Darboux polynomials and first integrals of natural polynomial Hamiltonian systems
Copyright policy )Darboux polynomials and first integrals of natural polynomial Hamiltonian systems
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.
14 pages, typos corrected
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Nonlinear Sciences - Exactly Solvable and Integrable Systems, FOS: Physical sciences, Dynamical Systems (math.DS), Mathematical Physics (math-ph), 37Jxx, integrability, 34Cxx, Relations of dynamical systems with symplectic geometry and topology, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 37Jxx;13Nxx;34Cxx, 13Nxx, Hamiltonian systems, Darboux polynomials, Mathematics - Dynamical Systems, Exactly Solvable and Integrable Systems (nlin.SI), Symmetries, invariants of ordinary differential equations, Mathematical Physics
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Nonlinear Sciences - Exactly Solvable and Integrable Systems, FOS: Physical sciences, Dynamical Systems (math.DS), Mathematical Physics (math-ph), 37Jxx, integrability, 34Cxx, Relations of dynamical systems with symplectic geometry and topology, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 37Jxx;13Nxx;34Cxx, 13Nxx, Hamiltonian systems, Darboux polynomials, Mathematics - Dynamical Systems, Exactly Solvable and Integrable Systems (nlin.SI), Symmetries, invariants of ordinary differential equations, Mathematical Physics
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