# Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases - Nonlinear Sciences > Exactly Solvable and Integrable Systems

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases - Nonlinear Sciences > Exactly Solvable and Integrable Systems - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this paper the problem of classification of integrable natural Hamiltoniansystems with $n$ degrees of freedom given by a Hamilton function which is thesum of the standard kinetic energy and a homogeneous polynomial potential $V$of degree $k>2$ is investigated. It is assumed that the potential is notgeneric. Except for some particular cases a potential $V$ is not generic, if itadmits a nonzero solution of equation $V-\vd=0$. The existence of suchsolution gives very strong integrability obstructions obtained in the frame ofthe Morales-Ramis theory. This theory gives also additional integrabilityobstructions which have the form of restrictions imposed on the eigenvalues$\lambda 1, .,\lambda n$ of the Hessian matrix $V-\vd$ calculated at anon-zero $\vd\in\C^n$ satisfying $V-\vd=\vd$. Furthermore, we show thatsimilarly to the generic case also for nongeneric potentials some universalrelations between $\lambda 1, .,\lambda {n}$ calculated at various solutionsof $V-\vd=\vd$ exist. We derive them for case $n=k=3$ applying themultivariable residue calculus. We demonstrate the strength of the obtainedresults analysing in details the nongeneric cases for $n=k=3$. Our analysiscover all the possibilities and we distinguish those cases where known methodsare too weak to decide if the potential is integrable or not. Moreover, for$n=k=3$ thanks to this analysis a three-parameter family of potentialsintegrable or super-integrable with additional polynomial first integrals whichseemingly can be of an arbitrarily high degree with respect to the momenta wasdistinguished.

Author: Maria Przybylska

Source: https://arxiv.org/