Integrals, invariant manifolds, and degeneracy for central force problems in R n

Publisher Website
Google Scholar
Abstract
Since the notion of angular momentum is defined in any dimension by using the exterior product in Rn, one would guess that central force problems in any dimension are completely integrable, as it is known for n=2 or 3. This is proved explicitly in this paper, by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that these problems are always reduced to a two‐dimensional plane, and the invariant manifolds are topologically, the Cartesian product of those of the reduced problem times n−2 circle factors. It is also proved that for n≳2 these problems are degenerate in the sense that their Hamiltonians do not satisfy one of the hypotheses of the Kolmogorov–Arnold–Moser theorem for persistence of invariant tori, when one considers small perturbations.