Ring moving on paraboloid? Using Newton's law and rotational mechanics it is easy to write motion equations for a ring that moves on the flat surface, but for me hard thing is to write equations for a paraboloid. for example, the normal constantly changes due to the geometry of the paraboloid. I had 3 angles for a flat surface, but I think I need another angle here and it is hard to connect this angle to the previous 3. what pieces of advice can you give me?
 A: 
this animation is result  of simulation the equation of motion, I used Euler- Lagrange with non holonomic constraint equation (rolling condition).

Ring rolling on paraboloid surface
Paraboloid parameter equation
$$\boldsymbol R= \begin{bmatrix}
         x(\lambda~,\vartheta) \\
         y(\lambda~,\vartheta) \\
         z(\lambda~,\vartheta) \\
       \end{bmatrix}=
\left[ \begin {array}{c} a\sqrt {{\frac {\lambda}{h}}}\cos \left(
\vartheta  \right) \\ a\sqrt {{\frac {\lambda}{h}}}
\sin \left( \vartheta  \right) \\ \lambda
\end {array} \right]$$

*

*$a~$ Paraboloid radius

*$h~$ Paraboloid height

*$\lambda~> 0$ Paraboloid parameter

*$\vartheta~\in [0~,2\,\pi]~$ Paraboloid parameter


the contact point is located at the plane that created by the tangential vector to the line $~\lambda~$, vector $~\boldsymbol t_\lambda~$ and to the line $~\vartheta~$ ,vector $~\boldsymbol t_\vartheta$
with:
\begin{align*}
&\boldsymbol t_\lambda=\frac{\partial \boldsymbol R}{\partial \lambda}=\left[ \begin {array}{c} {\frac {a\cos \left( \vartheta  \right) }{
\sqrt {4\,\lambda\,h+{a}^{2}}}}\\  {\frac {a\sin
 \left( \vartheta  \right) }{\sqrt {4\,\lambda\,h+{a}^{2}}}}
\\  2\,{\frac {\sqrt {\lambda}\sqrt {h}}{\sqrt {4\,
\lambda\,h+{a}^{2}}}}\end {array} \right]~,
\boldsymbol t_\vartheta=\frac{\partial \boldsymbol R}{\partial \vartheta}=\left[ \begin {array}{c} -\sin \left( \vartheta  \right)
\\  \cos \left( \vartheta  \right)
\\  0\end {array} \right]~,
\boldsymbol t_n=\boldsymbol t_\lambda\times\boldsymbol t_\vartheta
\end{align*}
we put at the ring center the coordinate system $~\boldsymbol\xi_n~,\boldsymbol\xi_\lambda~,\boldsymbol\xi_\varphi$ the disk can rotate  about  $~\boldsymbol\xi_\varphi~$ axes with the angle $~\varphi(t)~$ , and  additional rotate with constant angle $~\psi~$ about the $~\boldsymbol\xi_n~$ axes.
obtaining the $~\xi~$ coordinate system
\begin{align*}
  &\boldsymbol\xi_\lambda=\boldsymbol S_\Psi\,\boldsymbol t_\lambda\\
  &\boldsymbol\xi_\varphi=\boldsymbol S_\Psi\,\boldsymbol t_\vartheta\\
  &\boldsymbol\xi_n=\boldsymbol S_\Psi\,\boldsymbol t_n\\
    &\text{with}\\
&\boldsymbol S_\Psi=
 \left[ \begin {array}{ccc} \cos \left( \psi \right) &-\sin \left(
\psi \right) &0\\  \sin \left( \psi \right) &\cos
 \left( \psi \right) &0\\  0&0&1\end {array} \right]\\
\end{align*}
thus the position vector to the center of the ring is
$\boldsymbol R_c=\boldsymbol R+\rho\,\boldsymbol \xi_n~$ where
$~\rho~$ is the disc radius.
The rolling condition (non holonomic )
\begin{align*}
  &g_n=\left[(\boldsymbol\omega\times\boldsymbol r)-\boldsymbol v\right]\cdot \boldsymbol \xi_\lambda=\left[\left(\dot\varphi\,\boldsymbol\xi_\varphi\times\
  \rho\,\boldsymbol\xi_n\right)-\boldsymbol v\right]\cdot \boldsymbol \xi_\lambda
=0\tag A\\
&\text{where}\\
&\boldsymbol v=\frac{\partial \boldsymbol R_c}{\partial \boldsymbol q}\,\boldsymbol{\dot{q}}\\
&\text{and}\\
&\boldsymbol q=\begin{bmatrix}
        \lambda \\
        \vartheta \\
\varphi
      \end{bmatrix}
\end{align*}
Kinetic and potential energy
\begin{align*}
 &\text{kinetic energy}\\
 &T=\frac{m}{2}\,\boldsymbol v\cdot\,\boldsymbol v+\frac{I_\varphi}{2}\dot{\varphi}^2\\
 &\text{potential energy}\\
 &U=-m\,g\,\left(\boldsymbol{R}_c\right)_z\\
\end{align*}
Euler Lagrange with non-holonomic constraint equations
\begin{align*}
&\mathcal{L}  =T-U\\
   &\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{\boldsymbol{q}}}\right)-
  \frac{\partial \mathcal{L}}{\partial \boldsymbol{q}}   =\left[\frac{\partial \boldsymbol{R}_c}{\partial \boldsymbol{q}}\right]^T\boldsymbol{f}_a+
   \left(\frac{\partial \boldsymbol{g}_n}{\partial \dot{\boldsymbol{q}}}\right)^T\boldsymbol{\chi}\tag 1
 \end{align*}
\begin{align*}
  &\ddot{\boldsymbol{g}}_n=\frac{\partial \boldsymbol{g}_n}{\partial \dot{\boldsymbol{q}}}\ddot{\boldsymbol{q}}+\frac{\partial \boldsymbol{g}_n}{\partial \boldsymbol{q}}\dot{\boldsymbol{q}}\tag 2
 \end{align*}
Where $~\boldsymbol g_n~$ ist the non holonomic equation (Eq. A).
equations (1) and (2) are 4 equations for 4 unknowns, the solution give you the differential equations $\ddot\lambda~,\ddot\vartheta~,\ddot\varphi~$ and
the constraint force $~\chi $.
