Path between two points in rolling terrain

[Loren Booda]

Show that there exists at least one trajectory connecting any two points on a frictionless, smooth (albeit curved) terrain.

 

[CaptainQuasar]

This actually sounds like more of a mathematics question, a proof even, rather than a physics question. Are you talking about a 2D terrain, a 3D terrain? Is a trajectory a parabola or something? What would constitute a path between two points that is not a trajectory?

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[arildno]

I don't understand what you are asking.
A connected set is not necessarily path-connected.

 

[Loren Booda]

CaptainQuasar

Take for instance an idealized putting green, one without friction or discontinuities, but with any combination of curvatures. Is it always possible to putt from one point on the green so that the ball rolls directly over any other point thereon? I may be overlooking the obvious.

arildno

A connected set is not necessarily path-connected.

Would it be in this case?

 

[Loren Booda]

arildno,

I now understand what you are saying. You are quite right. I withdraw my question.

 

[CaptainQuasar]

Ah, I see. So in this case a trajectory is a path that remains in contact with the surface and behaves like a particle with a gravity-like force applied to it? Is this a particular meaning of “trajectory” or am I ignorant of the real definition of that word?

I don't know if this is an answer or a misunderstanding of the problem, but in your putting example if there were any overhangs the golf ball would leave the surface, so in that sort of curved surface there would definitely be points which the golf ball could not roll over.

I'm not making some sort of objection to the problem, as arildno may think, I'm really not understanding it.

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[Ben Niehoff]

I think this is the most rigorous way to state the question:

Given some simply-connected, smooth, 2-dimensional metric space M and a point P on M, does every point of M have a geodesic that also passes through P?

 

[CaptainQuasar]

Ah, so as I suspected, this is essentially a topology problem that would require a proof as an answer? I've never studied topology so even if this is a trivial problem I'd have some learning to do to answer it but it looks interesting. Just looking up the words you used was very helpful, Ben. But that's basically what I meant by saying it looks like a math problem instead of a physics problem.

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[Loren Booda]

Let's adopt Ben's conjecture. Thanks for your clarity, Ben.