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.

Rescaled Jacobi amplitude – general solution for the mathematical pendulum

Jacobi amplitude function appeared yesterday in the episode Derivatives of Jacobi elliptic am, sn, cn, dn. We have derived a beautiful simple differential equation satisfied by this beautiful function

(1)   \begin{equation*}\left(\mathrm{am}'(u,m)\right)^2=1-m\,\sin^2(\mathrm{am}(u,m)),\quad \mathrm{am}(0,m)=0.\end{equation*}

Today we will see that, after rescaling, this is a perfect fabric for making the nonlinear outfit for the mathematical pendulum.
Wikipedia has smart animations showing pendulum’s motion for different kinetic energies, for instance

Initial angle of 45°

Pendulum with enough energy for a full swing.
It has also another little animation showing the angle \theta avrying with time.

But this last picture is not well adapted for a pendulum that is making full swings around the circle. Therefore I will refer to the picture that I was already using in The case of the swinging pendulum:

This last picture has many more features depicted than I will need. I will need only the angles \theta and \theta/2 and the length of the pendulum l. The mass of the swinging point P I will denote by \mu. Usually it is denoted by m, but we will use m for the square of the modulus m=k^2 of the Jacobi amplitude function \mathrm{am}(u,m).

For solving our pendulum problem we will only need conservation of energy.

When the pendulum swings, the angle \theta(t) changes with time t. We will use the dot to denote time derivative of \theta


The linear velocity V of the pendulum is V=l\dot{\theta}, therefore the kinetic energy E_k is

    \[E_k=\frac{\mu V^2}{2}=\frac{\mu l^2\dot{\theta}^2}{2}.\]

For the potential energy we will choose the zero of the potential at the bottom, Denoting by h the height of the mass with respect to the lowest level, we have

    \[ h= l(1-\cos\theta).\]

For \theta=0 we have h=0, for \theta=\pi we have h=2l. Here it is useful to introduce the half-angle \phi=\theta/2. We know from trigonometry that

    \[ 1-\cos \theta =2 \sin^2 \theta/2.\]


(2)   \begin{equation*} h = 2l\sin^2 \phi.\end{equation*}

Potential energy E_p is \mu g h, that is

(3)   \begin{equation*}E_p=2\mu gl \sin^2\phi.\end{equation*}

Maximum kinetic energy E_{k,max} is at the bottom, for \theta=0. At this point we have also minimum potential energy, since E_p=0 at the point. Maximal potential energy E_{p,max} is at the top: E_{p,max}=2\mu g l.

The character of the motion will depend on the ratio E_{p,max}/E_{k,max}. When this ratio is >1, there will be not enough kinetic energy to rise the swinging mass to the top, and the pendulum will oscillate back and forth. But when the ratio E_{p,max}/E_{k,max}<1, then
even at the top the mass will have a nonzero speed, and the pendulum will be making full circles. We denote this important ratio by m

    \[ m=k^2\stackrel{df}{=}\frac{E_{p,max}}{E_{k,max}}.\]


(4)   \begin{equation*} E_{k,max}=\frac{2\mu gl}{m}.\end{equation*}

We now write the conservation of energy equation:

    \[ E_k+E_p=E_{k,max}.\]

On the left we have total energy at time t. On the right we have total energy at the bottom, when there is only kinetic energy. Substituting the E_k,E_p,E_{k,max} with the corresponding expressions derived above we get

(5)   \begin{equation*} \frac{1}{2}\mu l^2 \dot{\theta}^2+2\mu gl \sin^2\phi=\frac{2\mu gl}{m}.\end{equation*}

Now, \theta=2\phi, therefore \dot{\theta}^2=4\dot{\phi}^2.
We also introduce \omega defined as

(6)   \begin{equation*}\omega=\sqrt{\frac{g}{l}}.\end{equation*}

This is the expression for the standard angular frequency for a linear pendulum, fo small oscillations.

