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2answers
48 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
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0answers
17 views

Definiton of invariant curve

What is the definition(s) of an invariant curve? What book should i read to get a better idea of their use in dynamical systems. Are there any defining features i should be aware of especially with ...
1
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0answers
23 views

Constants of motion for non-chaotric billiards

As I understand it, for a mechanical system, each symmetry leads to a constant of motion. For integrable systems, the number of constants of motion equals the number of degrees of freedom. So take 2D ...
2
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1answer
47 views

Turning a rhombus billiard into an equivalent barrier billiard

I am reading a research paper and the authors map the rhombus billiard (angles $60$-$120$ degrees) to an equivalent barrier billiard. They start with a rhombus standing upright and reflect in a ...
3
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1answer
73 views

Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
5
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1answer
122 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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1answer
73 views

Tool for generating mathematical billiards

i'm currently looking for some program which can generate arbitrary mathematical billiards and trajectories inside them. Any help would be much appreciated! Thanks/ Mats
19
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2answers
3k views

Perfectly centered break of a perfectly aligned pool ball rack

This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed). ...
3
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1answer
83 views

The forever moving billiard ball

Suppose I have a rectangular table, dimensions $x$ by $y$, and a billiard ball is positioned in the very center. For descriptive convenience, let us impose a coordinate system on this table with an ...