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.
Keywords3-body problems with two fixed centers "∞"-type solutions Lagrangian actions
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).
- Gordon W: A minimizing property of Keplerian orbits. Am J Math 1977, 99: 961-971. 10.2307/2373993MATHView ArticleGoogle Scholar
- Siegel C, Moser J: Lectures on Celestial Mechanics. Springer, Berlin; 1971.View ArticleGoogle Scholar
- Euler M: De motu coproris ad duo centra virium fixa attracti. Nov Commun Acad Sci Imp Petrop 1766, 10: 207-242.Google Scholar
- Euler M: De motu coproris ad duo centra virium fixa attracti. Nov Commun Acad Sci Imp Petrop 1767, 11: 152-184.Google Scholar
- Euler M: Probleme un corps etant attire en raison reciproque quarree des distances vers deux points fixes donnes trouver les cas ou la courbe decrite par ce corps sera algebrique. Hist Acad R Sci Bell Lett Berlin 1767, 2: 228-249.Google Scholar
- Palais R: The principle of symmetric criticality. Commun Math Phys 1979, 69: 19-30. 10.1007/BF01941322MATHMathSciNetView ArticleGoogle Scholar
- Gordon W: Conservative dynamical systems involving strong forces. Trans AMS 1975, 204: 113-135.MATHView ArticleGoogle Scholar
- Ambrosetti A, Coti Zelati V: Periodic Solutions for Singular Lagrangian Systems. Springer, Boston; 1993.View ArticleGoogle Scholar
- Hardy G, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge; 1952.Google Scholar
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