1, 2, 3 and butterfly effect

Posted on by arkajad

I know: my blog is somewhat chaotic. But what is chaos? From “Celestial Mechanics: The Waltz of the Planets” by Alessandra Celletti and Ettore Perozzi:


Chaos From the Greek Χάος, denoting primordial emptiness. The term is used to indicate the extreme sensitivity of a trajectory to the initial conditions, which implies unpredictable dynamical behaviour in the corresponding dynamical system.

With T-handle floating in space and flipping for no apparent reason – it looks like we are in a chaotic mode:

But are we? Where is “the extreme sensitivity of a trajectory to the initial conditions“? We have to address this question. The waltz of planets can be, in some respect, chaotic. But sometimes it is well organized. Like this waltz:

1,2,3 and 1,2,3, and my T-handle that replicates the cosmic behavior in the Dzhanibekov effect (see my previous posts) has principal moments of inertia 1,2,3. Why? What 1,2,3 has to do with chaos?

Time to explain. In general, when moons and planet spin, their motion is not chaotic. But sometimes it can be. Let us take my T-handle with principal moments of inertia I_1=1,I_2=2,I_3=3 (see my previous post).

Spin it about its shorter axis. It may show the strange flipping behavior – see the animated gif below (click to open it)

The “extreme sensitivity of a trajectory to the initial conditions” happens when the parameter d, that is the ratio of the doubled kinetic energy to squared angular momentum vector, is close to 1/I_2. The motion is governed by Jacobi elliptic functions \cn,\sn,\dn. The period \tau of these functions directly affects the time spent by the T-handle between flips. And this period is a very sensitive function of d, when d is close to 1/I_2.
Here is the graph of \tau for our T-handle:

The formula for \tau(d) involves elliptic K function. But for d very close to 1/I_2 there is an approximate (asymptotic) formula derived in the paper “Janibekov’s effect and the laws of mechanics” by A.G. Petrov and S. E. Volodin, Doklady Physics}, 58(8):349–353, 2013:

    \[\tau_0(d)=2\log\frac{16}{|\mu|},\]

where

    \[\mu=\frac{I_2(1-d I_2)(I_3-I_1)}{(I_3-I_2)(I_2-I_1)}.\]

Here is how the two formulas compare:

The blue curve is the simple asymptotic formula.

But why 1,2,3?

Why? Because 1,2,3 is optimal. But what is optimal?

How do we know if 9 to 5 is the most optimal time to work? – Quora
We don’t and we never will know, because everyone is different.

So, what did I optimize to get my 1,2,3? First I had a vague feeling that this is good

Genesis 1:31 God saw all that he had made, and it was very good …

But then I was able to prove that it was good by optimizing \tau(d). The dimensionless parameter \epsilon=1-d I_2 tells us how close we are to the critical value of d=1/I_2. Being very close, near the peak in the graphs above, we want to know what should be the values of I_1,I_2,I_3 for which the period \tau is maximal? We use the approximate formula above. To find where is the maximum of \tau, with \epsilon fixed, is to find where is the minimum of the ratio:

    \[\frac{I_2(I_3-I_1)}{(I_3-I_2)(I_2-I_1)}.\]

Here I_1,I_2,I_3 cannot be completely arbitrary. The moments of inertia should satisfy the inequality I_1+I_2\geq I_3. Suppose I_2=2. What should be I_1,I_2? I used Mathematica to get the answer:

If I assume that I_2=2, then I_1=1,I_3=3. If I assume that I_2=20, then I_1=10,I_3=30. Simple?

So, even if everyone is different, even if there is no optimal answer that would satisfy everybody, there is an optimal answer to my needs, and that is 1,2,3.

“In trying to please all, he had pleased none.”
― Aesop, Aesop’s Fables

With 1,2,3 we have more time to see the butterfly flying. By flipping its wings it can change dramatically the period of the flying T-handle on the other side of the planet.

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