December  2014, 34(12): 5229-5245. doi: 10.3934/dcds.2014.34.5229

Some remarks on symmetric periodic orbits in the restricted three-body problem

1. 

Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Germany

Received  March 2013 Revised  March 2014 Published  June 2014

The planar circular restricted three-body problem (PCRTBP) is symmetric with respect to the line of masses and there is a corresponding anti-symplectic involution on the cotangent bundle of the 2-sphere in the regularized PCRTBP. Recently it turned out that each bounded component of an energy hypersurface with low energy for the regularized PCRTBP is fiberwise starshaped. This enables us to define a Lagrangian Rabinowitz Floer homology which is related to periodic orbits symmetric for the anti-symplectic involution in the regularized PCRTBP and hence to symmetric periodic orbits in the unregularized problem. We compute this homology and discuss the properties of the symmetric periodic orbits.
Citation: Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
References:
[1]

R. F. Arenstorf, Periodic solutions of the restricted three body prblem repesenting analytic continuations of Keplerian Elliptic motions, Am. J. Math., 85 (1963), 27-35. doi: 10.2307/2373181.

[2]

P. Albers, U. Frauenfelder, O. van Koert and G. Paternain, The contact geometry of the restricted 3-body problem, Comm. Pure Appl. Math., 65 (2012), 229-263. doi: 10.1002/cpa.21380.

[3]

P. Albers, J. Fish, U. Frauenfelder, H. Hofer and O. van Koert, Global surfaces of section in the planar restricted 3-body problem, Archive for Rational Mechanics and Analysis, 204 (2012), 273-284. doi: 10.1007/s00205-011-0475-2.

[4]

P. Albers, J. Fish, U. Frauenfelder and O. van Koert, The Conley-Zehnder indices of the rotating Kepler problem, Mathematical Proc. of the Cambridge Philosophical Society, 154 (2013), 243-260. doi: 10.1017/S0305004112000515.

[5]

A. Abbondandolo, A. Portaluri and M. Schwarz, The homology of path spaces and Floer homology with conormal boundary conditions, J. Fixed Point Theory Appl., 4 (2008), 263-293. doi: 10.1007/s11784-008-0097-y.

[6]

A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz-Floer homology, J. Topology Analysis, 1 (2009), 307-405. doi: 10.1142/S1793525309000205.

[7]

R. Barrar, Existence of periodic orbits of the second kind in the restricted problem of three bodies, Astronom. J., 70 (1965), 3-4. doi: 10.1086/109672.

[8]

G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Matem. Palermo, 39 (1915), 265-334. doi: 10.1007/BF03015982.

[9]

K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Annales scientifiques de l'ENS, 43 (2010), 957-1015.

[10]

K. Cieliebak, U. Frauenfelder and O. van Koert, The Cartan geometry of the rotating Kepler problem, arXiv:1110.1021

[11]

A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547.

[12]

A. Floer, The unregularizaed gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813. doi: 10.1002/cpa.3160410603.

[13]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., 42 (2004), 2179-2269. doi: 10.1155/S1073792804133941.

[14]

U. Frauenfelder and J. Kang, From Gradient Flow Lines to Finite Energy Planes, in preparation.

[15]

H. Hofer, Pseudo-holomorphic curves in symplectisations with application to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.

[16]

A. Harris and G. P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom., 34 (2008), 115-134. doi: 10.1007/s10455-008-9111-2.

[17]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictily convex energy surfaces, Ann. Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.

[18]

H. Hofer, K. Wysocki and E. Zehnder, Finite Energy Foliations of tight three-spheres and Hamiltonian dynamics, Ann. Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.

[19]

W. Klingenberg, Lectures on Closed Geodesics, Die Grundlehren der Math. Wissenschaften, 230, Springer-Verlag, 1978.

[20]

Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, 2002. doi: 10.1007/978-3-0348-8175-3.

[21]

Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow (II), Chinese Ann. Math. Ser. B, 21 (2000), 89-108. doi: 10.1007/BF02731963.

[22]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded symmetric domains, Advanced in Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.

[23]

W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle, arXiv:1010.4190, Calc. Var. Partial Differential Equations, 42 (2011), 355-404. doi: 10.1007/s00526-011-0391-1.

