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. Zehnder1 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 a billion.
You should try to visualize this in its four-dimensional phase space (p 1, p 2, q 1, q 2), where p is momentum and q is position. The potential energy of the system is V = 2π2(q 12/ T 12 + q 22/ T 22), and its total energy, or Hamiltonian, is H = 1/2(p 12 + p 22) + V. The surfaces of constant energy are three-dimensional ellipsoids, around which wind the trajectories of the motion. If T 1/ T 2is rational, all these trajectories are closed, corresponding to periodic motions. If T 1/ T 2is irrational, then there are only two closed trajectories for any energy, corresponding to the two fundamental modes of vibration; all other trajectories are open, filling the whole constant-energy surface. This picture extends simply to any number of degrees of freedom.
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