Abstract
It was discovered by Gordon (Am J Math 99(5):961–971, 1977) that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this note is to give a direct proof of these results already proved by authors in Hu and Sun (Adv Math 223(1):98–119, 2010), Hu et al. (Arch Ration Mech Anal 213(3):993–1045, 2014) through a self-contained and explicit computation of the Conley–Zehnder index through crossing forms in the Lagrangian setting. The techniques developed in this paper can be used to investigate the higher dimensional case of Keplerian ellipses, where the classical variational proof no longer applies.
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Notes
The symmetry of the restriction of the bilinear map \(\omega _0(T\cdot , \cdot )\) onto \(L_0 \times L_0\) is consequence of the fact that L is Lagrangian.
This index was defined in a slightly different manner by authors in [14].
More generally if \(f(t)>0\) for every \(t \in [\alpha , \beta ] \subseteq [0,T]\), then the perturbed path has no crossing instants in \([\alpha , \beta ]\). In fact the equation:
$$\begin{aligned} f(t_\varepsilon )=\dfrac{2(1-\cos \varepsilon )}{\sin \varepsilon },\qquad \forall \, t \in [\alpha ,\beta ] \end{aligned}$$has no solution as the left hand side is negative whilst the right hand side is positive.
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Acknowledgements
The third name author wishes to thank all faculties and staff at the Queen’s University (Kingston) for providing excellent working conditions during his stay and especially his wife, Annalisa, that has been extremely supportive of him throughout this entire period and has made countless sacrifices to help him getting to this point.
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A. Portaluri: Authors are partially supported Prin 2015 “Variational methods, with applications to problems in mathematical physics and geometry” No. 2015KB9WPT_001.
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Kavle, H., Offin, D. & Portaluri, A. Keplerian Orbits Through the Conley–Zehnder Index. Qual. Theory Dyn. Syst. 20, 10 (2021). https://doi.org/10.1007/s12346-020-00430-0
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DOI: https://doi.org/10.1007/s12346-020-00430-0