The critical turning points in the solutions of the magnetic-top equations of motion

Summary

The solutions of the equations of motion of the magnetic top, whose orientation is described by the Euler angles θ, φ,χ are expressed through the integrals of motion and the dependence of their properties on these integrals of motion are studied. We find that when the initial canonical momentap _χ andp _φ have equal absolute values, the angular velocity θ undergoes a discontinuous change of sign when the turning point of the orbit in θ are at the poles (cos θ₁ = 1 or cos θ₂ = -1). These discontinuities in θ can be compensated by discontinuities in φ and χ (by π or-π) so that the linear velocity components of the frame axes are continuous.