We show here that a recently developed criterion for the stability of conservative Hamiltonian systems can be extended to Hamiltonians with weak time dependence. In this method, the geodesic equations contain the Hamilton equations of the original potential model through an inverse map in the tangent space in terms of a geometric embedding. The second covariant derivative of the geodesic deviation generates a dynamical curvature, resulting in a (energy dependent) local criterion for unstable behavior different from the usual Lyapunov criterion. We show by direct simulation that our geometrical criterion predicts correctly the stability/instability of motions, sometimes contrary to indications of the local Lyapunov method.
Received 22 November 2012Accepted 15 May 2013Published online 05 June 2013
Lead Paragraph: In this paper, we discuss a geometrical embedding method for the analysis of the stability of time-dependent Hamiltonian systems using geometrical techniques familiar from general relativity. This method has proven to be very effective in numerous examples, predicting correctly the stability/instability of motions, sometimes contrary to indications of the Lyapunov method. For example, we show that although the application of local Lyapunov analysis predicts the completely integrable Kepler motion to be unstable, this geometrical analysis predicts the observed stability. The general theory of the structure and application of this method for the time-dependent potential problem is given in the present paper, with criteria for its applicability in these cases. As examples, we study in more detail the restricted three body problem as well as to the two-dimensional Duffing oscillator with a time-dependent coefficient. The first represents an important class of geophysical and astrophysical problems, and the second, the result of a perturbed bistable system with analogues in electric circuit theory as well. In both cases, the local Lyapunov analysis fails to predict the correct limiting behaviour.
We wish to thank S. Shnider, Eran Kalderon, A. Belenkiy, P. Leifer, I. Aharonovitch, and Avi Gershon for helpful discussions.
Article outline: I. INTRODUCTION II. REVIEW OF STABILITY ANALYSIS FOR TIME INDEPENDENT HAMILTONIANS III. GENERALIZATION TO WEAKLY TIME-DEPENDENT POTENTIALS IV. EXAMPLES A. The time-dependent 2-dimensional Duffing oscillator B. The restricted three-body problem V. CONCLUSIONS
12.D. Anosov, “ Geodesic flows on closed Riemannian manifolds with negative curvature,” in Proceedings of the Steklov Institute of Mathematics (American Mathematical Society, Providence, Rhode Island, 1969), Vol. 90.
13.A. Neishtadt and A. Vasiliev, Chaos17, 043104 (2007).
21.The notation corresponds to the quantities in accordance with our interpretation of the Hamilton manifold.
22.Note that performing parallel transport on the local flat tangent space of the Gutzwiller manifold (for which and gij are compatible), the resulting connection, after raising the tensor index (as in (11)) to reach the Hamilton manifold, is exactly the “truncated” connection (17).
23.Substituting the conformal metric (5) into (18), and taking into account the constraint that both trajectories and xℓ have the same energy E, one sees that (18) becomes the orbit deviation equation based on (5), the Hamilton equations generated by H. Thus, the introduction of the second covariant derivative (20) in the framework of the embedding general geometric structure carries more information on stability than the stability analysis applied to the Hamilton equations derived directly from (1) (which, as it is easy to see, corresponds to a Lyapunov analysis).
24.A. Gershon and L. P. Horwitz, J. Math. Phys.50, 102704 (2009);
24.L. P. Horwitz, A. Gershon, and M. Schiffer, Found. Phys.41, 141 (2011).
27.W. Siegert, Local Lyapunov Exponents, Lecture Notes in Mathematics Vol. 1963 (Springer, 2009).
28.It has been shown by E. Kalderon, L. P. Horwitz, R. Kupferman and S. Shnider, Chaos 23, 013120 (2013). “On a geometrical characterization of chaos in Hamiltonian dynamics” (submitted), that, although the relation (11) is not integrable, the result (23) can be obtained rigorously.
29.Y. B. Zion and L. P. Horwitz, Phys. Rev. E81, 046217 (2010).