> ...The problem is that for the globally-optimal route, the initial
motion is purely vertical,
> so for any finite coefficient of friction the ball will /slip/.
Actually, the optimal initial motion is vertical ONLY if the initial
velocity is zero
> Consider the globally optimal route from A to B. Obviously B must be
lower than A, and
> indeed the entire route must be > lower than A. The interesting
thing is that for /some/
> conditions, part of the route is even lower than B, which means the
vertical component
> of the motion is non-monotonic. (Note that this feature has been
missing from the
> piecewise-linear "ramp" models mentioned in this thread.) The
question is: What are
> the necessary and sufficient conditions for which the optimal route
spends part of its time
> below B? The usual spherical-cow approximations apply: Uniform
gravitational field,
> negligible friction, et cetera
Again for an initial velocity of zero only - I guess that a terminal
ascent phase occurs only
where the horizontal displacement of the target B exceeds the vertical
descent to B
by a ratio pi/2 (using the definition of the cycloid)