On the period of the periodic orbits of the restricted three body problem

On the period of the periodic orbits of the restricted three body problem We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $$2 T=k\pi +\int _\Omega g$$ 2 T = k π + ∫ Ω g where k is an integer, $$\Omega $$ Ω is the region enclosed by the periodic orbit and $$g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$$ g : R 2 → R is a function that only depends on the constant C known as the Jacobian constant; it does not depend on $$\Omega $$ Ω . This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around $$L_4$$ L 4 such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $$L_5$$ L 5 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Celestial Mechanics and Dynamical Astronomy Springer Journals

On the period of the periodic orbits of the restricted three body problem

Loading next page...
 
/lp/springer_journal/on-the-period-of-the-periodic-orbits-of-the-restricted-three-body-1PUfaCryxn
Publisher
Springer Netherlands
Copyright
Copyright © 2017 by Springer Science+Business Media Dordrecht
Subject
Physics; Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics
ISSN
0923-2958
eISSN
1572-9478
D.O.I.
10.1007/s10569-017-9766-8
Publisher site
See Article on Publisher Site

Abstract

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $$2 T=k\pi +\int _\Omega g$$ 2 T = k π + ∫ Ω g where k is an integer, $$\Omega $$ Ω is the region enclosed by the periodic orbit and $$g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$$ g : R 2 → R is a function that only depends on the constant C known as the Jacobian constant; it does not depend on $$\Omega $$ Ω . This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around $$L_4$$ L 4 such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $$L_5$$ L 5 .

Journal

Celestial Mechanics and Dynamical AstronomySpringer Journals

Published: Apr 19, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off