The Euler-angle parametrization and the heavy symmetric top system

Euler-angle parametrization. Let $R_z,R_y$ be the coordinate rotation matrices of The coordinate rotation matrices $R_z(a)$ and $R_y(a)$: values in $SO(3)$, composition law, and derivatives. The Euler-angle parametrization is the map $\mathrm{\Theta }:\mathbb{R}\times (0,\pi )\times \mathbb{R}⟶SO(3),\mathrm{\Theta }(\phi ,\theta ,\psi ):=R_z(\phi )R_y(\theta )R_z(\psi ),$ which is $SO(3)$-valued by The coordinate rotation matrices $R_z(a)$ and $R_y(a)$: values in $SO(3)$, composition law, and derivatives and $SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$. The numbers $(\phi ,\theta ,\psi )$ are the Euler angles and the open set $\mathbb{R}\times (0,\pi )\times \mathbb{R}$ is its domain; $\theta$ is restricted to the open interval $(0,\pi )$ so that $\sin \theta >0$, with $\pi$ defined as twice the smallest positive cosine zero (Pi as twice the smallest positive zero of cosine) and positivity supplied by Pi is the first positive zero of sine. No injectivity and no chart or diffeomorphism property of $\mathrm{\Theta }$ is asserted or used here, and no periodicity of the angles is imposed — the real parameters stand for themselves.

Parameters. Fix reals $A>0$, $C>0$, $m>0$, $d>0$ and $g>0$; they are mathematical parameters, not identified with any physical quantity in this item.

The heavy-top function. Write $q=(\phi ,\theta ,\psi )$ for a point of the domain and $v=({\phi }^{\mathrm{\prime }},{\theta }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ for a triple of reals. The heavy-top function is $L(q,v):=\frac{1}{2}A({\theta }^{\mathrm{\prime }2}+{\phi }^{\mathrm{\prime }2}{\sin }^2\theta )+\frac{1}{2}C({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta {)}^2-mgd\cos \theta ,$ using the sine and cosine of Sine and cosine defined by their real power series. Repeated application of the sine/cosine derivative identities (The derivatives of sine and cosine are cosine and minus sine) and the finite sum, product and chain rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$) gives continuous partial derivatives of every order, so it is $C^{\mathrm{\infty }}$ on the domain times ${\mathbb{R}}^3$ ($C^k$ Euclidean maps and diffeomorphisms); its relevant partial derivatives are, explicitly, ${\mathrm{\partial }}_{{\phi }^{\mathrm{\prime }}}L=A{\phi }^{\mathrm{\prime }}{\sin }^2\theta +C\cos \theta ({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta ),{\mathrm{\partial }}_{\phi }L=0,$ ${\mathrm{\partial }}_{{\theta }^{\mathrm{\prime }}}L=A{\theta }^{\mathrm{\prime }},{\mathrm{\partial }}_{\theta }L=A{\phi }^{\mathrm{\prime }2}\sin \theta \cos \theta -C({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta ){\phi }^{\mathrm{\prime }}\sin \theta +mgd\sin \theta ,$ ${\mathrm{\partial }}_{{\psi }^{\mathrm{\prime }}}L=C({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta ),{\mathrm{\partial }}_{\psi }L=0.$

The heavy-top system. Let $I\subseteq \mathbb{R}$ be an interval with at least two points. A heavy-top trajectory for these parameters is a twice-differentiable curve $q=(\phi ,\theta ,\psi ):I\to \mathbb{R}\times (0,\pi )\times \mathbb{R}$ such that, with $q^{\mathrm{\prime }}=({\phi }^{\mathrm{\prime }},{\theta }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ its derivative, the three second-order equations $\frac{d}{dt}(A{\phi }^{\mathrm{\prime }}{\sin }^2\theta +C\cos \theta ({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta \left)\right)=0,\frac{d}{dt}\left(A{\theta }^{\mathrm{\prime }}\right)=A{\phi }^{\mathrm{\prime }2}\sin \theta \cos \theta -C({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta ){\phi }^{\mathrm{\prime }}\sin \theta +mgd\sin \theta ,$ $\frac{d}{dt}\left(C\right({\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta \left)\right)=0$ hold on $I$, all functions being evaluated at $t$. These displayed equations are the definition of the system; they are written out so that no separate definition of a variational principle is needed. The first and third are the statements that ${\mathrm{\partial }}_{{\phi }^{\mathrm{\prime }}}L$ and ${\mathrm{\partial }}_{{\psi }^{\mathrm{\prime }}}L$ are constant along the trajectory.

• The system as displayed is what later items use. The three equations above are the system studied in Reduction of the heavy symmetric top system: conserved momenta, one-dimensional $\theta$ equation and conserved reduced energy; that item derives the conservation laws and the reduced one-dimensional equation directly from these displayed equations. Naming $L$ records the intended reading, but no variational derivation is invoked anywhere on this page, and none of the claims below depends on one.

• The angular-velocity interpretation is a separate lemma. That $L$ is the kinetic-minus-potential reading of a rotating body with principal moments $A,A,C$ under a uniform field, and that the variables $(\phi ,\theta ,\psi )$ are the same angles that parametrize the rigid placement, is the content of Angular velocity in Euler angles: ${\omega }_3={\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta$ and ${\omega }_1^2+{\omega }_2^2={\theta }^{\mathrm{\prime }2}+{\phi }^{\mathrm{\prime }2}{\sin }^2\theta$ and of the physical items that consume this pair, not of this definition.

• No physical law is asserted here. That a physical symmetric top satisfies such a system, and what the parameters $A,C,m,d,g$ mean physically, are postulates or derived statements of the consuming physics pages.