Direct methods for the search of the second invariant

Abstract We discuss the direct methods that can be used to search for the second invariant of a system defined by the Hamiltonian H = 1 2 (p x 2 ) + p y 2 + A(x, y)p x + B(x, y)p y + 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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