Norton's dome is described here: The Norton's dome suggesting non-determinism in classical physics. Basically, it is a dome with a ball on top. The dome is shaped such that the ball can start rolling down at any future time.
Using a dome is unnecessary and causes confusion, such as shown in this blog post which talks about the "nonconservation of energy" and "flying off the dome", both of which are irrelevant to demonstrating indeterminism in Newtonian mechanics.
The core of the argument can be reduced to a particle moving on $[0, \infty)$: Let a particle of mass $m=1$ be at the origin. There is a central force field with force $F(x)$, then the particle's equation of motion $x(t)$ satisfies $x''=F(x(t))$.
The Nortan situation happens when there exists $F(x), x(t), t_0$, such that $x(t) = 0$ for all $t<t_0$, but $x(t) > 0$ for some $t \geq t_0$.
Norton's dome was designed so that $$F(x) = \sqrt{x}, x(t) = \frac{1}{144} (t-t_0)^4$$
The objection given by Malament in Norton’s Slippery Slope are:
that $F$ is not smooth enough, because it's not Lipschitz.
That the phase space of the ball's motion is incomplete, because the ball could "fly off the dome".
By using a central force field directly, and not using a dome at all, objection 2 is gone. Objection 1 can be dealt with by using a smoother function than $\sqrt{x}$.
We can work backwards by starting with some $x(t)= g(t)$, then calculate what $F$ must be: $F(t) = F(g(t)) = g''(t)$, so that $$F = g''\circ g^{-1}$$
An example would be for $g(t) = \cases{0 \text{, when } t\leq0\\ e^{-1/t^2}\text{, when } t>0}$. Then it's easy to calculate that the corresponding central force is $$F(x) = x ((\ln{x})^3+3(\ln{x})^2)$$.
This $F$ is Lipschitz-continuous and $C^1$, nullifying objection 1.
Presumably, Malament would object that $F$ is not smooth. So here's the question I want to know: Is there an invertible function $g:[0,\infty)\rightarrow [0,\infty)$, such that $g$ and $g''\circ g^{-1}$ are both smooth?
