Non-integrability and chaos of a conservative compound pendulum

By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e., the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.