Kruskal's Perturbation Method

A general method for eliminating the angle variable from the equations of a perturbed periodic motion and for deriving an ``adiabatic invariant'' J has been given by Kruskal and, for a special class of Hamiltonian systems, McNamara and Whiteman have shown (to order ε2) that J is related to a set of invariants I obtained from the expansion of Poisson-bracket relations. In this work, an order-by-order algorithm for Kruskal's method is introduced, and a new set of invariants Z1 is obtained. It is shown that these invariants bear a close relation to those obtained from the Poisson-bracket expansion and, in the special case investigated by McNamara and Whiteman, the relation between I and Z1 may be brought to the same form as the relation between I and J derived by those authors. Finally, the relationship between Z1 and J is examined, and arguments are presented that in certain cases the two are equal to all orders.