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Darboux Polynomials and First Integrals of Natural Polynomial Hamiltonian Systems

Andrzej J. Maciejewski·Maria Przybylska
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Abstract

We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals.

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Andrzej J. Maciejewski, Maria Przybylska. 2004-05-03. Darboux Polynomials and First Integrals of Natural Polynomial Hamiltonian Systems. https://doi.org/10.1016/j.physleta.2004.04.034

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