Past exam of the options II course of the University of Oxford / 2005 / H / 2 / Solution

Write the position of a mass element relative to the centre of mass as $r$. Expand its squared velocity and integrate:The mixed term vanishes because $\int rdm=0$. Alsousing the scalar triple-product identity. ThereforeIn principal axes the inertia tensor is diagonal, so the second term is $\frac{1}{2}(A{\omega }_1^2+B{\omega }_2^2+C{\omega }_3^2)$. Substituting the three Euler-angle components given in the source gives its displayed kinetic-energy expression directly.

For $A=B$, the first two component squares have opposite mixed terms and sum to ${\dot{\theta }}^2+{\dot{\varphi }}^2{\sin }^2\theta$. Put $n=\dot{\psi }+\dot{\varphi }\cos \theta$. Using the orientation-dependent potential printed in the source therefore gives the formal modelThe cyclic coordinates $x,y,\psi$ have conserved momenta $M\dot{x},M\dot{y},Cn$. Consequently $\dot{x},\dot{y},n$ are constant. The coordinate $\varphi$ is also cyclic, withFor imposed fixed tilt $\theta =\alpha \in (0,\pi /2)$, conservation of $n$ and $p_{\varphi }$ forces $\dot{\varphi }=\Omega$ constant. Impose the fixed-height condition $z=h$ in this formal constrained model. Its $\theta$ equation isSet the tilt constant and divide by its nonzero sine. This givesA real polynomial root requires its quadratic discriminant to be nonnegative:When this condition holds the two algebraic rates are $(Cn\pm \sqrt{C^2{n}^2-4AMgh\cos \alpha })/(2A\cos \alpha )$.

There is a physical inconsistency in the printed data that must be kept distinct from this algebra. If $(x,y,z)$ are literally Cartesian coordinates of the centre of mass in uniform gravity, the gravitational potential is $Mgz$, not $Mgz\cos \theta$. The printed Lagrangian consequently does not follow from that Cartesian-height interpretation. Moreover its unconstrained $z$ equation is $\ddot{z}=-g\cos \theta$, so constant height at the indicated tilt requires a constraint reaction. The requested rate and discriminant follow from the printed orientation-dependent potential with fixed height imposed; a standard heavy-top interpretation instead uses a height proportional to an axis distance times $\cos \theta$, with the reference point and corresponding inertia specified consistently. The kinetic-energy derivation is independent of this source inconsistency.