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The Conley–Zehnder indices of the rotating Kepler problem

Published online by Cambridge University Press:  03 October 2012

PETER ALBERS
Affiliation:
Mathematisches Institut, Büro 308Westfälische Wilhelms-Universität MünsterEinsteinstrasse 62, D-48149 Münster, Germany. e-mail: [email protected]
JOEL W. FISH
Affiliation:
Max Planck InstituteInselstrasse 22, D-04103 Leipzig, Germany. e-mail: [email protected]
URS FRAUENFELDER
Affiliation:
Department of Mathematics and Research Institute of Mathematics, Seoul National University, Building 27, room 403 San 56-1, Sillim-dong, Gwanak-gu, Seoul 151-747, South Korea. e-mail: [email protected]
OTTO VAN KOERT
Affiliation:
Department of Mathematics and Research Institute of MathematicsSeoul National University, Building 27, room 402 San 56-1, Sillim-dong, Gwanak-gu, Seoul 151-747, South Korea. e-mail: [email protected]

Abstract

We determine the Conley–Zehnder indices of all periodic orbits of the rotating Kepler problem for energies below the critical Jacobi energy. Consequently, we show the universal cover of the bounded component of the regularized energy hypersurface is dynamically convex. Moreover, in the universal cover there is always precisely one periodic orbit with Conley–Zehnder index 3, namely the lift of the doubly covered retrograde circular orbit.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[Abb03]Abbondandolo, A.On the Morse index of Lagrangian systems. Nonlinear Anal. 53 (2003), no. 3–4, 551566.CrossRefGoogle Scholar
[AFF+11]Albers, P., Fish, J. W., Frauenfelder, U., Hofer, H. and van Koert, O.Global surfaces of section in the planar restricted 3-body problem. Arch. Ration. Mech. Anal. 204, no. 1 (2012), 273284.CrossRefGoogle Scholar
[AFvKP12]Albers, P., Frauenfelder, U., van Koert, O. and Paternain, G.The contact geometry of the restricted 3-body problem. Comm. Pure Appl. Math. 65 (2012), no. 2, 229263.CrossRefGoogle Scholar
[Bar65]Barrar, R.Existence of periodic orbits of the second kind in the restricted problem of three bodies. Astronom. J. 70 (1965), 34.CrossRefGoogle Scholar
[CFvK11]Cieliebak, K., Frauenfelder, U. and van Koert, O. The Cartan geometry of the rotating Kepler problem. 2011, arXiv:1110.1021, to appear in Publ. Math. Debrecen.Google Scholar
[Dui76]Duistermaat, J. J.On the Morse index in variational calculus. Advances in Math. 21 (1976), no. 2, 173195.CrossRefGoogle Scholar
[HWZ98]Hofer, H., Wysocki, K. and Zehnder, E.The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math. (2) 148 (1998), no. 1, 197289.CrossRefGoogle Scholar
[LC20]Levi–Civita, T.Sur la régularisation du problème des trois corps. Acta Math. 42 (1920), no. 1, 99144.CrossRefGoogle Scholar
[McG69]McGehee, R. P. Some homoclinic orbits for the restricted three-body problem. Ph.D. Thesis. The University of Wisconsin-Madison (1969).Google Scholar
[Mos70]Moser, J.Regularization of Kepler's problem and the averaging method on a manifold. Comm. Pure Appl. Math. 23 (1970), 609636.CrossRefGoogle Scholar
[RS93]Robbin, J. and Salamon, D. A.The Maslov index for paths. Topology 32 (1993), no. 4, 827844.CrossRefGoogle Scholar
[Web02]Weber, J.Perturbed closed geodesics are periodic orbits: index and transversality. Math. Z. 241 (2002), no. 1, 4582.CrossRefGoogle Scholar