Euler equations (Professional)

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The Nahm Equations: Euler's equations
Level: Beach \| Hill \| Mountain

1 Euler's Equations for a spinning top
- 1.1 Integrals of Motion
- 1.2 Solution

In the 2x2 case we take quite an obvious ansatz to the Nahm equations using the Pauli sigma matrices:

$T_i (t) = - \frac{i}{2} f_i(t) \sigma_i$

this gives the equations

$\begin{cases} \frac{d }{dt} f_1 (t) = f_2 (t) f_3 (t)\\ \frac{d }{dt} f_2 (t) = f_3 (t) f_1 (t) \\ \frac{d }{dt} f_3 (t) = f_1 (t) f_2 (t) \end{cases}$

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Euler's Equations for a spinning top

These are related to Euler's equations for a spinning top

$\begin{cases} \frac{d }{d \tau} g_1= (\lambda_3 - \lambda_2) g_2 g_3 \\ \frac{d }{d \tau} g_2= (\lambda_1 - \lambda_3) g_3 g_1 \\ \frac{d }{d \tau} g_3= (\lambda_2 - \lambda_1) g_1 g_2 \\ \end{cases}$

by the transformation

$\tau \rightarrow it,\ \ \ \ \ \ g_1\rightarrow \frac{1}{\sqrt{(\lambda_3-\lambda_1)(\lambda_2-\lambda_1)}} f_1,\ \ \ \ \ \ \ \ g_2\rightarrow\frac{1}{\sqrt{(\lambda_1-\lambda_2)(\lambda_3-\lambda_2)}} f_2, \ \ \ \ \ \ \ \ g_3\rightarrow\frac{1}{\sqrt{(\lambda_2-\lambda_3)(\lambda_1-\lambda_3)}} f_3$

This can be thought of as a transformation between $S O (3)$ and $S U (2)$ .

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Integrals of Motion

The integrals of motion corresponding to total energy and angular momentum are

$2E =\frac{M_1^2}{I_1}+\frac{M_2^2}{I_2}+\frac{M_3^2}{I_3} = \lambda_1 g_1^2 + \lambda_2 g_2^2 + \lambda_3 g_3^2$

$M^2 = M_1^2 + M_2^2 + M_3^2 = g_1^2 + g_2^2 + g_3^2$

These are two surfaces in $\mathbb{R}^3$ , a sphere and an ellipsoid. This tells us the solutions, which lie on the intersection of these surfaces, have no poles.

However the integrals of motion for our system are not ellipsoids in $\mathbb{R}^3$ but hyperboloids. Say taking $\lambda_1=-\frac{1}{2},\lambda_2=\frac{1}{2}$ and $\lambda_3=0\$ we get the integrals of motion

$f_2^2-f_1^2$

and

$f_1^2+f_2^2-2f_3^2$

which are the constants obtained in Curves and Lax pairs.

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Solution

This tells us the solutions do have poles.

If we take $f_1^2 \le f_2^2 \le f_3^2$ the solutions to these equations are

$f_1 (t) = \pm \frac{D cn_k(D t)}{sn_k(D t)}$ ,

$f_2 (t) = \pm \frac{D dn_k(D t)}{sn_k(D t)}$ ,

$f_3 (t) = \pm \frac{D}{sn_k(D t)}$

where $sn_k\,$ , $cn_k\,$ and $dn_k\,$ are the Jacobi elliptic functions. We have that $0 \le k \le 1$ and $D$ is a function of $k$ and a constant of the system.