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 ?.
ball is in a (stable) equilribium when the net force on
it (gravity + constraint) = zero (hence it is stationary)
describe the behavior of physical, chemical, biological, and social systems
systems behave so as to minimize “resources”
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
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 toPrinciple of Minimum Distance
reflection of B through S
Our thrifty bee will fly in straight towards to the pools surface at P, drink, then fly straight to B.Principle of Minimum Distance
1. line segments AP and PB lie in a normal plane to S
2. incidence angle APA’ = reflection angle BPB’
reflection of B through S
330 BC our enlightened bee is reincarnated as Euclid, writes The Elements, and (reputedly) in the Catopricastates that rules 1 & 2 govern the refection of light
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?)
Appolonius 262-190BC studied conic sections, pools surface at P, drink, then fly straight to B.
curves (ellipses, parabolas, and hyberbolas)formed
by intersecting planes with conical surfaces.Conic Sections
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).
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
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.
Choose P to minimize dist (AP)+dist (BP)+dist (CP) pools surface at P, drink, then fly straight to B.Steiner’s Problem and 120 Degree Angles
pages 92-100 in The Parsimonius Universe,
Steffan Hildebrandt and Anthony Tromba
1. Light rays change direction when they traverse interfaces between substances such as air and water. Explain this phenomena using a minimum principle.Tutorial Problems
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.