Abstract
It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle \begin{document}$ \mathit{\theta }$\end{document} .
Highlights
IntroductionThe planar three-body problem studies the motion of three masses m1, m2, m3 moving in a fixed plane satisfying Newton’s law of gravitation: miqi
The planar three-body problem studies the motion of three masses m1, m2, m3 moving in a fixed plane satisfying Newton’s law of gravitation: miqi = ∂ ∂qiU (q), i = 1, 2, 3, (1)q1 where qi =, a row vector in the xy plane, is the position of mi, q = q2q3 is a 3 × 2 matrix and the kinetic energy K and the potential energy U are given as follows: K ≡ K (q) 1 2 mi|qi|2, i=1U (q) m1m2 |q1 − q2|
The purpose of this paper is to show the existence of a new set of orbits in the planar equal-mass three-body problem
Summary
The planar three-body problem studies the motion of three masses m1, m2, m3 moving in a fixed plane satisfying Newton’s law of gravitation: miqi. For each θ ∈ [0.084π, 0.183π], there exists a nontrivial and collisionfree minimizing path P0, θ ≡ P0, θ(t ∈ [0, 1]) connecting the two configuration sets QS and QE in (3), and it can be extended to a periodic or quasi-periodic orbit. {q∈P (Qs,Qe)} 0 where K(q(t)) is the standard kinetic energy, U (q(t)) is the standard potential energy, and P(Qs, Qe) is defined as follows: If one wants P to be a part of a periodic solution, the boundaries must be special 9) with straight lines, where the positions qi(t) (i = 1, 2, 3) satisfy qi(t) = qi,k + 10 t It follows that the test path Ptest is a continuous and piecewise smooth function.
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