Abstract

Lie group symmetries and invariants of the generalized Hénon–Heiles equations are found. The coupled second-order equations are invariant under translations in time, in general, and the stretching group (dilation) if the linear terms in the ‘‘force’’ are absent. The equivalent set of four coupled first-order equations is found to be invariant under a one-parameter group for three cases and the group generators are given. Three different approaches are reported: the ‘‘classical method’’ for determining Lie group symmetries, a modified method for finding Lie group symmetries with vector fields and the direct method for calculating the invariants. For the Hénon–Heiles equations the direct method is the most efficient.

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