Standing on the shoulders of giants – Reboot

Standing on the shoulders of giants may be good, even very-very good. But at the same time it can be very dangerous. First of all it is a dangerous balancing act. It is easy to fall.

But what if the giant gets a hiccup? Or stumble? You may die. Or someone can die because of aftershocks.

In Standing on the shoulders of giants I quoted two papers that I used in working out my computer simulations of Dzhanibekov’s effect. These were

Ramses van Zon, Jeremy Schofield, “Numerical implementation of the exact dynamics of free rigid bodies“, J. Comput. Phys. 225, 145-164 (2007)
Celledoni, Elena; Zanna, Antonella.E Celledoni, F Fassò, N Säfström, A Zanna, The exact computation of the free rigid body motion and its use in splitting methods, SIAM Journal on Scientific Computing 30 (4), 2084-2112 (2008)

The algorithms were working smoothly. But recently I decided to use quaternions instead of rotation matrices. And in recent post Introducing geodesics, I have presented a little animation:

It looks impressive, but if we pay attention to details, we see something strange:

We have a nice curly trajectory, but there are also strange straight line spikes. On animation they are, perhaps, harmless. But if something like this happens with the software controlling the flight of airplanes, space rockets or satellites – people may die. There is a BUG in the algorithm. Yes, I did stay on shoulders of giants, but perhaps they were not the right giants for my aims? And indeed this happens. The method I used is not appropriate for working with quaternions.

That is why I have to do the reboot, and look for another giant. Here is my giant:
Antonella Zanna, professor of Mathematics, University of Bergen, specialty Numerical Analysis/Geometric Integration.

I am going to use the following paper:

Elena Celledoni, Antonella Zanna, et al. “Algorithm 903: FRB–Fortran routines for the exact computation of free rigid body motions.” ACM Transactions on Mathematical Software, Volume 37 Issue 2, April 2010, Article No. 23, doi:10.1145/1731022.1731033.

The paper is only seven years old. And it has all what we need. As in Standing on the shoulders of giants we will split the attitude matrix Q into a product: Q=Q_1Q_0, but this time we will do it somewhat smarter. Celledoni and Zanna denote the angular momentum vector in the body frame using letter \mathbf{m}=(m_1,m_2,m_3), and they assume that \mathbf{m} is normalized: m=\sqrt{\mathbf{m}^2}=1. They assume, as we do, I_1<I_2<I_3, and use T to denote the kinetic energy. Let us first look at the following part of their paper (which is available here):

Though it is not very important, I must say that it is very nice of the two authors that they acknowledge Kosenko. I was trying to study Kosenko’s paper, but I gave up not being able to understand it. Then wen notice that the case considered is that of d>1/I_2 using the notation in my posts. That means “high energy regime”.
The angular momentum \mathbf{m} is constant (fixed) in the laboratory frame owing to the conservation of angular momentum. Suppose we orient the laboratory frame so that its z axis is oriented along the angular momentum vector. Then, in the laboratory frame angular momentum has coordinates (0,0,1). Suppose we construct a matrix Q_0 that transforms \mathbf{m}=(m_1,m_2,m_3) into (0,0,1). The attitude matrix Q transforms vector components in the body frame into their components in the laboratory frame. In particular it transforms \mathbf{m}=(m_1,m_2,m_3) into (0,0,1). Therefore,if we write Q=Q_1Q_0, the matrix Q_0 must transform (0,0,1) into itself. Therefore Q_0 must be a rotation about the third axis, that is it is of the form:

(1)   \begin{equation*}Q_1= \begin{bmatrix}\cos\psi&-\sin\psi&0\\\sin\psi&\cos\psi&0\\0&0&1\end{bmatrix}.\end{equation*}

All of that we have discussed already in Standing on the shoulders of giants. But now we are going to choose Q_0 differently.

So suppose we have vector \mathbf{m} of unit length. What would be the most natural way of constructing a rotation matrix that rotates \mathbf{m} into (0,0,1)? Simple. Project \mathbf{m} vertically onto (x,y) plane. Draw a line \mathbf{n} on (x,y) plane that is perpendicular to the projection. Rotate about this line by the angle \theta between \mathbf{m} and \mathbf{k}=(0,0,1).
Let us now do the calculations. Vector \mathbf{m} has components (m_1,m_2,m_3). Its projection on (x,y) plane has components (m_1,m_2,0). Orthogonal vector in (x,y) plane has components (m_2,-m_1,0) (to check orthogonality calculate scalar product), So the normalized vector \mathbf{n} has components

    \[\mathbf{n}=(\frac{m_2}{m_p}, -\frac{m_1}{m_p},0),\]



