterday I have made an error in my calculations). Here is what I came with today. In fact, just a while ago, I have found this problem discussed, though not from the angle we need, in CBSE Class XI Supplementary Textual Material in POhysics, Unit IV, Motion in a Vertical Circle. I am borrowing a picture from this site:

It fits perfectly our needs, except that we use letter $\mu$ for the mass. Letter $m$ we use for the second argument of Jacobi elliptic functions discussed in previous posts. For a pendulum on a string to work the centrifugal force $\mu\dot{\theta}^2 l$ experienced by the rotating mass must be greater or equal to the radial component of the gravitational force:
\[\mu\dot{\theta}^2l\geq -\mu g \cos \theta.\]
The mass $\mu$ cancels out, and $g/l = \omega^2,$ thus
\begin{equation}\dot{\theta}^2\geq -\omega^2 \cos \theta.\end{equation}
For $\dot{\theta}$ we use Eq. (1) from Cosmoplanetary pendulum, but without fixing the value of $\omega$:
\begin{equation}\dot{\theta}^2=4\omega^2(\frac{1}{m}-\sin^2\frac{\theta}{2}).\end{equation}
Using the last two equations we arrive at the inequality
\[4(\frac{1}{m}-\sin^2\frac{\theta}{2})\geq-\cos \theta.\]
Now \[\cos \theta=1-2\sin^2\frac{\theta}{2},\]
therefore
\[4(\frac{1}{m}-\sin^2\frac{\theta}{2})\geq 2\sin^2\frac{\theta}{2}-1\]
or
\begin{equation}6 \sin^2\frac{\theta}{2}\leq \frac{4}{m}+1.\end{equation}
Suppose first that $m<1.$ Then $\theta$ varies from $0$ to $2\pi$ and the left hand side has the largest value for $\sin^2\frac{\theta}{2}=1.$ Therefore \[\frac{4}{m}+1\geq 6,\] or \begin{equation}m\leq\frac{4}{5}.\end{equation} We are done with $m<1.$ When $m>1,$ we already know from the discussion in previous posts that for the highest point of the pendulum we have $\sin^2\frac{\theta}{2}=\frac{1}{m}.$
Therefore
\[ \frac{6}{m}\leq \frac{4}{m}+1,\]
or
\begin{equation}m\geq 2.\end{equation}

It follows that the forbidden region is for $m$ between $4/5$ and $2$ – as correctly predicted by Bjab in his comment yesterday.

We now come to the second issue – that of imaginary time. Here is the exercise form notes by John Baez: The Pendulum, Elliptic Functions and Imaginary Time. There we find the following “exercise”:

11. Show that making the replacement
\[t\mapsto it\]
in Newton’s law is equivalent to reversing the sign of all forces.

In the present problem, this amounts to reversing the force of gravity, making it pull the pendulum up. But an upside-down pendulum is just another pendulum. Therefore the function $\mathrm{sn}(it, k)$ must also be periodic as a function of $t$. This suggests that $\mathrm{sn}(z, k)$, as a function of $z\in\mathbb{C}$, is periodic in both the real and imaginary directions. And it’s true!

So, the pendulum gives a physical explanation of the fact that elliptic functions are periodic in two directions on the complex plane!

12. Prove, as rigorously as you can, that $\mathtm{sn}(z, k)$ is periodic in two directions. You can do this either by fleshing out the above argument, or by studying the integral in equation (3) and worrying about those branch points. In fact we have
\[\mathrm{sn}(z + 4K, k) = \mathrm{sn}(z, k),\quad \mathrm{sn}(z + 2iK’,k) = \mathrm{sn}(z, k)\]
where for$ 0 < k < 1$ \[K=\int_0^1\frac{dy}{\sqrt{(1-y^2)(1-k^2y^2}}\] and \[K'=\int_0^{1/k}\frac{dy}{\sqrt{(y^2-1)(1-k^2y^2}}.\]

It seems that the first nine exercises from these notes (for MATH 241 course) has been solved by Toby Barlets and posted. But not 11 and 12. Can we make sense of Baez’idea of “upside-down pendulum is just another pendulum” and use it to derive the formula with 2K’ imaginary period?

Who can do it?

Cosmoplanetary pendulum

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First about Kozyrev’s mirrors. I have started to study the subject. That is a real adventure in a real jungle! First of all it seems they are not “Kozyrev’s”. They have been invented and investigated by V. P. Kaznacheev, but Kaznacheev was very much impressed by Kozyrev’s work, so he decided to call the mirrors with Kozyrev’s name. The work, as I understand, have started in Novosibirsk under the guidance of Kaznacheev for many years. It is now being continued at the “International Scientific Research Institute of Cosmoplanetary Anrthropoecology“. The web site of the institute is all in Russian, but the publication page of the “institute” has some English text. In particular you can find there these links:

http://www.altaibooks.com/trofimov2016.pdf
http://www.altaibooks.com/trofimov2012.pdf

These are interviews in English with A. Trofimov, who was a collaborator of Kaznacheev, and now is the head of the Institute. Here are some excerpts form the 2016 interview, so that you can form an idea about what kind of research is being conducted in Novosibirsk now.

