Edwards's Solution of Pendulum Oscillation

image

energy surface

position and momentum

time double helix

Hamiltonian, α = 2 H =

2

p

+

2

q

- (q p

2

)\),



2

p

+

2

q

 1 - 1/4

2

q



2

p

+ 1/2 (1 - cos(2 q))

energy α

0.5

time evolution

0.

initial condition

0.

The masterful derivation by Harold Edwards [1] finally brings the vision of Abel to the wider audience it deserves. In addition to its elegance, the article provides a constructive approach for improving computations along elliptic curves. The new and simple addition rules have been widely appreciated, although less so for the ingenious

ψ

function introduced in [1, Section 15]. As with the much earlier Weierstrass

℘

function, the Edwards

ψ

function determines time-dependent solutions for a range of interesting Hamiltonian systems [2]. This Demonstration shows three interrelated examples, including one that describes the oscillation of a plane pendulum. The Edwards

ψ

function is truly an amazing and beautiful, doubly periodic, meromorphic function!