The cosinus of the angle \theta is m_3. Its sinus is \sqrt{1-m_3^2}=m_p.
In Spin – we know that we do not know we have derived a general formula for a rotation about an angle \theta about axis defined by a unit vector \mathbf{n}:

    \[R=I+\sin\theta\, W(\vec{n})+(1-\cos\theta)\,W(\vec{n})^2,\]

where for any vector \mathbf{v}

(2)   \begin{equation*}W=W({\vec{v})=\begin{pmatrix}0& -v_3&v_2\\v_3&0&-v_1\\-v_2&v_1&0.\end{pmatrix}\end{equation*}

Applying the formula to our case, simple algebra leads to:

(3)   \begin{equation*}Q_0=\begin{bmatrix}(m_2^2 + m_1^2 m_3)/m_p^2& (m_1 m_2 (-1 + m_3))/m_p^2& -m_1\\  (m_1 m_2 (-1 + m_3))/m_p^2& (   m_1^2 + m_2^2 m_3)/m_p^2& -m_2\\m_1& m_2& m_3\end{bmatrix}.\end{equation*}

We can easily check that indeed Q_0 is an orthogonal matrix with determinant 1, Q0 acting on (m_1,m_2,m_3) vector gives (0,0,1). One can also check that the vector (-m_2,m_1,0) is invariant under Q_0 – as it should be as it is on the rotation axis. REDUCE program that checks it all is here.

At this point we have to make a break. It is not yet clear what is the relation of my Q_0 to the paper quoted, what is the difference between this and the old version, and why is this version better?

We will continue climbing on the shoulders of giants in the following notes. But, please, remember, Standing on shoulders of giants (like “Nobel Prize Winner” Obama), is risky:

Photo of Kellyanne Conway kneeling on Oval Office couch sparks…

When China Rules the World

The title of the post can be found on Wikipedia with additional information. But this post is more personal. Indeed China rulez! Few weeks ago I received the following email:

Quantum World-2017

Time: 16th-18th October, 2017

Place: Changsha, Hunan Province, China

Dear Dr. Arkadiusz Jadczyk,

On behalf of the organizing committee of CQW-2017, we sent you a letter a few days ago invited you to join us and give a speech at first Annual Conference of Quantum World (CQW-2017), which will be held on 16th-18th October, in Changsha, Hunan Province, China. It seems you have not received that letter yet, so I am writing again to extend our sincere invitation. As we have learnt your valuable contribution to Asymptotic Formula for Quantum Harmonic Oscillator Tunneling Probabilities…, we believe your inspirational speech and participation will highlight this congress a lot!

Under the theme “From E=MC2 to Quantum Industry”, the first Annual Conference of Quantum World (CQW-2017) aims at 200+ oral presentations in Quantum Physics and Mechanics, Quantum Information Science, Quantum Chemistry, Quantum Optics, Quantum Materials, and Quantum System, Quantum Engineering and Application, which cover hot topics with both theoretical and experimental contributions.

The conference venue Changsha, as the capital of Hu’nan province, is a beautiful, creative, historical and cultural city with comfortable climate, unique scenery and convenient transportation. It will give you a special experience on the colligation and integration of the Huxiang Culture with the modern civilization.

Worth mentioning that partial scientific program with speakers’ profile and excellent speech titles has been updated on website, kindly click here to view and give us your valuable advice.

Look forward to your kind reply with positive response.

Sincerely yours,

CQW-2017 Organizing Committee

Of course I was surprised, because I do not expect anybody but few experts in the whole world care about my paper. Looks to me like a huge conference industry.

China opens world’s ‘highest and longest glass bottomed bridge’

In my blog post Lorentz transformation from an elementary point of view – from blogging to science publishing I wrote about a paper that came as the result of blogging. This is the second paper that I wrote together with prof. Jerzy Szulga, a continuation of our previous paper, A Comment on “On the Rotation Matrix in Minkowski Space-time” by Ozdemir and Erdogdu,, Reports on Mathematical Physics, 74(1), 2014, 39-44,
DOI: 10.1016/S0034-4877(14)60056-2. Today I received from my coauthor a message with a copy of another invitation from China:

Dear Dr. ….,

I’m writing to follow-up my last invitation as below, would you please give me a tentative reply? Thank you very much. I apologize for the inconvenience if the letter disturbed you more than once.

It is our great pleasure and privilege to welcome you to join the 8th World Gene Convention-2017, which will take place in Macao, China during November 13-15, 2017. We would like to welcome you to be the chair/speaker in Theme 902: Agriculture, Food and Plant Biotechnology while presenting about A Comment on “On the Rotation Matrix in Minkowski Space-Time” by Ozdemir and Erdogdu…….

