Cosmoplanetary pendulum

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:

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


which is the same as


We set \omega=1 and solve for \dot{\theta}

(1)   \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?

Nonlinear pendulum period and Kozyrev’s mirrors

Yesterday, while thinking about Dzhanibekov effect, gyroscopes and elliptic functions, I checked my mailbox and read the following email from one of the readers:

Dear Ark,

Can you write about Kozyriev mirrors in your future posts? Rossiya 1 did a documentary awhile ago still available here:
Kozyrev Mirrors_Breakthrough into the Future – (english subtitles)

What are your thoughts of what might be going on in these experiments? Do the Kozyriev mirror experiments have to do with time traveling and accessing the information field? Why a “concave” mirror or cylinder? What is special about this shape and the materials used? Would it be detrimental for people to experiment with these mirrors? Any scientific or informal thoughts on this subject will be most welcomed.

The truth is: in my imagination gyroscopes and Kozyrev’s mirrors are completely different subjects. But then I asked myself: or aren’t they?

After small search I have downloaded from the Internet THE SCIENCE OF TORSION, GYROSCOPES AND PROPULSION. It starts with


My critical article on Shipov’s “4D gyroscopes” is mentioned there, but the works and ideas of Kozyrev are also mentioned.
Wikipedia article on Kozyrev with his mirrors quotes “Akimov, A.E., Shipov, G. I., Torsion fields and their experimental manifestations, 1996” – the subject closely related to the “unconventional physics” of spinning objects. So, perhaps at some deeper level the two subjects are closely related? With this in mind I will have to read what is available about research done with Kozyrev’s mirrors. At present I know next to nothing, and what I once knew I have mostly forgotten. But I will keep it in mind, study, and in the future return the strange properties of space, vacuum, structured aether and Kozyrev mirrors. For now, however, I need to finish what I have started – nonlinear mathematical pendulum. Today we will discuss the expressions for its period.

Pendulum period: m<1

In the previous post Rescaled Jacobi amplitude – general solution for the mathematical pendulum, we have derived a general formula for time evolution of a nonlinear mathematical pendulum

(1)   \begin{equation*}\theta(t)=2\mathrm{am}(\frac{\omega t}{k},m),\end{equation*}

where m=k^2 is the ratio E_{p,max}/E_{k,max} of maximal potential energy to maximal kinetic energy, and \omega=\sqrt{g/l}, where g is the gravitational acceleration, and l is the length of the pendulum.
The inequality m<1 means that the pendulum has sufficient kinetic energy to swing full circles. Lets us recall the graph of the amplitude function \mathrm{am}(u,m)

For m<1 the value of \mathrm{am}(u,m) grows from left to right. That is clear: the angle \theta constantly increases. But when it riches 2\pi the pendulum, in fact makes a full circle. Therefore the period T of our pendulum is calculated from the formula

(2)   \begin{equation*}2\pi=2\mathrm{am}(\frac{\omega T} {k},m).\end{equation*}

We recall from Jacobi amplitude- realism or cubism that \mathrm{am} is the inverse function of F, the incomplete elliptic integral of the first kind given by (see also Wikipedia: Elliptic integral)

(3)   \begin{equation*}F(\phi,m)=\int_0^\phi\frac{d\theta}{\sqrt{1-m\,\sin^2\theta}}.\end{equation*}

Therefore Eq. (2) is equivalent to

    \[ \frac{\omega T} {k}= F(\pi,m).\]

The function under integral in Eq. (3) has the symmetry property that tells us that F(\pi,m)=2F(\pi/2,m). The value F(\pi/2,m) is usually given the name: the complete elliptic integral of the first kind, and it is often denoted with the capital letter K=K(m). Thus we obtain:

(4)   \begin{equation*}T=\frac{2Kk}{\omega},\quad  0<m<1.\end{equation*}

The case of m=0 is uninteresting, as it means either the pendulum of zero potential, or of infinite kinetic energy. In the case of m=1, we have, in fact, two possible solutions. One is with \theta constant, \theta=\pi. That is very unstable, like a pencil that stands on its tip. There is also second solution, one given by our formula with m=1. The motion is non-periodic, there is just one flip all around the circle, and it takes infinite time.

Pendulum period: m>1

For m>1 we have to return to the definition of the amplitude function – Eqs (1),(2) in Jacobi elliptic cn and dn:

(5)   \begin{equation*} \mathrm{am}(u,m)=\arcsin \matherm{sn}(u,m)=\arcsin\left( \frac{1}{k}\mathrm{sn}(ku,1/m)\right).\end{equation*}

We can see from the graph above that for m>1 the function \matrm{am} oscillates periodically. Since, taking into account simplification of multiplying and dividing by k, we get

    \[\mathrm{am}(\frac{\omega t} {k},m)=\arcsin\left(\frac{1}{k}\mathrm{sn}(\omega t,1/m)\right),\]

it follows that the period T of \theta is the same as the period of the function t\mapsto \mathrm{sn}(\omega t,1/m), and it is the same as the period of the function \mathrm{sn}(\omega t,1/m).
It is therefore given by the formula

    \[\omega T=F(2\pi,1/m),\]


(6)   \begin{equation*} T=\frac{4K(1/m)}{\omega},\quad m>1. \end{equation*}

For very small oscillation (very small kinetic energies) m is very large and 1/m is close to zero. The integrand in the definition of F(\phi,1/m) can be replaced by the constant 1, so that, for very large m, F(\phi,1/m)\approx\phi. Therefore K can be replaced by \pi/2 and Eq. (6) reduces to

(7)   \begin{equation*} T=\frac{2\pi}{\omega},\quad m>>1. \end{equation*}

This is the standard formula for the linear pendulum with small oscillation. It was known to Galileo.

International Journal of Unconventional Science

International Journal of Unconventional Science is being translated form Russian to English. Here is what is already available:

  • / International Journal of Unconventional Science | Журнал Формирующихся Направлений Науки \ Issue #E1
  • The last issue, for the time available only in Russian: №12-13 has, in particular, an article by S. Kernbach and O. Kernbach “On Symbols and Memes”. It discusses, in particular, whether esoteric symbols may indeed be related to some “unconventional physics”.

    Kernbach and Zhigalov publish there an article about “new inquisition”

    С. Кернбах, В. Жигалов, А. Смирнов. ‘Молот ведьм’ reloaded: новая инквизиция в борьбе с инакомыслием.

    Vlad Zhigalovs reviews the recent conference on “torsion fields and information theoretic interactions”

    В.А. Жигалов. О конференции «Торсионные поля и информационные взаимодействия — 2016».

    Lot of goodies. I hope all of these will be available in English pretty soon.