Pendulum, elliptic functions, and relative cohomology classes

Abstract

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed and an identity between elliptic functions is found whose proof can be based on symplectic geometry and global relative cohomology. Alternatively it can be reduced to a well known identity between elliptic functions. Normal canonical action-angle variables are also constructed around the stable equilibrium and a corresponding identity is exhibited.