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• [Professor Masur]: But I thought I would mention one other

• problem that arises that you can ask

• what's called a periodic orbit? When I

• say orbit, that might be "start off at

• a point, you bounce off the walls and

• after some long number of times; can you

• always find a path that repeats itself?"

• So let me give one very, very classical

• example of that and that goes back to

• the squareif you start, let's say on the

• bottom, and move off at a right angle,

• 90 degrees, you move to the top. Now when you

• get to the top, because these are

• parallel, this angle is also 90 degrees,

• and so the billiard path says you

• rebound where angle of reflection equals

• angle of incidence, and that means that

• when you come to the side you will

• bounce off exactly the way that you came

• in; you will come to the side and

• bounce right back. You come back along

• this line and then you will bounce and

• repeat yourselfso that's what's called

• periodic. There are other periodic lines.

• So, for example, you could start at a

• 45-degree angle, everything would be

• 45 degrees, and you would come back right

• where you started and then repeat

• yourself. In fact, if you started any

• point, at any angle that is a rational

• number – a quotient of two integersif you started with an angle of 27 degrees

• starting here, you will come back after a certain number of bounces.

• That's true for a square; and, in fact, one can prove

• for any rational polygon, were all the angles are fractions of 180 degrees,

• or P over Q times 180 degrees,

• again I like to write it like that,

• you can always find a periodic orbit.

• [Brady]: From any point? [Professor Masur]: No not from any point.

• There's some periodic orbit, from somewhere.

• Some periodic orbit from somewhere.

• There might be; so there will be

• some directSome point, in some

• direction, where there will be a periodic orbit.

• [Brady:] For example on this side there are none, but there is one sitting over here somewhere.

• [Professor Masur:] Well usually what happens ifIt, it

• In most cases, thatThat's possible;

• in fact you can always find periodic

• orbits that will hit every side. In most

• cases a periodic orbit will hit a side.

• There might be other points on that side,

• where the orbit in that same direction, is not periodic.

• So in other words, you start at a point on a side and move in a direction

• it might close up, meaning periodic,

• but if you move the point a little bit and headed off in the same direction,

• what could happen is that it isn't periodic;

• what might happen is it hits a corner

• and then you don't know what to do.

• That's different from the square. In the square no matter where I

• started, if I went at 45 degrees, it would be periodic,

• but in a general,

• rational polygon that may not be the case.

• I want to talk about triangles now

• that are not necessarily rational, but just triangles.

• If they are acute,

• which means all of the angles are 90 degrees or less,

• what you could do is: you could take the vertex of the triangle

• and drop a perpendicular down to the opposite side to get a point,

• and then drop this perpendicular down to the opposite side and get a point so these are right angles.

• This triangle joining those points gives a periodic

• orbit; meaning if I start here I go to

• the line towards here this angle will

• equal that angle and I'll bounce like that.

• And then i'll get to this point and

• this angle will equal this angle and

• I'll bounce like that.

• And I come back, this angle equal this angle, and I'll repeat myself.

• This example was has been known for hundreds of years.

• I don't think it was ever thought of as billiards, but anybody in high school or

• even junior high could do this inIn trigonometry.

• When you do obtuse triangles,

• you can't drop a perpendicular because if I started at this vertex and drop the perpendicular,

• it wouldn't be on that side.

• That's because this angle is bigger than 90 degrees and, in fact,

• for something as very simple as in a

• triangle, obtuse triangle, one does not

• know whether there are periodic orbits.

• So this is a famous unsolved problem in

• the subject of, what this is called is, dynamical systems.

• Well so again if the obtuse triangle is rational,

• then there are periodic orbits.

• So I'm talking about obtuse triangles. I think I gave an example

• 90 times square root of twothat particular one I would

• not be able to tell you if it had a periodic orbit.

• There are some

• Some reason where people, by very, very hard

• work, have shown where if this angle is less than a hundred degrees,

• then there are periodic orbits. But that was very,

• very hard work and if it was a hundred and ten degrees;

• again I'm assuming not rational;

• then we don't know the answer.

• If it's rational we always know, if all the

• angles, if this is a hundred and eleven,

• and this is 36.

• What's left? 33 degrees there will be.

• But if one of the angles is an irrational

• number then we don't know.

• [Brady:] So what, does greatness await if someone can crack that?

• [Professor Masur:] Well, it's… It's um

• There is a very prominent, one of the most prominent

• mathematicians in the United States or

• in the world in the subject of

• dynamical systems whose name is

• Professor Katok at Penn State has

• listed this as one of the five

• outstanding problems in dynamical systems.

• But really there's nothing you

• could do. You can try desperately to

• solve it, but if it hasn't been solved

• for a hundred years, you probably aren't going to.

• And you know it's only given to one

• person, so to speak, to solve a particular

• one of these problems. So we're used to

• it and here's an atmosphere of resignation.

[Professor Masur]: But I thought I would mention one other

# 週期性軌道的問題 - Numberphile(數字愛好者) (Problems with Periodic Orbits - Numberphile)

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林宜悉 發佈於 2021 年 01 月 14 日