Lecture notes - Institut für Mathematik - TU Berlin

12 ULRICH PINKALL (NOTES BY G. PAUL PETERS)Let ω denote the angle of rotation represented by λ, then one maycheck thatH(ϕ n−1 , ϕ n ) = (1 + tan 2 δ 2 )(1 + tan2 ˜δ2 )(tan ω 2 + 1)−1 .The evolution T of the Verlet–integrator of the pendulum equation (VP)is the so called standard map of chaos theory, which should be interpretedin the context of the KAM–theorem as a perturbation of theintegrable discrete pendulum equation (DP).Figures 1 and 2 show (for K = .04 and K = .3, respectively) orbitsin phase space obtained by the Runge–Kutta method, the Verlet–integrator (VP), and the integrable discrete pendulum equation (DP).The integrability of (DP) expressed in Theorem 9.4 implies that itsorbits have no choice but to lie on the level sets of the constant of themotion H.□Figure 1. Phase space of the pendulum equation(Runge–Kutta, VP, DP)Figure 2. Phase space of the pendulum equation(Runge–Kutta, VP, DP)Institut für Mathematik, Technische Universität Berlin, Straße des17. Juni 136, 10623 Berlin, GermanyE-mail address: peters@math.tu-berlin.de, pinkall@math.tu-berlin.de