[24]

W. J. Merry, Rabinowitz Floer Homology and Mañé Supercritical Hypersurfaces, Ph.D. thesis, University of Cambridge, 2011.

[25]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.

[26]

Y.-G. Oh, Symplectic topology as the geometry of of action functional. I - Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.

[27]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. doi: 10.1016/0040-9383(66)90002-4.

[28]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W.

show all references

References:
[1]

R. F. Arenstorf, Periodic solutions of the restricted three body prblem repesenting analytic continuations of Keplerian Elliptic motions, Am. J. Math., 85 (1963), 27-35. doi: 10.2307/2373181.

[2]

P. Albers, U. Frauenfelder, O. van Koert and G. Paternain, The contact geometry of the restricted 3-body problem, Comm. Pure Appl. Math., 65 (2012), 229-263. doi: 10.1002/cpa.21380.

[3]

P. Albers, J. Fish, U. Frauenfelder, H. Hofer and O. van Koert, Global surfaces of section in the planar restricted 3-body problem, Archive for Rational Mechanics and Analysis, 204 (2012), 273-284. doi: 10.1007/s00205-011-0475-2.

[4]

P. Albers, J. Fish, U. Frauenfelder and O. van Koert, The Conley-Zehnder indices of the rotating Kepler problem, Mathematical Proc. of the Cambridge Philosophical Society, 154 (2013), 243-260. doi: 10.1017/S0305004112000515.

[5]

A. Abbondandolo, A. Portaluri and M. Schwarz, The homology of path spaces and Floer homology with conormal boundary conditions, J. Fixed Point Theory Appl., 4 (2008), 263-293. doi: 10.1007/s11784-008-0097-y.

[6]

A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz-Floer homology, J. Topology Analysis, 1 (2009), 307-405. doi: 10.1142/S1793525309000205.

[7]

R. Barrar, Existence of periodic orbits of the second kind in the restricted problem of three bodies, Astronom. J., 70 (1965), 3-4. doi: 10.1086/109672.

[8]

G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Matem. Palermo, 39 (1915), 265-334. doi: 10.1007/BF03015982.

[9]

K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Annales scientifiques de l'ENS, 43 (2010), 957-1015.

[10]

K. Cieliebak, U. Frauenfelder and O. van Koert, The Cartan geometry of the rotating Kepler problem, arXiv:1110.1021

[11]

A. Floer, Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547.

[12]

A. Floer, The unregularizaed gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813. doi: 10.1002/cpa.3160410603.

[13]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., 42 (2004), 2179-2269. doi: 10.1155/S1073792804133941.

[14]

U. Frauenfelder and J. Kang, From Gradient Flow Lines to Finite Energy Planes, in preparation.

[15]

H. Hofer, Pseudo-holomorphic curves in symplectisations with application to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.

[16]

A. Harris and G. P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom., 34 (2008), 115-134. doi: 10.1007/s10455-008-9111-2.

[17]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictily convex energy surfaces, Ann. Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.

[18]

H. Hofer, K. Wysocki and E. Zehnder, Finite Energy Foliations of tight three-spheres and Hamiltonian dynamics, Ann. Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.

[19]

W. Klingenberg, Lectures on Closed Geodesics, Die Grundlehren der Math. Wissenschaften, 230, Springer-Verlag, 1978.

[20]

Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, 2002. doi: 10.1007/978-3-0348-8175-3.

[21]

Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow (II), Chinese Ann. Math. Ser. B, 21 (2000), 89-108. doi: 10.1007/BF02731963.

[22]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded symmetric domains, Advanced in Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.

[23]

W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle, arXiv:1010.4190, Calc. Var. Partial Differential Equations, 42 (2011), 355-404. doi: 10.1007/s00526-011-0391-1.

[24]

W. J. Merry, Rabinowitz Floer Homology and Mañé Supercritical Hypersurfaces, Ph.D. thesis, University of Cambridge, 2011.

[25]

J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.

[26]

Y.-G. Oh, Symplectic topology as the geometry of of action functional. I - Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.

[27]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. doi: 10.1016/0040-9383(66)90002-4.

[28]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W.

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