Analytical study of the critical behavior of the nonlinear pendulum

摘要

The dynamics of the simple pendulum, a small bob ofnmass m tied to a fixed point O by a massless, rigid rod ofnlength u0001, is analyzed in most introductory physics textbooks.nWhen it oscillates with a small amplitude, the approximationnsin u0001u0005u0001, where u0001 is the angular displacement from the vertical,nyields a linear equation of motion whose solution is ansinusoidal function of time with a constant period T0n=2u0002u0006u0001/g, with g the local acceleration of gravity.1 This approximatensolution works well for u0001u00037°, for which T0 givesnan error less than 0.1% compared to the exact period.2–4 Fornlarger amplitudes the nonlinear nature of the pendulum oscillationsnbecomes apparent. Although the sinusoidal solutionnremains a good approximation to the exact solution of thennonlinear equation of motion for small amplitudes, the periodnincreases rapidly with the amplitude.5