If the suggested thematic session is not your current focused core, you may look through the whole sessions and transfer another one that fit your interest (more info about the program is available at

Under our SAB members’ contributions and endeavor, BIT’s 7th World Gene Convention-2016 (WGC-2016), successfully held in Shanghai, China during November 13-15, 2016. Totally, there were nearly 300 world-renowned experts, professors, laboratory principals, project leaders and representatives of well-known enterprises attended the WGC-2016. Participants from the international enterprises, academic and research institutions enjoyed the three days scientific program. Depending on the warmly support and good suggestions from all of the participants, we are confident in organizing WGC-2017 which would be better and more successful than WGC-2016.

WGC-2017 features a very strong technical program, mainly focused on: breakthroughs in gene, advances genomics & genetics, new research of DNA and RNA, focus on basic research, the frontier research of life sciences, new biotherapy discovery, emerging areas for medicine applications, robust technology development, and cutting-edge Biotechnology. It aims to provide a platform for all experts from academia, industry and national labs to discuss latest hot researches and achievements. Attendees will hear world-class speakers discussing the challenges and opportunities facing the gene, biotechnology and life sciences field. The business & academic experts who are from home and abroad will give excellent speeches.

In addition to the dynamic scientific program, you will benefit from the wonderful experience in Macao, China. Macao is an international free port. It’s famous for light industry, tourism, and hotel. Macao is also one of the most developed and richest regions in the world, this is a city of amazing and fascinating cultural wealth. The unique blend of European and Oriental cultures existing here creates a pleasurable and laid back atmosphere in a truly beautiful city. We hope you will enjoy your stay in this beautiful city with all its feature, beauty, architecture and hospitality!

We expect your precious comments or suggestions; also your reference to other speakers will be highly appreciated. We look forward to receiving your replies on the following questions:

1. What is the title of your speech?

2. Do you have any suggestions about our program?

For more information, please visit the conference website

We look forward to see you in Macao in 2017 for this influential event.

Sincerely yours,

Ms. Teresa Xiao

Organizing Committee of WGC-2017

It is only Chinese people can have such a broad and brave imagination and vision of the future. To connect Lorentz and spin groups of matrices with genes and agriculture – it is a real feat.

Introducing geodesics

Why are geodesics important? Probably because they are very simple. They are, in a sense, the simplest possible paths. Quoting from Richard Buckminster Fuller Domes and archives, 1960, 1965

Fuller inspired by his observations of nature. The inventor applies the concept of the geodesic line (the shortest line joining two points on a surface) to construct the most balanced, lightweight and resistant structure possible. His domes are a synthesis of all of the inventor’s fundamental precepts, combining a reasoned and aesthetic use of technological progress with a holistic conception of man’s relationship to nature. Such was the reputation of the inventor in the scientific domain that a family of carbon-based molecules with a geodesic structure was named after him: Buckminster fullerenes, later changed to fullerenes. Many of these molecules have played a role in recent nanotechnology discoveries.

Quoting from Wikipedia – Geodesics on an ellipsoid :

Geodesics on an ellipsoid of revolution
There are several ways of defining geodesics (Hilbert & Cohn-Vossen 1952, pp. 220–221). A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid’s surface (somewhat more than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points.

If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Newton (1687) showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid and, in this case, the equator and the meridians are the only closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), then only three geodesics are closed and one of these is unstable.
On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by X = 0, Y = 0, and Z = 0.

I am not playing with geodesics on three axial ellipsoid. But studying unexpected flips of asymmetric spinning top in zero gravity brings us very close to this circle of ideas. We do have three axial ellipsoid, because we have three different moments of inertia I_1<I_2<I_3. And we do have geodesics, even if we have to wait a little bit to see them nicely introduced in this series of posts. Histories of rotations in the rotation group are geodesics. We have seen a bunch of them in two previous posts. These were closed. Nothing particularly interesting – who did not see a circle? True, we have seen a bunch of circles spanning a torus, but these were generated artificially by a rotating observer.

But no it is time to move beyond the safe mode. If you have a spinning top in free space, and if you struck it, like Peggy Whitson

it will start to nutate. And when all three moments of inertia are different, this nutation is non-periodic. Geodesics have infinite length. They are certainly not the shortest connections between points in the group, in any sense. One geodesic line is wandering through the three-dimensional sphere S^3 sometimes almost returning to the starting point, then traveling far away. Strange are these trajectories.

Below is a part of one such trajectory, for I_1=1,I_2=2,I_3=3 and d=0.4. I am rotating it in animation, so that you can have a better view of its 3D structure. Of course, as in previous posts, this is stereographic projection from S^3.

Geodesic on three sphere
Geodesic of the left-invariant metric in the group \mathrm{SU}(2)

Of course I will have to explain the details …