Category: Dzhanibekov effect

Meeting with remarkable circles

Posted on by arkajad

According to the web site “How stuff works”

Our universe is devoid of perfect circles.
…
Only in the abstract world of pure mathematics can we find our perfect circle — a world of points and infinitely-thin lines with no room for particle inconsistencies or spherical oblateness.
…
In our world there is no true beauty, but we have an innate understanding and longing for the true form of beauty as it exists beyond the limits of our reality. There’s no true justice here, but we have a sense of it because the unreachable ideal exists in the realm of forms.

And this is what we are after: the true beauty. And nothing is going to stop us. Here it is. We stumbled upon it in our analysis of the internal mathematical structure of the Dzhanibekov effect. It is one of the simplest and at the same time one of the most amazing effects hidden in the laws of physics that were in principle known for centuries, but there is a difference between knowing and understanding.

In “Meetings with Remarkable Men” G. I. Gurdjieff wrote:

YES, PROFESSOR, KNOWLEDGE AND UNDERSTANDING ARE QUITE DIFFERENT. Only understanding can lead to being, whereas knowledge is but a passing presence in it. New knowledge displaces the old and the result is, as it were, a pouring from the empty into the void.
One must strive to understand; this alone can lead to our Lord God.
And in order to be able to understand the phenomena of nature, according and not according to law, proceeding around us, one must first of all consciously perceive and assimilate a mass of information concerning objective truth and the real events which took place on earth in the past; and secondly, one must bear in oneself all the results of all kinds of voluntary and involuntary experiencing.

Yes, we want to know, because knowledge protects (while ignorance endangers), but we also want to understand. For that math is needed. Here are, one more time, our remarkable circles, now rotating in front of you as in the Spring Fashion Show.

Click on image to start the rotation animation

These are the two remarkable limit circles connected by the bridge.

And how do we get these limit circles?

Simple we go to the limits.

Recall the formulas from The mysterious paths on the three-sphere. Remember, we set I_1=1,\,I_2=2,\,I_3=3.

(1)   \begin{eqnarray*} A_1&=&\sqrt{\frac{I_1(I_3-I_2)}{I_2(I_3-I_1)}}=1/2,\\ A_2&=&1,\\ A_3&=&\sqrt{\frac{I_3(I_2-I_1)}{I_2(I_3-I_1)}}=\frac{\sqrt{3}}{2},\\ B&=&\frac{1}{I_2}\,\sqrt{\frac{(I_2-I_1)(I_3-I_2)}{I_1I_3}}=\frac{1}{2\sqrt{3}},\\ \delta&=&\frac{\sqrt{I_2(I_3-I_1)}-\sqrt{I_1(I_3-I_2)}}{\sqrt{I_3(I_2-I_1)}}=\frac{1}{\sqrt{3}}. \end{eqnarray*}

Define:

(2)   \begin{eqnarray*} l_1(t)&=& A_1\,\mathrm{sech }(B t),\\ l_2(t)&=&A_2\,\tanh( B t),\\ l_3(t)&=&A_3\,\mathrm{sech }(B t), \end{eqnarray*}

where \mathrm{sech }(x)=1/\cosh(x).
Define

(3)   \begin{equation*} \psi(t)=\frac{t}{I_2}+2\arctan\left(\delta \tanh(B t/2)\right). \end{equation*}

Define:

(4)   \begin{eqnarray*} q_0(t)&=&\frac{\sqrt{1+l_1(t)}\cos\frac{\psi(t)}{2}}{\sqrt{2}},\\ q_1(t)&=&\frac{\sqrt{1+l_1(t)}\sin\frac{\psi(t)}{2}}{\sqrt{2}},\\ q_2(t)&=&\frac{l_3(t)\cos\frac{\psi(t)}{2}+l_2(t)\sin\frac{\psi(t)}{2}}{\sqrt{2(1+l_1(t))}},\\ q_3(t)&=&\frac{-l_2(t)\cos\frac{\psi(t)}{2}+l_3(t)\sin\frac{\psi(t)}{2}}{\sqrt{2(1+l_1(t))}}. \end{eqnarray*}

We want to know what happened in the distant path, and what will happen in the distant future. In other words we are interested in the asymptotic behavior at t\rightarrow +\infty and t\rightarrow -\infty.

Let us start with l_1(t).

