"Rolling Geodesics": Designing a $k$-putt green I am interested in what might be called rolling geodesics, paths
of physical particles confined to a surface in $\mathbb{R}^3$
under certain force conditions.
Here I will pose a specific (but imprecise)
recreational question that addresses some aspects these curves:
Design a wickedly difficult golf green.

         
         


Let $D \subset \mathbb{R}^2$ be a unit-radius disk in the plane.
Let $h(x,y)$ be a continuous, smooth function defined over $D$;
$h(x,y)$ represents the height of a golf green surface $S$ over $(x,y) \in D$.
At a point $c \in S$ is centered a golf hole/cup of diameter $\delta$ larger than
the diameter of a golf ball, which is sitting at some spot $b \in S$ on the green.
The hole should project strictly interior to $D$, as should $b$.
The challenge is to design the shape of the green's surface
$S=h(x,y)$ to make it extremely difficult to hole a putt.
A putt imparts an instantaneous velocity $v_0$ to the ball at $b$,
with the goal that the ball rolls under the influence of gravity and
rolling friction with the grass so that its point of contact ends up suspended
over the cup hole.  Assume that, regardless of its velocity $v(t)$
at that moment, the ball is "sunk," i.e., it falls into the cup.
Some additional assumptions:


*

*
The surface gradient is bounded, e.g., $|\nabla h| \le 1$.
(Perhaps 1 is too generous. Rarely is a green sloped more than $10^\circ$.)


* Friction suffices to hold a carefully placed
ball stationary-stable at any point $b \in S$.


* The initial velocity $v_0$ is bounded by the requirement that
the ball never lifts-off the surface airborn at any point.
(See the earlier MO question,
Gently falling functions.)
This rolling-constraint might be difficult to accommodate, in which
case an upperbound $v_{\max}$ could be assumed.


* Assume that $D$'s fringe is surrounded by such a deep drop-off that a
ball that leaves the domain of the green can never return.



Here are two specific questions:

Q1.  Is there a surface $S$
  with cup/pin placement $c \in S$ such that,
  for some starting point $b \in S$, it is impossible
  to 1-putt from $b$?

If the answer to Q1 is Yes, then:

Q2.
  Given any integer $k>1$, is there an $S$ and $c \in S$ that
  requires $\ge k$ putts to sink the ball from some $b \in S$ into the cup?

Insights under any assumptions (either more or less realistic) welcomed!
(Credit to Alejandro López-Ortiz, who posed several geometric
golf questions [but not this specific one] at a conference in August 2011.)
Addendum.  Here is Jaap Eldering's example:

         
         


 A: Some (heuristic) speculation, separate from my previous answer. I'm ignoring the velocity/loss of contact constraint and consider the ball as a point particle, ignoring angular momentum etc. as well.
I think that under these conditions it is always possible to make a 1-putt. Let me assume a sufficiently high initial speed $v_0$ such that the ball never falls still. The problem can then be viewed as a geodesic flow on $T^1 D$. At each point $x \in D \cong S$, the speed is determined by energy conservation, while the (unit) direction of the velocity is specified by $v \in T_x^1 D$. The metric on $T^1 D$ will be dermined by Newton's laws and the height function $h$.
Now let $b$ be the pin position and consider the geodesic flow on $T^1_b D$: the set of all possible directions in which we can shoot the ball. This is topologically a circle and will stay a circle under the continuous flow (assuming that $h$ is sufficiently smooth). Now my speculation is that this circle can be viewed as a "wavefront" that will keep expanding away from $b$ (since the velocity is high enough), and that it will have fully crossed the boundary of $D$ at some time. By continuity, each point on the green then must have been on (I think precisely one) geodesic curve.
A: Your constraint on the velocity that the ball may not loose contact with the surface leads to impossible putts: take a vulcano shaped green with the hole at the very top and put a small rim at the foot around the vulcano, for example, take a radially symmetric $h(r) = 10 \exp(-r^2) + \exp(-1000(r-0.8)^2)$. Place the ball (just) outside the rim. A minimum speed to putt is given by potential height energy. On the other hand, if the top of the rim is sharp enough (high second derivative of $h$), then the ball will lift off from the surface there.
You could add a constraint on second derivatives of $h$ to prevent such problems.
A: I wonder if surface like this could thwart a 1-putt, with an appropriate velocity limit
$v_0 \le v_{\max}$?

        


The cup $c$ is directly ahead of the ball at $b$, directly along the line indicated by the
arrow.  Clearly putting straight or left of that arrow sends the ball off to the left.
It seems that reasonable velocity conditions might ensure that angles $\epsilon$ right of the
arrow would still roll left of $c$ from the influence of the left slope of the second knoll,
and that aiming further rightward would roll off the right slope of that second knoll.
