Continuous symmetric perturbations of planar power law forces

Abstract

We show the existence of periodic solutions for continuous symmetric perturbations of certain planar power law problems.In this paper we study continuous symmetric perturbations of planar power law problems of the form. In particular, if = 1, the unperturbed problem is Kepler's problem.We prove the existence of periodic solutions of perturbed problems as above, close to a given circular orbit of the unperturbed problem. We have two cases. When = 1 (that is, the unperturbed problem is Kepler's problem) we will require that the perturbed problems are symmetric with respect to the x and y axes. For = 1, we will require just one symmetry.

On the existence of periodic orbits for the fixed homogeneous circle problem

“…0.2). Theorem A will follow essentially from a very geometric result about perturbations of certain central forces [1]. Our next result shows the existence of infinitely many "figure eight" periodic orbits in any vertical plane.…”

“…Note that U(r, 1 [1], for sufficiently small, r (t), 0 t t, intersects transversally the negative x-axis in exactly one point r (t ) and y (t) > 0, 0 < t < t (Fig. 4.12).…”

“…It follows thatṙ(t 1 ) is not horizontal, that is, the intersection of r with E n at t 1 >t(x, v); r(t, x, v) ∈ E n }. Note that t 2 exists by the intermediate value theorem and Proposition 3.1.…”

“…(e.g. Proposition 1.5 of [1] which r(t, x , v ), 0 t a, intersects E n transversally. By the uniqueness of the intersections, t (x k , v k ) = t 1 (x k , v k ), which implies that t 1 is continuous.…”

“…By the continuous dependence of the solutions (e.g. see Proposition 1.5 of [1] z v, (t)). Note that r v, (t) is a solution of r = −∇V (r, 1 ).…”