The Mathematics of Billiards Washington University Math Circle - - PowerPoint PPT Presentation

▶
the mathematics of billiards
SMART_READER_LITE
LIVE PREVIEW

The Mathematics of Billiards Washington University Math Circle - - PowerPoint PPT Presentation

Jun 07, 2023 114 likes •602 views

The Mathematics of Billiards Washington University Math Circle Chris Cox March 6, 2016 Chris Cox Wash U Stl The Mathematics of Billiards One thing you could do but we wont: play real billiards! Chris Cox Wash U Stl The Mathematics

slide-1
SLIDE 1

The Mathematics of Billiards

Washington University Math Circle Chris Cox March 6, 2016

Chris Cox Wash U Stl The Mathematics of Billiards

slide-2
SLIDE 2

One thing you could do but we won’t: play real billiards!

Chris Cox Wash U Stl The Mathematics of Billiards

slide-3
SLIDE 3

Instead, focus on one key idea: “specular reflection”

Chris Cox Wash U Stl The Mathematics of Billiards

slide-4
SLIDE 4

Warm up problem:

Draw the path of ONE billiard on a rectangular table through several collisions. (1)

Chris Cox Wash U Stl The Mathematics of Billiards

slide-5
SLIDE 5

Warm up problem:

Draw the path of a billiard on a rectangular table through several collisions.

Chris Cox Wash U Stl The Mathematics of Billiards

slide-6
SLIDE 6

Warm up problem:

Draw the path of a billiard on a rectangular table through several collisions.

Chris Cox Wash U Stl The Mathematics of Billiards

slide-7
SLIDE 7

Warm up problem:

Draw the path of a billiard on a rectangular table through several collisions.

Chris Cox Wash U Stl The Mathematics of Billiards

slide-8
SLIDE 8

Rectangle billiards

One type of behavior: 41 collisions

Chris Cox Wash U Stl The Mathematics of Billiards

slide-9
SLIDE 9

Rectangle billiards

One type of behavior: 221 collisions

Chris Cox Wash U Stl The Mathematics of Billiards

slide-10
SLIDE 10

Rectangle billiards

One type of behavior: 362 collisions

Chris Cox Wash U Stl The Mathematics of Billiards

slide-11
SLIDE 11

Rectangle billiards

Another type: “periodic orbit” of period 10

Chris Cox Wash U Stl The Mathematics of Billiards

slide-12
SLIDE 12

Specular collisions

Next problem: some more interesting shapes!

Chris Cox Wash U Stl The Mathematics of Billiards

slide-13
SLIDE 13

Rectangle billiards

Actually the first one might be periodic too:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-14
SLIDE 14

Specular collisions

Next problem: some more interesting shapes!

Chris Cox Wash U Stl The Mathematics of Billiards

slide-15
SLIDE 15

Triangle billiards

Chris Cox Wash U Stl The Mathematics of Billiards

slide-16
SLIDE 16

Specular collisions

Things to notice:

▸ The model works for curved

boundaries, not just lines

▸ This model assumes there is no

friction, no loss of energy, and no spinning

Chris Cox Wash U Stl The Mathematics of Billiards

slide-17
SLIDE 17

Specular collisions

For curves, we use the tangent line:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-18
SLIDE 18

Moon billiards

Chris Cox Wash U Stl The Mathematics of Billiards

slide-19
SLIDE 19

Worksheet Question 4: What do billiards look like in a circular table?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-20
SLIDE 20

Worksheet Question 4: What do billiards look like in a circular table?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-21
SLIDE 21

Question 1: What do billiards look like in a circular table?

That one looks like this:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-22
SLIDE 22

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-23
SLIDE 23

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-24
SLIDE 24

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-25
SLIDE 25

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-26
SLIDE 26

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-27
SLIDE 27

Question 1: What do billiards look like in a circular table?

With different direction:

Chris Cox Wash U Stl The Mathematics of Billiards

slide-28
SLIDE 28

Question 2: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-29
SLIDE 29

Three types of behavior

▸ Periodic ▸ Nice but non-periodic ▸ Chaotic

Chris Cox Wash U Stl The Mathematics of Billiards

slide-30
SLIDE 30

Undergraduates Research Billiard Dynamics

Chris Cox Wash U Stl The Mathematics of Billiards

slide-31
SLIDE 31

Yakov Sinai awarded the Abel Prize

Chris Cox Wash U Stl The Mathematics of Billiards

slide-32
SLIDE 32

Can you make a periodic circle billiard?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-33
SLIDE 33

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-34
SLIDE 34

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-35
SLIDE 35

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-36
SLIDE 36

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-37
SLIDE 37

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-38
SLIDE 38

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-39
SLIDE 39

Question: What is wrong with my circle?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-40
SLIDE 40

Some really interesting tables

Worksheet question 6: ellipses and moons and stadia and mushrooms

Chris Cox Wash U Stl The Mathematics of Billiards

slide-41
SLIDE 41

The mushroom billiard

Figure by Carl Dettmann Chris Cox Wash U Stl The Mathematics of Billiards

slide-42
SLIDE 42

The ergodic game

How does our sampling rule change the “average”?

Chris Cox Wash U Stl The Mathematics of Billiards

slide-43
SLIDE 43

A new rule:“no-slip” circles

Chris Cox Wash U Stl The Mathematics of Billiards

slide-44
SLIDE 44

Applications of specular collisions

This simple model has many applications:

▸ Modeling fluids (Lorentz gases)

Chris Cox Wash U Stl The Mathematics of Billiards

slide-45
SLIDE 45

Applications of specular collisions

This simple model has many applications:

▸ Modeling fluids (Lorentz gases) ▸ Brownian motion

Chris Cox Wash U Stl The Mathematics of Billiards

slide-46
SLIDE 46

Applications of specular collisions

This simple model has many applications:

▸ Modeling fluids (Lorentz gases) ▸ Brownian motion ▸ Heat transfer (How do things cool off?)

Chris Cox Wash U Stl The Mathematics of Billiards

slide-47
SLIDE 47

Applications of specular collisions

This simple model has many applications:

▸ Modeling fluids (Lorentz gases) ▸ Brownian motion ▸ Heat transfer (How do things cool off?) ▸ Diffusion (How do mixtures spread out?)

Chris Cox Wash U Stl The Mathematics of Billiards