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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

Jungsoo Kang 1,

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.
Keywords: symmetric periodic orbits, The planar circular restricted three-body problem, Rabinowitz Floer homology..
Mathematics Subject Classification: Primary: 58F05, 58E05; Secondary: 53C3.
Citation: Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.2307/2373181.  Google Scholar

[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.  doi: 10.1002/cpa.21380.  Google Scholar

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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.  doi: 10.1007/s00205-011-0475-2.  Google Scholar

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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.  doi: 10.1017/S0305004112000515.  Google Scholar

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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.  doi: 10.1007/s11784-008-0097-y.  Google Scholar

[6]

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

[7]

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

[8]

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

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

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K. Cieliebak, U. Frauenfelder and O. van Koert, The Cartan geometry of the rotating Kepler problem,, , ().   Google Scholar

[11]

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

[12]

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

[13]

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

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

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

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

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

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

[19]

W. Klingenberg, Lectures on Closed Geodesics,, Die Grundlehren der Math. Wissenschaften, 230 (1978).   Google Scholar

[20]

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

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

[22]

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

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W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle,, , 42 (2011), 355.  doi: 10.1007/s00526-011-0391-1.  Google Scholar

[24]

W. J. Merry, Rabinowitz Floer Homology and Mañé Supercritical Hypersurfaces,, Ph.D. thesis, (2011).   Google Scholar

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

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.  doi: 10.2307/2373181.  Google Scholar

[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.  doi: 10.1002/cpa.21380.  Google Scholar

[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.  doi: 10.1007/s00205-011-0475-2.  Google Scholar

[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.  doi: 10.1017/S0305004112000515.  Google Scholar

[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.  doi: 10.1007/s11784-008-0097-y.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

K. Cieliebak, U. Frauenfelder and O. van Koert, The Cartan geometry of the rotating Kepler problem,, , ().   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.  doi: 10.1007/s10455-008-9111-2.  Google Scholar

[17]

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

[18]

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

[19]

W. Klingenberg, Lectures on Closed Geodesics,, Die Grundlehren der Math. Wissenschaften, 230 (1978).   Google Scholar

[20]

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

[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.  doi: 10.1007/BF02731963.  Google Scholar

[22]

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

[23]

W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle,, , 42 (2011), 355.  doi: 10.1007/s00526-011-0391-1.  Google Scholar

[24]

W. J. Merry, Rabinowitz Floer Homology and Mañé Supercritical Hypersurfaces,, Ph.D. thesis, (2011).   Google Scholar

[25]

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

[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.   Google Scholar

[27]

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

[28]

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

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