[NOTES/CM-09010] A Heavy Top ---- Special Cases

By a heavy top one means a rigid body moving under influence of gravity with one of its points fixed. The body axes are chosen to be the principal axes relative to the fixed point of the body. The other quantities such as Euler angles, angular velocity angular momentum etc all refer to this choice of body axes. We shall be primarily interested here in this problem of motion of a heavy top the following cases arise.

1. The case of an asymmetric top, \(I_1 > I_2 > I_3\) is the most general case. One can find three constants of motion. The Euler equations for the rigid body can be solved and angular velocities can be expressed in terms of elliptic functions. The solution for the time dependence of the Euler angles is fairly complex and the answers can be written down in terms of elliptic functions and theta functions.
2. The Euler top is case of asymmetric top without an external torque. In thiis case there  are three constants of motion which make the system integrable.
3. In case of Lagrange top, also called symmetric top, there is an axis of symmetry and the centre of mass lies on the symmetry axis. Choosing the \(Z\)-axis as the symmetry axis, a symmetric top has \(I_1 = I_2 \ne I_3\). In this case the Euler angles \(\phi, \psi\) become cyclic and corresponding canonical momenta are conserved. There are three constants of motion. Energy conservation along with constancy of \(p_\phi, p_\psi\) can be used to reduce the solution to quadratures.
4. The case \(I_1 = I_2 = 2I_3\) is known as Kowalevskaya top. It is like a symmetric top, but the centre of mass does not lie on the axis of symmetry, the \(Z\)-axis. The centre of mass lies in the \(X-Y\) plane away from the \(Z\) axis. In this case also there are three constants of motion which make the model integrable.