Polynomial Integrability Of The Three-degrees Of Freedom Hamiltonian Systems With Homogeneous Potential

This paper mainly study the polynomial integrability of the Hamiltonian sys-tem with three degrees of freedom and homogeneous potential. With three degrees of freedom, we characterize the analytic integrability of nature Hamiltonian sys-tems with Hamiltonian function H having homogeneous potential V(q1,..., qm) of degree k. Usually the homogeneous potential of degree k is given either by a polynomial, or by an inverse of a polynomial.For Hamiltonian function itself is always a first integral of the Hamiltonian systems, we can know from Liouville completely integrable theorem that a m-degrees of freedom Hamiltonian system is completely integrable, if it has the other (m - 1) first integrals which are functionally independent with the Hamiltonian function H. Then the differential equations can be solved by the m first integrals.In the past few years, we already have quite accurate results on finding the other functionally independent first integral I, with a two-degrees of freedom Hamiltonian system and homogeneous potentials V(q1,q2) of degree k= -3, -2, -1,0,1,2,3,4. Furthermore, some discusses about two-degree of freedom Hamiltonian system and high-degree homogenous potentials work out some special examples.But results of high-degree of freedom Hamiltonian system and homogenous potentials seem quite rare.Considering two-degree of freedom Hamiltonian system, if the degree of homo-geneous potential is k= - 1,0,1, we can directly know that the Hamiltonian sys-tem is completely integrable. If the degree of homogeneous potential is 2≤ k≤ 5, Hietarinta [Phys. Lett. A 96(1983),273-278] firstly discussed the problem and proved that the Hamiltonian system with homogeneous potential of degree k= 2 is completely integrable. Later on Maciejewski and Przybylska [Phys. Lett. A,327(5- 6) (2004),461-473] showed all the homogeneous potentials so that the Hamiltonian system is completely integrable. After that Maciejewski and Przybylska [J. Math. Phys.46 (6) (2005) 062901] give out all the homogeneous potentials, except the class of V =1/2αq12(q1+iq2)2+1/4(q12+q22)2, that the Hamiltonian system is completely integrable. And Llibra, Mahdi and Valls [J. Math. Phys.52 (2011),012702,9 pp] worked out the unsolved class of potentials and proved the Hamiltonian system is completely integrable with only several kinds of potential. If the degree of homo-geneous potential is κ = -2, Llibra, Mahdi and Valls [J. Math. Phys. Lett. A 375 (2011),1845-1849] got the result that the Hamiltonian system is completely integrable under strong limited conditions. Later, the same three people [Phys. D 240 (2011) 1928--1935] solve the situation of homogeneous potential of degree κ=-3.In this paper, we study the polynomial integrability of nature Hamiltonian systems with three degrees of freedom having a homogeneous potential. Here we focus on the results of their polynomial integrability for κ= -2, -1,0,1,2, and completely discuss all integrable situations of these five kinds of potentials. In the case of κ= -1,0,1,2, the Hamiltonian system is completely integrable and we can find the exact formulas of the three functionally independent first integrals. But in the case of κ= -2, only several systems of strict contraints are completely integrable.