We humans tend to think the way to wisdom is by looking outwards, but it’s not. We already have wisdom. We need only to look inward. Dr. Alexander Trofimov, Director of the Institute for Scientific Research in Cosmoplanetary Anthropoecology (ISRICA) in Novosibirsk, Russia, has spent a lifetime honing his skills at looking inward.
….
Now, he tells me, he and his team of inner-space cosmonauts have another breakthrough—they have begun actual communication with extraterrestrials.
….
At ISRICA, at our New International Center of Kozyrev’s Space Investigations we are pursuing correct scientific accompaniment of these contacts. We are open for cooperation with anyone with clean aims.
…..
I want say some words to observers who work with other Mirror devices, but not genuine Kozyrev’s Mirrors: Don’t attempt to exploit our experiment for money and business! This is dangerous for future of our civilization!
…..
We have now photo confirmation of the effectiveness of our technology: the Mirror’s quantum superposition of human consciousness. We ‘ll be glad to look at photos and other results of communications from our colleagues.

There are different research projects within the institute, so that the “Kozyrev’s mirrors” project may well be a smoke screenI It is not clear where the Institute is getting money from. Some people working on alternative mirrors speculate that, perhaps, from China (Chineese seems to be, in particular, interested in “magical properties of water”). Years ago money was coming from the government and other secret sources. Nowadays it seems to be a total mess. But I do not have a reliable information.

The page with “News” from the Institute has a beautiful picture of NGC 6357 galaxy that is called “War and Peace”.

The Institute is said to be working on “Starry world without war on Earth”.
Perhaps I should add that the official address of the institute points to just one office in a big building in Novosibirsk that is renting offices:

аренда помещения в Новосибирске, в …Новосибирск, Академика Лаврентьева проспект, д. 6/1, 3000 м2.

As far as I know there is no strong evidence that the (para)psychological/medical phenomena that are observed with the help of aluminium and other mirrors have anything to do with mirrors at all. There are claims that similar effects can be obtained by just imagining being within a mirror – much like placebo effects. The effects seem to depend also on “the operator”. There seems to be a war between the group in Novosibirsk and competitors who also claim to be able to see the future in their version of Kozyrev’s mirrors via “energo-informatic interactions”:

So much about Kozyrev’s mirrors today. Now back to mathematics and physics of the swinging pendulum. According to cosmoplanetary philosophy it may be not the right thing to do, because we should be concentrating our efforts on saving our endangered eco and info space so much polluted by us, human beings, who instead of concentrating on how to “be” are concentrating on how to “have”. But I do not want to have the pendulum, I simply want to know how it works. And after finding it out – I may want to ask further questions like “what is inertia?”, “what is space?” and “what is time?”. But first things first.

It would be a shame to leave our pendulum and move back to gyroscopes without showing first the phase diagram. It is featured on Wikipedi’a page on Pendulum:

Here is another illustration from the book “The chaotic pendulum” by Moshe Gitterman

It would be nice to know how these people got these pictures and what they mean. And we are quite ready for it, it is in fact very very easy. Easier than solving differential equations. In fact these are, in a sense, pictures of the differential equations.

In Rescaled Jacobi amplitude – general solution for the mathematical pendulum we have derived differential equation (Eq. (7)) for $\phi=\theta/2$
\[\frac{m}{\omega^2}\dot{\phi}^2=1-m\,\sin^2\phi,\]
which is the same as
\[\frac{m}{\omega^2}\dot{\theta}^2=4(1-m\,\sin^2\frac{\theta}{2}).\]
We set $\omega=1$ and solve for $\dot{\theta}$
\begin{equation}\dot{\theta}=\pm 2\sqrt{\frac{1}{m}-\sin^2\frac{\theta}{2}}.\end{equation}
We have the angular velocity $\dot{\theta}$ as a function of $\theta,$ and we can plot these functions for different values of $m.$ Here is what I got using Mathematica:

Here $m$ and $1/m$ are changing from 1 to 10. Nice picture, I think.

One thought occurs to me: our mathematical pendulum has an arm. It is supposed to be of constant length. Suppose we want to verify our predictions for the period in real experiments. If we make the arm out of steel, it will be hard to make it weightless. But if we make it out of string, then, I think, there will be some minimal angular velocity needed for the string not to collapse under gravity?