Lagrangian actions on 3-body problems with two fixed centers
© Wang and He; licensee Springer. 2012
Received: 1 October 2011
Accepted: 28 February 2012
Published: 28 February 2012
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.
Mathematical Subject Classification 2000: 34C15; 34C25; 70F10.
1 Introduction and Main Result
2 The Proof of Theorem 1.1
Using Palais'S symmetrical Principle , it's easy to prove the following variational Lemma:
Lemma 2.1 The critical point of f(q) in Λ is the noncollision periodic solution winding around q1 counter-clockwise and q2 clockwise one time during one period.
It's easy to see
Lemma 2.3 is a weakly closed subset of the Hilbert space W1,2(ℝ/ℤ, ℝ2).
Lemma 2.4 f(q) is coercive and weakly lower-semicontinuous on the closure of Λ.
Proof. By q(-t) = -q(t) and q(t) ∈ W1,2(ℝ/ℤ, ℝ2), we have . By Wirtinger's inequality, we know f(q) is coercive. By Sobolev's embedding Theorem and Fatou's Lemma, f is weakly lower-semi-continuous on the weakly closed set of W1,2.
Lemma 2.5 Let X be a reflexive Banach space,M ⊂ X be weakly closed subset,f : M → R be weakly lower semi-continous and coercive (f(x) → +∞ as ∥x∥ → +∞), then f attains its infimum on M.
So if the minimizer of f(q) on has collision at some moment, then Gordon's Lemma tell us the minimum value is +∞ which is a contradiction.
The most interesting case α = 1 is the case for Newtonian potential, we try to prove the minimizer is collision-free, but it seems very difficult, here we give some remarks.
There is equality only for a certain hyperelliptic curve.
Let , then , that is φ is strictly convex.
Let φ'(s) = 0, we solve it to get is the critical point for φ(s), and , which is the maximum value for φ(s) on s > 0 since φ is convex and φ(s) → +∞ as s → 0+.
then the minimizer of f(q) on is collision-free.
The authors would like to thank the anonymous referees for their valuable suggestions which improve this work. This work was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZA172).
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