Transcript Equilibrium1 - Department of Mathematics

EFFICIENCY & EQUILIBRIUM
Lecture 1 : Geometry – Nature’s Poetry
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
SPS2171 Presentation 21/01/2008
What do you see in this soap bubble ?
How do bubbles differ from films ?
How do the bubbles pack together ?
Where did this design originate ?
What angles do these sufaces make ?
What angles do the surfaces make ?
Compare these surfaces
with these ?
Where did these designs originate ?
Equilibrium
ball is in a (stable) equilribium when the net force on
it (gravity + constraint) = zero (hence it is stationary)
height
constraint
equilibria
Efficiency Principles
describe the behavior of physical, chemical,
biological, and social systems
systems behave so as to minimize “resources”
within constraints
balls move to minimize their local height
soap surfaces move to minimize their local area
bees and cells build (hives and skeletons)
to minimize material
Principle of Minimum Distance
A thrifty bee located at point A wants to get a drink
(on the planar surface S of a pool) before flying to
point B, what point P on the surface S should it fly to
Minimize : d  distance (A  P)  distance (P  B)
A
plane S
B
~
B
P
reflection of B through S
~
Thinks : d  distance(A  P)  distance(P  B)
~
Decides : fly straight t owards B
Principle of Minimum Distance
~
Our thrifty bee will fly in straight towards B to the
pools surface at P, drink, then fly straight to B.
1. line segments AP and PB lie in a normal plane to S
2. incidence angle APA’ = reflection angle BPB’
A
plane S
B
P
A'
B'
~
B
reflection of B through S
330 BC our enlightened bee is reincarnated as Euclid,
writes The Elements, and (reputedly) in the Catoprica
states that rules 1 & 2 govern the refection of light
Principle of Minimum Distance
Euclid’s laws of reflection were well understood by
Archimedes who, in 214BC used parabolic mirrors
to ignite Roman warships during the 2nd Punic War.
About 100AD Heron states his law that “light must
always take the shortest path” (hmm – what took them
400 years to discover something obvious to a bee?)
Conic Sections
Appolonius 262-190BC studied conic sections,
curves (ellipses, parabolas, and hyberbolas) formed
by intersecting planes with conical surfaces.


Their optical properties were long understood.
1605 Johannes Kepler discovers his three laws, the
first states that planets move in ellipses. This leads
Newton to develop his theory of gravitation which
he combines with his calculus to show that all
celestial bodies move in conic orbits (nonbound
objects move in parabolas and hyperbolas).
Principle of Minimum Action
1710 Gottfried Wilhelm Leibniz publishes “On the
Kindness of God, the Freedom of Man, and the
Origin of Evil” in which he develops the principle
that our world is organized to be
the best of all possible worlds
He also states in a letter the following principle :
nature always minimizes action
action  energy  time
Principle of Minimum Action
This principle, mistakenly attributed to Maupertuis,
implies that a particle moving (only) under its own
inertia and constrained to move on a surface, moves
with constant speed along a path that minimizes the
distance between any two points on the path.
These paths are called geodesics. For motion on
a plane they are straight lines, for motion on a
spherical surface they are arcs of great circles.
Euler, Lagrange, and Hamilton developed the
calculus of variations and classical mechanics from
this principle which underlines Einstein’s relativity.
Steiner’s Problem and 120 Degree Angles
Choose P to minimize dist (AP)+dist (BP)+dist (CP)
C
P
A
pages 92-100 in The Parsimonius Universe,
Steffan Hildebrandt and Anthony Tromba
B
Tutorial Problems
1. Light rays change direction when they traverse
interfaces between substances such as air and water.
Explain this phenomena using a minimum principle.
2. Make a conical surface out of paper, place two dots
on it and then draw a geodesic connecting them. Test
your drawing by cutting and flattening the cone.
3. Build a physical devise to choose point P in
Steiner’s Problem. Hint: use 3 rings fastened on the
edges of a table at vertices A,B,C; 3 equal weights
attached to 3 strings that are tied at a single knot.
4. Investigate & Explain : mean curvature, minimal
surfaces, surface tension, 1st Fields Medal problem.