Linear stability test for Hamiltonian orbits

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Abstract
The following test can distinguish regular from irregular orbits, even in marginal cases. A small sphere in phase space, initially centered at the starting point of the orbit, evolves into an ellipsoid by virtue of the Hamiltonian equations of motion. This ellipsoid rotates and vibrates "erratically" as its center moves along the classical trajectory. However, the proposed test involves only consecutive shapes of the ellipsoid as the orbit happens to pass in the vicinity of its initial point. The major axis of the ellipsoid then increases with time in a way which is roughly linear for regular orbits, and roughly exponential for irregular ones. Moreover, if the orbit is regular, the behavior of the angle between the major axis and the tangent to the trajectory indicates the smoothness of the invariant torus containing the orbit. This angle tends to a constant if the torus is very smooth, and oscillates if the torus has a complicated shape in the vicinity of the initial point.