Abstract
In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form
where q = (q 1, …, q n ) ∈ ℂn, p = (p 1, …, p n ) ∈ 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 F 1, … F m . Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral F m+1 such that all integrals F 1, … F m+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.
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Maciejewski, A.J., Przybylska, M. Partial integrability of Hamiltonian systems with homogeneous potential. Regul. Chaot. Dyn. 15, 551–563 (2010). https://doi.org/10.1134/S1560354710040106
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DOI: https://doi.org/10.1134/S1560354710040106