Generalized Euler 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 SO(3) to SU(2).
Euler's equations for a spinning top
The Lagrangian for a free spinning top is
with the principal moments of inertia and the components of angular velocity in the comoving system of coordinates.
The Euler equations are obtained by transforming to the moving frame as follows:
or using the momenta and where :
This system has two first integrals:
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 where
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.
The reduced Euler system
Euler's equations can be reduced to a simpler system where are scalar functions by using
As before the integrals of motion are
It can be verified that these are constants of the motion by differentiating with respect to τ.
If we take the solutions to these equations are
where , and are the Jacobi elliptic functions. We have that and D is a function of k and a constant of the system . There are two possibilties for the signs: all negative or two positive.
By using the Pauli matrices
we can promote the scalar functions fi to matrices:
Substituting these into the equations results in
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. iσi are skew hermitian traceless matrices and thus lie in the Lie algebra . In this way we see that scalar functions have been promoted to matrix functions in .
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
- and permutations
then the ansatz
gives the Euler equations in the reduced form
- and permutations.
This ansatz may be used to solve the Nahm equations in the k = 2 case.
Exercise. Verify this. Note that the Pauli sigma matrices satisfy