Construction of a quasiconserved quantity in the Hénon-Heiles problem using a single set of variables.

Abstract

The problem of finding the coefficients of a simple series expansion for a quasiconserved quantity K for the H\'enon-Heiles Hamiltonian H using a single set of variables is solved. In the past, this type of approach has been problematic because the solution to the equations determining the coefficients in the expansion is not unique. As a result, the existence of a consistent expression for K to all orders had not previously been established. We show how to deal with this arbitrariness in the expansion coefficients for K in a consistent way. Due to this arbitrariness, we find a class of expansions for K, in contrast to the single unique expansion for K generated by the normal-form approach of Gustavson [Astron. J. 71, 670 (1966)]. It may be possible to devise a criterion for deciding which one of our expansions is ``optimally convergent,'' although we do not deal with this question here. We proceed by introducing a single set of dynamic variables that have simple symmetry properties and that also ``diagonalize'' the problem of finding the coefficients of K. No canonical transformations are required. A straightforward constructive procedure is given for generating the power series to any order for quantities having the symmetry of the Hamiltonian that are formally conserved. This leads to a very practical method for calculating a quasiconserved quantity in the H\'enon-Heiles problem. A comparison is made through several orders of the terms generated by this approach and those generated in the original Gustavson expansion in normal form.