Race to the Bottom Brachistochrones and Simple Harmonic Motion 1 Race to the Bottom

Least Time Path from A to B Race to the Bottom 2 x y A B g

The Brachistochrone Problem Greek for “shortest time” first expressed by Galileo solved by multiple Bernoulli’s led to calculus of variations and the Euler-Lagrange equations Race to the Bottom 3

Formulation Race to the Bottom 4

Formulation (cont.) Race to the Bottom 5

Formulation (cont.) The answer is a cycloid: Race to the Bottom 6

Some Example Times Race to the Bottom 7 ShapeTime straight line1.189 quadratic1.046 cubic1.019 ellipse1.007 cycloid1.003

Isochronous Race to the Bottom 8 x y A B g C Time from C to B is the same as the time from A to B

Simple Harmonic Motion Race to the Bottom 9 x=0

Disturb the Particle Race to the Bottom 10

Disturb the Particle (cont.) Race to the Bottom 11

Equation of Motion Race to the Bottom 12

Why Do We Obsess About SHM? The preceding line of argument applies to almost any system which is in equilibrium and is slightly disturbed. An extremely large class of systems: pendulums the Earth after an earthquake you and me stars SHM solutions show up everywhere Race to the Bottom 13

When Good Methods Fail The assumptions of SHM break down when the disturbance becomes too large. Consider a simple pendulum: Race to the Bottom 14

Large-amplitude Pendulum Race to the Bottom 15

Huygens Pendulum Race to the Bottom 16 period independent of amplitude

Pendulum in Phase Space Race to the Bottom 17