Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4

Abstract

Abstract. In the analytic case we prove the conjecture of A.J. Maciejewski and M. Przybylska that appeared in J. Math. Phys. 46 (6) (2005) 062901 regarding Hamiltonian systems with a homogeneous polynomial potential of degree 4. The proof of the conjecture completes the characterization of all the analytic integrable Hamiltonian system with a homogeneous polynomial potential of degree 4.

Integrability conditions of geodesic flow on homogeneous Monge manifolds

“…The integrability of Hamiltonian systems with homogeneous potentials has been well studied (see, for example, [2, 6, 7, 14, 21-24, 31, 32]), and in fact all integrable homogeneous potentials of order three and four have been classified (see [12,15,16]). Throughout this work, by 'integrable' we mean complete meromorphic integrability in the sense of Liouville (defined precisely below).…”

Computing necessary integrability conditions for planar parametrized homogeneous potentials

“…Some prior works used a similar strategy, but it was unclear which cases were possible to tackle, in particular for singular ones. The approach was not fully automatized and this explains that results were only available for special families of potentials, for instance polynomials of small degree (3 or 4) [13,14,11,12], as the number of singular cases grows very fast (already 44 for polynomials of degree 5). By contrast, our treatment is unified and fully automated, and it allows not only to retrieve (and sometimes correct) known results, but more importantly, to treat potentials of degrees previously unreached (up to 9).…”

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree −2

“…, 8 given in Table 2 are the nonequivalent integrable homogeneous potentials of degree 4. In [9] we proved that for the family (3) only the potentials V 9 and V 10 of Table 2 are integrable.…”

“…It is well-known (see for instance Proposition 1 of [9]) that the study of the existence of analytic first integrals of a weight-homogeneous polynomial differential system reduces to the study of the existence of a weighthomogeneous polynomial first integrals. This fact together with Theorem 3 states the following main theorem.…”