Regular and Chaotic Dynamics

, Volume 15, Issue 4–5, pp 551–563 | Cite as

Partial integrability of Hamiltonian systems with homogeneous potential

Special Issue: Valery Vasilievich Kozlov-60


In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form
$$ H = \frac{1} {2}p^T Mp + V(q), $$
where q = (q1, …, qn) ∈ ℂn, p = (p1, …, pn) ∈ ℂn, are the canonical coordinates and momenta, M is a symmetric non-singular matrix, and V (q) is a homogeneous function of degree k ∈ ℤ*. We assume that the system admits 1 ⩽ m < n independent and commuting first integrals F1, … Fm. Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral Fm+1 such that all integrals F1, … Fm+1 are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain n body problem on a line and the planar three body problem.

Key words

integrability non-integrability criteria monodromy group differential Galois group hypergeometric equation Hamiltonian equations 

MSC2000 numbers

70Hxx 70Fxx 70F07 37J30 


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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.J. Kepler Institute of AstronomyUniversity of Zielona GóraZielona GóraPoland
  2. 2.Toruń Centre for AstronomyN. Copernicus UniversityToruńPoland
  3. 3.Institute of PhysicsUniversity of Zielona GóraZielona GóraPoland

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