Euler equations (Professional)
In the 2x2 case we take quite an obvious ansatz to the Nahm equations using the Pauli sigma matrices:
this gives the equations
Euler's Equations for a spinning top
These are related to Euler's equations for a spinning top
by the transformation
This can be thought of as a transformation between SO(3) and SU(2).
Integrals of Motion
The integrals of motion corresponding to total energy and angular momentum are
These are two surfaces in , 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 but hyperboloids. Say taking and we get the integrals of motion
which are the constants obtained in Curves and Lax pairs.
This tells us the solutions do have poles.
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.