Reduction of the heavy symmetric top branch: pivotal angular momentum, Euler-angle equations, the two conserved momenta and the one-dimensional $\theta$ equation

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only for the finiteness of the defining integrals of the model's inertial quantities. Adopt the heavy symmetric top branch Adoption of the heavy symmetric top branch: axisymmetric rigid material, a fixed pivot material point, a uniform gravitational field and the balance of angular momentum about the pivot with parameters $A_c>0$, $C_c>0$, $d>0$, $g>0$ and total mass $m>0$; use the notation of that postulate: the pivot material point ${\rho }_p=-d{e}_3$ is held at the origin, so $c(t)=dR(t)e_3$; the pivot reaction contributes no torque about the origin; and the external gravitational field is $G=-g{e}_3$ with total torque about the origin ${\tau }_p(t)=-mgd(R(t)e_3)\times e_3$. Put $A:=A_c+m{d}^2>0,C:=C_c>0,$ the transverse moment of the model about the pivot and its axial moment. Let $x(t,\rho )=c(t)+R(t)\rho$ be a motion of the branch with time window $I_0$, and suppose that there are functions $\phi ,\theta ,\psi :I_0\to \mathbb{R}$ of class $C^2$ with $\theta (I_0)\subseteq (0,\pi )$ and $R(t)=\mathrm{\Theta }(\phi (t),\theta (t),\psi (t))=R_z(\phi (t))R_y(\theta (t))R_z(\psi (t))(t\in I_0),$ the Euler-angle description of The Euler-angle parametrization and the heavy symmetric top system. Then:

1. Angular momentum about the pivot. With $\omega$ the body angular velocity of the motion and $\tilde{I}:=A(e_1{e}_1^{\mathsf{T}}+e_2{e}_2^{\mathsf{T}})+C{e}_3{e}_3^{\mathsf{T}}$ (the diagonal matrix $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(A,A,C)$ in the principal basis) one has $J_p(t):=\int x(t,\rho )\times {\mathrm{\partial }}_{t}x(t,\rho )d\nu (\rho )=R(t)\tilde{I}\omega (t),$ and the balance of Adoption of the heavy symmetric top branch: axisymmetric rigid material, a fixed pivot material point, a uniform gravitational field and the balance of angular momentum about the pivot clause (v) reads $J_p^{\mathrm{\prime }}={\tau }_p$ with ${\tau }_p(t)=-mgd(R(t)e_3)\times e_3,{\tau }_p(t)\cdot e_3=0.$
2. Body-frame pivot equations. With $\gamma (t):=R(t{)}^{\mathsf{T}}{e}_3$, $\tilde{I}{\omega }^{\mathrm{\prime }}+\omega \times \tilde{I}\omega =-mgd(e_3\times \gamma )\text{on }{I}_0,$ that is, componentwise in the principal basis, $A{\omega }_1^{\mathrm{\prime }}=(A-C){\omega }_2{\omega }_3+mgd{\gamma }_2,A{\omega }_2^{\mathrm{\prime }}=(C-A){\omega }_1{\omega }_3-mgd{\gamma }_1,C{\omega }_3^{\mathrm{\prime }}=0.$ Under the Euler-angle description of the statement, $\gamma =(-\sin \theta \cos \psi ,\sin \theta \sin \psi ,\cos \theta ),{\omega }_3={\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta \mathrm{.}$
3. Reduction to one dimension. Define on $I_0$ $b:=C({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta ),a:=A{\phi }^{\mathrm{\prime }}{\sin }^2\theta +b\cos \theta \mathrm{.}$ Then $b$ and $a$ are constant on $I_0$; consequently ${\phi }^{\mathrm{\prime }}=\frac{a-b\cos \theta }{A{\sin }^2\theta },{\psi }^{\mathrm{\prime }}=\frac{b}{C}-\frac{(a-b\cos \theta )\cos \theta }{A{\sin }^2\theta };$ the reduced energy $E:=\frac{1}{2}A{\theta }^{\mathrm{\prime }2}+\frac{(a-b\cos \theta {)}^2}{2A{\sin }^2\theta }+\frac{b^2}{2C}+mgd\cos \theta$ is constant on $I_0$; and with the effective potential $U(\theta ):=\frac{(a-b\cos \theta {)}^2}{2A{\sin }^2\theta }+\frac{b^2}{2C}+mgd\cos \theta$ one has $A{\theta }^{\mathrm{\prime }\mathrm{\prime }}=-U^{\mathrm{\prime }}(\theta )$ on $I_0$.
4. The heavy-top system. Equivalently, $(\phi ,\theta ,\psi )$ is a heavy-top trajectory of The Euler-angle parametrization and the heavy symmetric top system with parameters $(A,C,m,d,g)$, and clauses 1-4 of Reduction of the heavy symmetric top system: conserved momenta, one-dimensional $\theta$ equation and conserved reduced energy hold for it.

The statement assumes the Euler-angle description as a hypothesis on the interval: no claim is made that every motion of the branch admits such a description, that a description extends across the poles $\theta \in \{0,\pi \}$ (where the sleeping-top and hanging configurations lie), or that the description is unique. On every subinterval on which a description with $\theta \in (0,\pi )$ exists, the conclusions hold there. The statement asserts no friction or driving torque at the pivot, no measurement and no empirical value.

• Where $A=A_c+m{d}^2$ comes from. The transverse moment of the model about the pivot is the transverse moment $A_c$ about the centre of mass shifted by $m{d}^2$; this is why the reduction uses $A$ and the supplier's parameter is identified with the pivot moment, not with $A_c$. The axial moment is unchanged because the shift is along the symmetry axis.

• The role of the chart. The Euler-angle description is a hypothesis; the chart is singular at the poles $\theta =0,\pi$, where the sleeping top and the hanging configuration lie, and the statement says nothing there. The definitions of $a$, $b$, $E$ and $U$ depend on the chosen description (the angles are determined only up to the chart's ambiguities), while the physical conclusions of clause 3 are invariants of the motion.

• No empirical content. Clause 3 is a conditional deduction from the adopted branch; in particular the sleeping-top threshold, the stability of the upright top and the interpretation of precession in any concrete apparatus are not asserted here, and no value of $A_c$, $C_c$, $m$, $d$ or $g$ is measured.