科研之友
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摘要
We study integrability in the Liouville sense for natural Hamiltonian systems with a homogeneous rational potential V(q). Strong necessary conditions for the integrability of such systems were obtained by analysis of differential Galois groups of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions on eigenvalues of the Hessians V ''(d) calculated at non-zero solutions d of the equation grad V(d) = d. Such solutions are called proper Darboux points.
It was recently proved that for generic polynomial homogeneous potentials, there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The existence of such relations for rational potentials seems to be a difficult question. One of the reasons is the presence of points of indeterminacy of the potential and its gradient. Nevertheless, for two degrees of freedom we prove that such a relation exists. This result is important because it allows us to show that the set of admissible values for Hessian eigenvalues at a proper Darboux point for potentials satisfying the necessary conditions for integrability is finite. In turn, this provides a tool for classification of integrable rational potentials.
It was also recently shown that for polynomial homogeneous potentials, additional necessary conditions for integrability can be deduced from the existence of improper Darboux points, that is, points d that are non-zero solutions of grad V(d) = 0. These new conditions also take the form of arithmetic restrictions imposed on eigenvalues of V ''(d). In this paper we prove that for rational potentials, improper Darboux points give the same necessary conditions for integrability.
- 出版日期2013-4-15
