The Geometry of Phase Space

Geometry of Hamiltonian Chaos

L. Horwitz, et al.

PRL 98, 234302 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e234302

This group from Israel has recast classical Hamiltonian dynamics in the language of differential geometry. In doing so, they have discovered a new criterion for chaotic motion: the deviation of parallel geodesics. The geodesics of the system are classical trajectories. If nearby geodesics diverge, it indicates an instability in the system. This is a different measure than the Lyapunov exponent, although it’s possible there is a deeper connection between the two. I would speculate that the Lyapunov exponents could be derived from this geometric theory.

The general idea behind the method is to rewrite the Hamiltonian using a metric in the space of conjugate momenta. From this metric, one can derive the connection and curvature. From the curvature, one can calculate geodesic deviation. However, the authors find that the geodesic equation and the Hamiltonian equations of motion do not agree. It turns out that the geodesics in the dual manifold (defined by dx^{i} = g^{ij} dx_{j}) do indeed agree with the Hamiltonian equations of motion. The authors simply state this fact without discussion. There must be a deeper significance of this result. Or maybe the authors just started their calculation on the wrong manifold. Regardless of the connection, the geodesics on the dual manifold and the Hamiltonian equations of motion are mathematically equivalent; thus, one has a geometric way of thinking about phase space trajectories.

The new connection is anti-symmetric, so there is torsion. I don’t know much about torsion. In my course on general relativity, one of the postulates we used in deriving Einstein’s equations was the “no torsion condition.” It has something to do with twist of the geodesics. It is irrelevant in the derivation of the authors because the antisymmetric terms cancel from the geodesic equation, leaving only a symmetric connection.

In the end, the authors find that the curvature derived from this connection determines the stability of an orbit. If the curvature is negative, geodesics diverge and the system is chaotic. If curvature is positive, geodesics converge, and the system is stable. This seems like a very useful tool for analyzing unstable systems. The authors point out that this approach does not require one to use approximations of the true equations of motion, unlike other methods.

The idea behind the paper is quite interesting. My one complaint with the paper is that the phrase “a complete discussion will be given elsewhere” was used at least four times, but the authors don’t indicate where. It would be interesting to see these discussions, as well as a discussion of the connection between the curvature studied by the authors and the Lyapunov exponents.