With all these substitutions and simplifications Eq. (5) can be written in the following form:

(7)   \begin{equation*}\frac{m}{\omega^2}\dot{\phi}^2=1-m\,\sin^2\phi.\end{equation*}

The last equation is almost identical with the equation (1) satisfied by the amplitude function, except for the coefficient \frac{m}{\omega^2} in front of \dot{\phi}^2 on the left. But this can be easily accomodated by changing the time scale. With m=k^2 we notice that

    \[\frac{k^2}{\omega^2}\left(\frac{d}{dt}\,\mathrm{am}(\frac{\omega t}{k},m)\right)^2=\left(\mathrm{am}'(\frac{\omega t}{k},m)\right)^2=1-m\,\sin^2(\mathrm{am}(\frac{\omega t}{k},m)).\]

Comparing with Eq. (7) we see that the solution of the pendulum equation is

(8)   \begin{equation*}\phi(t)=\mathrm{am}(\frac{\omega t}{k},m),\end{equation*}

and therefore

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


In the next post we will look closer at this solution and try to understand its meaning.

The case of the swinging pendulum

In the previous posts we have learned about Jacobi elliptic functions am, sn,cn, dn. It is time now to see them in action. But before we use them for unveiling the mysteries of the cosmic Dzhanibekov effect, we first give them a test run by using them for solving the swinging pendulum case.

Pendulum of Death
A classic Death Trap, tracing its origins to Edgar Allan Poe’s “The Pit and the Pendulum”. The hero is strapped to a table beneath a blade mounted on a pendulum. When activated, the pendulum descends slightly with each swing, gradually getting closer to the hero.

And yes, indeed, pendulum can be dangerous. Galileo thought that the period of the pendulum does not depend on the amplitude. We can read in Wikipedia:

Galileo Galilei (Italian pronunciation: [ɡaliˈlɛːo ɡaliˈlɛi]; 15 February 1564 – 8 January 1642) was an Italian polymath: astronomer, physicist, engineer, philosopher, and mathematician, he played a major role in the scientific revolution of the seventeenth century.
He has been called the “father of observational astronomy”, the “father of modern physics”, the “father of scientific method”,[7] and the “father of science”.

Galileo claimed that a simple pendulum is isochronous, i.e. that its swings always take the same amount of time, independently of the amplitude. In fact, this is only approximately true, as was discovered by Christiaan Huygens.

We do not know for sure if Galileo really believed that the pendulum is isochronous. Perhaps he was observing the swinging lamp inside the Duomo in Pisa. To some extent he could check whether the period depends on the amplitude or not, but it is rather hard to imagine him pushing the hanging lamp into extremely wide swings. Perhaps he has done some calculations. But, perhaps, he was very careful not to publish anything that would go again the philosophical doctrines he has chosen to subscribe to.

From David Hill, “Pendulums and planes: What Galileo didn’t publish“, Nuncius 9, pp.
499-515 (1994)

“I conclude with some reflections on these results. Galileo evidently knew a good deal more about circular descent than he cared to publish, and he published some things (the isochronism of the circular pendulum) which he knew to be false. Good as the Third and Fourth Days of Two New Sciences are, he could certainly have presented a far more penetrating analysis of these matters. From our perspective, the revisions and additions would have deepened and extended his treatment of the new science of motion. But from his perspective, I conjecture, these ideas causes serious problems. They do not cohere comfortably with a basic tenet of Copernican science: the perfection and uniformity of circular motion.”

The above conclusion of David Hill is speculative. And we are not studying the history of science here, we are studying mathematics as applied to physics. Therefore we need to solve the pendulum problem exactly. But, of course, we will make reasonable idealizations. Thus, for example, we will neglect the fact that the arm has some elastic properties, we will neglect air resistance, we will neglect the fact that the Earth is round and that there are earthquakes in Italy On the other hand while quite often when studying the pendulum one considers only very small amplitudes, we will allow our pendulum to have as large swings as one wishes them to have – even through the roof. Only then our elliptic functions will truly reveal their glamour. So, let’s go to the business.

We will need a picture, and I will borrow the picture as well as my inspiration from “Elliptic Functions and Applications” by Derek Lawden. Here it is:

The pendulum on the picture is hanging from the point O in the center of the circle. This is a mathematical pendulum. The arm OP is inessential. What is essential is the point mass m, here depicted as the point P. The radius of the circle is the length of the pendulum, l. Here, with this picture, we are assuming the the initial push given to the pendulum is not too large. Thus the pendulum, when swinging to the right, can reach at most the point B. The current angle of the pendulum is \theta, the maximal angle is \alpha. Later on we will deal with the case of a pendulum with enough energy to complete full circles, but for now let us restrict our attention to “tamed pendulums”.

However, even when if our pendulum is tamed, to avoid the swing of the sharp edge that right now is approaching us in the pit, I will stop now, and will continue with the math tomorrow.