INTEGRABLE AND NON-INTEGRABLE NON-KAM HAMILTONIANS AND MAGNETIC FIELD TOPOLOGY
The integrability of Hamiltonians H(P-1, P-2,Q-1,Q-2) = P-1G-1(Q-1,Q-2) + P-2G-2(Q-1,Q-2), with arbitrary analytic G-1 and G-2 2pi-periodic in Q-1 and Q-2, is analytically investigated. Such H cannot be separated into two parts, H = H-0 + H-1, such that the KAM theorem would apply for /H-1/ << /H-0/. For G-2 = const such Hamiltonians correspond to toroidal magnetic fields with constant rotational transform. Integrability is then equivalent to the existence of closed magnetic surfaces. The winding number omega of the Q-1, Q-2 flow (i.e. the rotational transform) is rational in "tongue-like" domains in (w-2/w-1,A) diagrams. Here w-i = <G-i> is the average over both Q-1 and Q-2, G-i = w-i + F-i = 1,2, and A is an amplitude parameter of F-i (F-i = O for A = O). Integrability is proved almost everywhere in the complementary domains, namely where omega is sufficiently irrational. In the generic case ("conditional") non-integrability is proved for the class delta-G-1/delta-Q-1 + delta-G-2/delta-Q-2 = O in the tongues, which in this case shrink to lines with omega = w-1/w-2. It is shown that if the number of dimensions in the Hamiltonian were larger than two, qualitatively different results would be expected.
Bibliographic Reference: WRITE TO MAX-PLANCK-INSTITUT FUER PLASMAPHYSIK, 8046 GARCHING BEI MUENCHEN (GERMANY), MENTIONING REPORT IPP 6-257, 1986
Record Number: 1989124128900 / Last updated on: 1987-01-01
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