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 … Show more
“…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.…”
Section: ) (Ii) Dist(s(t) C)mentioning
confidence: 89%
“…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).…”
Section: Proof Of the Corollary To Theorem B: Symmetric Figure Eight mentioning
confidence: 96%
“…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.…”
Section: Claim 32mentioning
confidence: 99%
“…(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.…”
Section: Claim 32mentioning
confidence: 99%
“…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 ).…”
Section: Proposition 53 Let C Be a Circle In The Xz-plane With Cenmentioning
We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and very near the homogeneous circle, as well as eight and spiral periodic orbits.
“…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.…”
Section: ) (Ii) Dist(s(t) C)mentioning
confidence: 89%
“…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).…”
Section: Proof Of the Corollary To Theorem B: Symmetric Figure Eight mentioning
confidence: 96%
“…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.…”
Section: Claim 32mentioning
confidence: 99%
“…(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.…”
Section: Claim 32mentioning
confidence: 99%
“…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 ).…”
Section: Proposition 53 Let C Be a Circle In The Xz-plane With Cenmentioning
We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and very near the homogeneous circle, as well as eight and spiral periodic orbits.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.