Invariant Tori through Direct Solution of the Hamilton-jacobi Equation*

We explore a method to compute invariant tori in phase space for classical non-integrable Hamiltonian systems. The procedure is to solve the HamiltonJacobi equation stated as a system of equations for Fourier coefficients of the generating function. The system is truncated to a finite number of Fourier modes and solved numerically by Newton’s method. The resulting canonical transformation serves to reduce greatly the non-integrable part of the Hamiltonian. Successive transformations computed on progressively larger mode sets would lead to exact invariant tori, according to the argument of Kolmogorov, Arnol’d, and Moser (KAM). The procedure accelerates the original KAM algorithm since each truncated Hamilton-Jacobi equation is solved accurately, rather than in lowest order. In examples studied to date the convergence properties of the method are excellent. One can include enough modes at the first stage to get accurate results with only one canonical transformation. The method is effective even on the borders of chaotic regions and on the separatrices of isolated broad resonances. We propose a criterion for the transition to chaos and verify its utility in an example with 1: degrees of freedom. We anticipate that the criterion will be useful as well in systems of higher dimension.