On one hand we easily find this kind of an advice:

Is it possible to follow more than one path?

If you are practising spiritual discipline under the guidance of a Master, it is always advisable to give up your connection with other paths. If you are satisfied with one Master but are still looking for another Master, then you are making a serious mistake.

On the other hand we find this

Be All You Can Be: Don’t Choose One Path, Choose Multiple Paths

[latexpage]

We will follow the second option, we will look for other paths, other than the one we already know. Other **but of the same quality**

This project has two parts. The first part is pure algebra. No pictures. Pictures will come in the second part, when we will already know what to picture.

Consider this scenario: we are looking at all possible trajectories of the dynamics of our free rigid body with $I_1

Eq. (\ref{eq:ek}) gives then

\begin{equation}1=I_1I_2\Omega_1^2+I_2^2\Omega_2^2+I_2I_3\Omega_3^2.\end{equation}

Comparing this with Eq. (\ref{eq:l}) the term $I_2^2\Omega_2^2$ cancels out and we get

\begin{equation}

I_1^2\Omega_1^2+I_3^2\Omega_3^2=I_1I_2\Omega_1^2+I_2I_3\Omega_3^2,

\end{equation}

or

\begin{equation}I_3(I_3-I_2)\Omega_3^2=I_1(I_2-I_1)\Omega_1^2.\end{equation}

Thus in order for the trajectory to be special the ratio $\Omega_3/\Omega_1$ must be special:

\begin{equation}

\frac{\Omega_3}{\Omega_1}=\pm\sqrt{\frac{I_1(I_2-I_1)}{I_3(I_3-I_2)}}.\end{equation}

For our particular rigid body that we often use, with $I_1=1,I_2=2,I_3=3,$ we should have

\begin{equation}

\frac{\Omega_3}{\Omega_1}=\pm

\frac{1}{\sqrt{3}}.\label{eq:rat}\end{equation}

For our special trajectory that we were plotting in the last post Circles of eternal return

we had

\begin{eqnarray}

\Omega_1(0)&=&l_1(0)/I_1=A_1=\frac{1}{2},\\

\Omega_3(0)&=&l_3(0)/I_3=A_3/3=\frac{1}{2\sqrt{3}}.

\end{eqnarray}

So it is just one particular way of satisfying Eq. (\ref{eq:rat}). There are, however other ways. These other ways need to be researched.

We will do it in the next posts.

Thanks!

I like Musashi, I even forced myself to watch all the parts of loong looking-like-not-properly-ended movie about him with Toshiro Mifune (Yes, the first part is o.k, but try the rest..)