How Does the Smaller Alignment Index (SALI) Distinguish Order from Chaos?

The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications. [Ch. Skokos, J. of Phys. A 34 (2001), 10029. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, in Proceedings of the 4th GRACM Congress on Computational Mechanics, ed. D. T. Tsahalis (Univ. Patras, Patras, 2002), Vol. IV, p. 1496; in Libration Point Orbits and Applications, ed. G. Gómez, M. W. Lo and J. J. Masdemont (World Scientific, 2003), in press, nlin.CD/0210053.] Basically it is observed that in chaotic regions the SALI goes to zero very rapidly, while it fluctuates around a nonzero value in ordered regions. In this paper, we make a first step forward explaining these results by studying in detail the evolution of small deviations from regular orbits lying on the invariant tori of an integrable 2D Hamiltonian system. We show that, in general, any two initial deviation vectors will eventually fall on the "tangent space" of the torus, pointing in different directions due to the different dynamics of the 2 integrals of motion, which means that the SALI (or the smaller angle between these vectors) will oscillate away from zero for all time.