We see that in the past and in the future l_1(t) effectively vanishes. The same for l_3(t). For l_2(t) we have:

Therefore in the past, asymptotically, \mathbf{L}=(0,-1,0), while in the future \mathbf{L}=(0,1,0).
What about \psi(t)?
Here is the plot of \tanh B t/2

And 2 \arctan \frac{1}{\sqrt{3}}=\pi/3. Therefore asymptotically, in the past,

(5)   \begin{equation*}\psi(t)=\frac{t}{2}-1.0472,\end{equation*}

and in the future

(6)   \begin{equation*}\psi(t)=\frac{t}{2}+1.0472.\end{equation*}

Therefore asymptotically at t\rightarrow -\infty

(7)   \begin{equation*}q(t)=\frac{1}{\sqrt{2}}(\cos \psi/2,\sin \psi/2,-\sin \psi/2,\cos \psi/2),\end{equation*}

while for t\rightarrow +\infty

(8)   \begin{equation*}q(t)=\frac{1}{\sqrt{2}}(\cos \psi/2,\sin \psi/2,\sin \psi/2,-\cos \psi/2),\end{equation*}

where \psi is given either by (5) or (6).
This will give us the two remarkable circles … In the next post. For now these circles should do:

Leshan Giant Buddha bridge Photography by Edy Petrova

Metamorphosis in action

Posted on by arkajad

Here is the deal:
There are two interlinked circles, like in a chain. You move around one of the circles for half of eternity almost without any change. Almost. Because one day you reach a threshold, there is an action potential, a quantum jump. And you move to the second circle, where you will continue for the second half of eternity.

This is the archetype, the essence, the kernel, of the Dzhanibekov effect that my series of posts is about.

The essential math is summarized in my previous post. Here comes the animation.

Click on the image to open gif animation. It will take time to load as it is 2.5 MB.

The upper image shows the unit quaternion representing the rigid body rotation. It moves first on one circle, then makes the jump to the second circle.
The lower image shows the rotating rigid body. Time on both images is synchronized. The quantum jump is the flip of the rotating rigid body.

Update: My Mathematica animation (almost the same as above) code that can be analyzed and should run with free Mathematica CDF Player (probably needs a recent one) is here.

The mysterious paths on the three-sphere

Posted on by arkajad

Wherever you are, whatever you do, there is a certain special direction that takes you out of the infinite labyrinth, and leads to the reincarnation cycle getting closer and closer to the ideal path. Ordinary people do not know about it. Warriors do know.

Beyond a certain point there is no return. This point has to be reached.
Franz Kafka

I am discussing geodesic lines on the three-sphere, geodesics of the left-invariant metric determined by asymmetric rigid body. There is a special, very special class of geodesics there. At every point there is a special direction. If you start the geodesic in this special direction – it has, at both ends, in the future and in the past – a limit cycle/circle. Enough esoteric talk. Let’s go to the math. The following math describes one such geodesic line. Other are obtained by left translations.

I_1,I_2,I_3 are moments of inertia of our rigid body, ordered as I_1<I_2<I_3.
We define

(1)   \begin{eqnarray*} A_1&=&\sqrt{\frac{I_1(I_3-I_2)}{I_2(I_3-I_1)}},\\ A_2&=&1,\\ A_3&=&\sqrt{\frac{I_3(I_2-I_1)}{I_2(I_3-I_1)}},\\ B&=&\frac{1}{I_2}\,\sqrt{\frac{(I_2-I_1)(I_3-I_2)}{I_1I_3}},\\ \delta&=&\frac{\sqrt{I_2(I_3-I_1)}-\sqrt{I_1(I_3-I_2)}}{\sqrt{I_3(I_2-I_1)}} \end{eqnarray*}

Define:

(2)   \begin{eqnarray*} l_1(t)&=& A_1\,\mathrm{sech }(B t),\\ l_2(t)&=&A_2\,\tanh( B t),\\ l_3(t)&=&A_3\,\mathrm{sech }(B t), \end{eqnarray*}

where \mathrm{sech }(x)=1/\cosh(x).
Define

(3)   \begin{equation*} \psi(t)=\frac{t}{I_2}+2\arctan\left(\delta \tanh(B t/2)\right). \end{equation*}

Define:

(4)   \begin{eqnarray*} q_0(t)&=&\frac{\sqrt{1+l_1(t)}\cos\frac{\psi(t)}{2}}{\sqrt{2}},\\ q_1(t)&=&\frac{\sqrt{1+l_1(t)}\sin\frac{\psi(t)}{2}}{\sqrt{2}},\\ q_2(t)&=&\frac{l_3(t)\cos\frac{\psi(t)}{2}+l_2(t)\sin\frac{\psi(t)}{2}}{\sqrt{2(1+l_1(t))}},\\ q_3(t)&=&\frac{-l_2(t)\cos\frac{\psi(t)}{2}+l_3(t)\sin\frac{\psi(t)}{2}}{\sqrt{2(1+l_1(t))}}. \end{eqnarray*}

The story is this. The body can rotate uniformly about its middle axis forever. Either left or right. These are two circles in the rotation group that, using its double cover, topologically is S^3. Stereographic projection maps circles into circles. So they become two circles in \mathbf{R}^3. These circles:

But there is another possibility, when this uniform rotation along the middle axis is only asymptotic. It happens in the infinite past and in the infinite future. But only approximately with the real life, and excluding the short metamorphosis period. This is the trajectory described by the formulas above. Here is the trajectory for time t from t=-20000 to t=20000:

And here is this trajectory together with asymptotic circles. Here the red circle is for the bad past, the blue circle, for the good future. Under microscope it reveals rich structure – infinite mystery.