Generalized Euler equations

From Islands

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

1 Euler's equations for a spinning top
2 The reduced Euler system
3 The Generalization
4 Euler equations from the Nahm equations

Starting with the Euler equations for a free spinning top, we can derive the Nahm equations. The Nahm equations can be thought of as a generalisation of the Euler equations from $S O (3)$ to $S U (2)$ .

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Euler's equations for a spinning top

The Lagrangian for a free spinning top is

$L = \frac{1}{2}( I_1 \Omega_1^2 + I_2 \Omega_2^2 + I_3 \Omega_3^2 )$

with $I_1, I_2, I_3\,$ the principal moments of inertia and $\Omega_1, \Omega_2, \Omega_3\,$ the components of angular velocity in the comoving system of coordinates.

The Euler equations are obtained by transforming to the moving frame as follows:

$\frac{d(I_i \Omega_i)}{dt} = \frac{dM}{dt} \rightarrow \frac{dM}{dt} + \Omega \times M =0$

and are

$\begin{cases} \frac{d }{d t} \Omega_1= \frac{I_2 - I_3}{I_1} \Omega_2 \Omega_3 \\ \frac{d }{d t} \Omega_2= \frac{I_3 - I_1}{I_2} \Omega_3 \Omega_1 \\ \frac{d }{d t} \Omega_3= \frac{I_1 - I_2}{I_3} \Omega_1 \Omega_2 \\ \end{cases}$

or using the momenta $g_1, g_2, g_3\,$ and where $\lambda_i = 1/I_i\,$ :

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

This system has two first integrals:

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

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

with $E$ and $M$ corresponding to the total energy and angular momentum of the system.

We can solve this system by using the two integrals of motion to eliminate two variables in the system of equations. Solutions are the Jacobi elliptic functions $s = \mathbf{sn}(\tau)$ where

$\tau = \int_0^s \frac{\mathbf{d}s}{\sqrt{(1-s^2)(1-k^2 s^2)}}$

The trajectory of M is described by the intersection of an ellipsoid and a sphere. A discussion of the different cases can be found in Mechanics by Landau and Lifshitz.

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The reduced Euler system

Euler's equations can be reduced to a simpler system where $f_1, f_2, f_3\,$ are scalar functions by using

$\begin{cases} t \rightarrow i \tau \\ g_1 (t) \rightarrow a_1 f_1 = \frac{1}{\sqrt{(\lambda_3-\lambda_1)(\lambda_2-\lambda_1)}} f_1 (i \tau) \\ g_2 (t)\rightarrow a_2 f_2 = \frac{1}{\sqrt{(\lambda_1-\lambda_2)(\lambda_3-\lambda_2)}} f_2 (i \tau)\\ g_3 (t) \rightarrow a_3 f_3 = \frac{1}{\sqrt{(\lambda_2-\lambda_3)(\lambda_1-\lambda_3)}} f_3 (i \tau) \end{cases}$

resulting in

$\begin{cases} \frac{d }{d \tau} f_1 (i \tau) = f_2 (i \tau) f_3 (i \tau)\\ \frac{d }{d \tau} f_2 (i \tau) = f_3 (i \tau) f_1 (i \tau) \\ \frac{d }{d \tau} f_3 (i \tau) = f_1 (i \tau) f_2 (i \tau) \end{cases}$

As before the integrals of motion are

$2 E = \lambda_1 a_1^2 f_1^2 + \lambda_2 a_2^2 f_2^2 + \lambda_3 a_3^2 f_3^2$

$M^2 = a_1^2 f_1^2 + a_2^2 f_2^2 + a_3^2 f_3^2$

It can be verified that these are constants of the motion by differentiating with respect to $τ$ .

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 $D = \sqrt{f_3^2 - f_1^2}$ . There are two possibilties for the signs: all negative or two positive.

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The Generalization

By using the Pauli matrices

$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \qquad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \qquad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

we can promote the scalar functions $f i$ to matrices:

$\begin{cases} \mathbf{1}_{2\times 2} f_1 = -2 i \sigma_1 T_1 \\ \mathbf{1}_{2\times 2} f_2 = -2 i \sigma_2 T_2 \\ \mathbf{1}_{2\times 2} f_3 = -2 i \sigma_3 T_3 \\ \end{cases}$

Substituting these into the equations $\dot{f}_1 = f_2 f_3 \ldots$ results in

$\begin{cases} \dot{T_1} = [T_2,T_3] \\ \dot{T_2} = [T_3,T_1] \\ \dot{T_3} = [T_1,T_2] \end{cases}$

iff

$\begin{cases} \left [T_1 + T_2 , \sigma_3 \right ] = 0 \\ \left [T_3 + T_1 , \sigma_2 \right ] = 0 \\ \left [T_2 + T_3 , \sigma_1 \right ] = 0 \\ \end{cases}$

where $[\, ,\,]\,$ is the commutator.

The Nahm Equations can thus be been seen as a generalization of Euler's equations.

The Pauli matrices correspond to infinitesimal rotations. $i σ i$ are skew hermitian traceless matrices and thus lie in the Lie algebra $\mathfrak{su(2)}$ . In this way we see that scalar functions have been promoted to matrix functions in $\mathfrak{su(2)}$ .

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Euler equations from the Nahm equations

We have shown above how starting with the Euler equations, we can arrive at the Nahm equations. The converse is simple to show: given the Nahm equations

$\dot{T}_1 = [T_2,T_3]\,$ and permutations