Elsevier

Physics Reports

Volume 147, Issue 2, March 1987, Pages 87-154
Physics Reports

Direct methods for the search of the second invariant

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Abstract

We discuss the direct methods that can be used to search for the second invariant of a system defined by the Hamiltonian H = 12(px2) + py2 + A(x, y)px + B(x, y)py + V(x, y). We give an extensive review of those systems that are known to have an invariant that is polynomial in the p's (most of these have A = B = 0). In addition we introduce the field of non-polynomial invariants by giving several new systems that have a rational or transcendental (in the p's) invariant (for these A and B are nonzero). The special case of integrability at a fixed value of the energy is also discussed.

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