Infeld, Einstein and blogging

For many years I was keeping my Polish blog. The discussions there were hot, it was rather normal to have over 100 of comments after each consecutive post. Now, that I have switched to English, everything instantly collapsed. Which was to be expected, nevertheless the collapse is a valid observation.

I am reading autobiography of Leopold Infeld, Polish theoretical physicist who worked with Albert Einstein. Wikipedia mentions that

“Infeld was one of the 11 signatories to the Russell–Einstein Manifesto in 1955, and is the only signatory never to receive a Nobel Prize.”

But that is not important here. What I want to point at is that Infeld was working with Einstein in Berlin, then in Princeton, then he had a position in Toronto. And yet in 1950 he returned to Poland. Why?

Infeld with Einstein

Reading his biography I am guessing that he was not among the best in Berlin, he was not among the best in Princeton, he was not the best, and no so important in Toronto. But after he has returned to Poland, he became important, the organizer of the Polish school of theoretical physics. He did a good and important job there.

I was “a somebody” on my Polish blog. There are not so many professors of theoretical physics that are blogging in Poland. Thus there were many readers, there were interesting contacts, correspondence, exchange of ideas, collaborations. But with my English blog I am nobody. There are thousands of similar blogs. The market is different. So far I have this luck that I have a dedicated collaborator who came from the Polish blog here. 99% of all discussion and stimulation comes from Him. Thank you Bjab!

This being said, let me finish handling the T-handle from the recent posts. What is still lacking is the case of d>1/I_2 that is m>1. The point is that it is better to reduce everything to the case of m<1, since one should not rely on the implementation of Jacobi elliptic functions for m>1. From my experience, Mathematica, for instance, can handle the case m>1, but it is not always reliable. Fortunately we can reduce the case of m>1 to that of m<1 and here is the complete algorithm:

We will use units in which the length L of the angular momentum vector is 1: L=1.
Given m>1 introduce \mu=1/m<1:

(1)   \begin{equation*} m=k^2>1,\mu=1/m,\end{equation*}

We recall the conversion formulas below

(2)   \begin{equation*} \sn(u,m)=\frac{1}{k}\sn(ku,\mu),\end{equation*}

(3)   \begin{equation*}\cn(u,m)=\dn(ku,\mu),\end{equation*}

(4)   \begin{equation*}\dn(u,m)=\cn(ku,\mu).\end{equation*}

We arrive at the following algorithm:

(5)   \begin{equation*}A_1 = \sqrt{I_1 (d I_3 - 1)/(I_3 - I_1)}\end{equation*}

(6)   \begin{equation*}A_2 = \sqrt{I_2 (1 - d I_1)/(I_2 - I_1)}\end{equation*}

(7)   \begin{equation*}A_3 = \sqrt{I_3 (1 - d I_1)/(I_3 - I_1)}\end{equation*}

(8)   \begin{equation*} B = \sqrt{(d I_3 - 1)(I_2 - I_1)/(I_1 I_2 I_3)}\end{equation*}

(9)   \begin{equation*}\mu= \frac{(1 - d I_1) (I_3 - I_2)}{(d I_3 - 1) (I_2 - I_1)}\end{equation*}

(10)   \begin{equation*} L_1(t) = A_1\, \dn(Bt, \mu)\end{equation*}

(11)   \begin{equation*} L_2(t) = A_2\, \sn(Bt, \mu)\end{equation*}

(12)   \begin{equation*} L_3(t) = A_3\, \cn(Bt, \mu)\end{equation*}

(13)   \begin{equation*} L_p(t) = \sqrt{L_1(t)^2 + L_2(t)^2}\end{equation*}

(14)   \begin{multline*} \psi(t) = t/I_3 + \left((I_3 - I_1)/\sqrt{\frac{I_1I_3}{I_2}(I_2-I_1)(d I_3 - 1)}\right)\\  \Pi\left(-\frac{I_3 (1 - d I_1)}{I_1 (d I_3 - 1)},\am\left(t  \sqrt{\frac{(I_2 - I_1) (d I_3 - 1)}{I_1 I_2 I_3}}, \mu\right), \mu\right)\end{multline*}

(15)   \begin{equation*}Q_0(t)=\begin{bmatrix}\frac{L_1(t)L_3(t)}{L_p(t)}&\frac{L_2(t)L_3(t)}{L_p(t)}&-L_p(t)\\-\frac{L_2(t)}{L_p(t)}&\frac{L_1(t)}{L_p(t)}&0\\ L_1(t)&L_2(t)&L_3(t)\end{bmatrix},\end{equation*}

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

(17)   \begin{equation*}Q(t)=Q_1(t)Q_0(t),\end{equation*}

To get a general solution one may need to change the signs of two components from (L_1,L_2,L_3) and/or shift the time: t\mapsto t+t_0.