Abstract
In this paper, we study the existence of figure ∞-type periodic solution for 3-body problems with strong-force potentials and two fixed centers, and we also give some remarks in the case with Newtonian weak-force potentials.
Highlights
Introduction and MainResult −1We assume two masses m1 = m2 = 2 are fixed at q1 = q2 = −q1 =, the third mass m3 is affected by m1 and m2 and moving according to the Newton’s second law and the general gravitational law [1,2], the position q(t) for m3 satisfies m3 q (t) =m1m3α(q1 − q) q1 − q α+2 +m2m3α(q2 − q) q2 − q α+2 (1:1)Equivalently, q(t) = α 2
The third mass m3 is affected by m1 and m2 and moving according to the Newton’s second law and the general gravitational law [1,2], the position q(t) for m3 satisfies m3 q (t) m1m3α(q1 − q) q1 − q α+2
We want to use variational minimizing method to look for periodic solution for m3 which winds around q1 and q2, let f (q) =
Summary
The third mass m3 is affected by m1 and m2 and moving according to the Newton’s second law and the general gravitational law [1,2], the position q(t) for m3 satisfies m3 q (t) m1m3α(q1 − q) q1 − q α+2 For the case a = 1, Euler [3,4,5] studied (1.1)-(1.3), but didn’t use variational methods to study periodic solutions. We want to use variational minimizing method to look for periodic solution for m3 which winds around q1 and q2, let f (q) =
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