Nonlinear modes of vibration

摘要

One of the most fundamental results in classical mechanics is that linear systems with n degrees of freedom have n fundamental modes of vibration, and that any motion of the systems can be obtained as a linear combination of these fundamental modes. This combination principle does not hold for nonlinear systems, but one suspects that a few periodic solutions will play a part akin to the fundamental modes. In one case, that suspicion has now been confirmed: H. Hofer, K. Wysocki and E. Zehnder have proved that so-called convex systems with two degrees of freedom must have either two periodic orbits for a given energy, or infinitely many — they cannot have three, or twenty thousand, or abillion.