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Re: Billiards
Posted:
Nov 22, 2009 4:22 PM
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> Hello,
> suppose to have a ball on a smooth rectangular billiard
> table. When the particle hits the boundary it reflects
> from it without loss of speed. I have two questions.
>
> (I) Given a point P on the billiard table, and a
> positive integer M, does there exist a trajectory
> P(t), with P(0)=P, such that the ball passes again
> through P after having hit the boundary at least M
> times? (I require, to be clear, that the trajectory
> passes again through P)
>
> (II) Given two distinct point P and Q, a direction u and
> a positive number e, does there exist a trajectory P(t),
> with P(0)=P, such P(t)=/=P for any t=/=0, and for some
> s>0 we have P(s)=Q with angle(u,v(s))<e?
>
> I think the answer is positive to both the questions,
> but unfortunately I couldn't give a proof.
>
> My Best Regards,
> Maury Barbato
Hint: tile the plane with billiard tables. Each is either
a translate of the original or a reflection (across the x or
y axis or both) of a translate. A trajectory corresponds to
a straight line in this plane.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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