Paraboloid with Friction

@Karan Chatrath has posted a nice solution summary. I will post mine as well, just to give some extra coverage and perspective.

Derive an acceleration constraint equation:

$z = x^2 + y^2 \\ \dot{z} = 2 x \dot{x} + 2 y \dot{y} \\ \ddot{z} = 2x \ddot{x} + 2 y \ddot{y} + 2 \dot{x}^2 + 2 \dot{y}^2$

Now write the Newton's Second Law equations. In these, $\vec{u}_1$ is the normal vector which can be calculated as explained at this link , and $\vec{u}_2$ is a unit vector in the opposite direction as the particle velocity. $\vec{F}_g$ is the gravitational force. $N$ is the magnitude of the normal force.

$\ddot{x} = F_{g x} + N u_{1x} + \mu N u_{2x} \\ \ddot{y} = F_{g y} + N u_{1y} + \mu N u_{2y} \\ \ddot{z} = F_{g z} + N u_{1z} + \mu N u_{2z}$

Plug these three equations into the acceleration constraint equation to solve a single equation for $N$ . Then knowing $N$ , calculate the acceleration components. This approach requires a bit of algebraic manipulation, but does not require any calls of a linear algebra routine. Numerical integration takes care of the rest.

Consider the position of the particle to be:

$\vec{r} = x \ \hat{i} + y \ \hat{j} + (x^2 + y^2) \ \hat{k}$

$\implies \vec{v} = \frac{d\vec{r}}{dt} = \dot{x} \ \hat{i} + \dot{y} \ \hat{j} + (2x\dot{x} + 2y\dot{y}) \ \hat{k}$

The unit velocity vector can be denoted as:

$\hat{v} = \frac{\vec{v}}{\lvert \vec{v}\rvert}$

A unit vector normal to the inner surface directed towards the increasing Z direction is:

$\Phi = z - x^2-y^2=0$ $\hat{n} = \frac{\nabla \Phi}{\lvert \nabla \Phi\rvert}$

Let the normal force magnitude be $N$ . Let the gravity vector be:

$\vec{g} = -10 \ \hat{k}$

The equations of motion of the particle according to Newton's law are:

$\boxed{m \frac{d\vec{v} }{dt} = m\vec{g} + N \ \hat{n} - \mu N \hat{v}}$

This gives rise to three equations and there are three unknowns which are $\ddot{x}$ , $\ddot{y}$ and $N$ . The equations are linear with respect to the unknown variables. The resulting equations are very large and require significant simplification, and are of a nonlinear nature, so they cannot be solved analytically. I have taken a numerical route. I have left out simplifications and the subsequent number crunching, in this solution. I will add those details later if requested.