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Some remarks on symmetric periodic orbits in the restricted three-body problem
1. | Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Germany |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz-Floer homology,, J. Topology Analysis, 1 (2009), 307.
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.
doi: 10.1086/109672. |
[8] |
G. D. Birkhoff, The restricted problem of three bodies,, Rend. Circ. Matem. Palermo, 39 (1915), 265.
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.
|
[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.
|
[12] |
A. Floer, The unregularizaed gradient flow of the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 775.
doi: 10.1002/cpa.3160410603. |
[13] |
U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology,, Int. Math. Res. Not., 42 (2004), 2179.
doi: 10.1155/S1073792804133941. |
[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. |
[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. |
[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. |
[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. |
[19] |
W. Klingenberg, Lectures on Closed Geodesics,, Die Grundlehren der Math. Wissenschaften, 230 (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.
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.
doi: 10.1016/j.aim.2005.05.005. |
[23] |
W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle,, , 42 (2011), 355.
doi: 10.1007/s00526-011-0391-1. |
[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. |
[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.
|
[27] |
R. S. Palais, Homotopy theory of infinite dimensional manifolds,, Topology, 5 (1966), 1.
doi: 10.1016/0040-9383(66)90002-4. |
[28] |
J. Robbin and D. Salamon, The Maslov index for paths,, Topology, 32 (1993), 827.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1086/109672. |
[8] |
G. D. Birkhoff, The restricted problem of three bodies,, Rend. Circ. Matem. Palermo, 39 (1915), 265.
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.
|
[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.
|
[12] |
A. Floer, The unregularizaed gradient flow of the symplectic action,, Comm. Pure Appl. Math., 41 (1988), 775.
doi: 10.1002/cpa.3160410603. |
[13] |
U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology,, Int. Math. Res. Not., 42 (2004), 2179.
doi: 10.1155/S1073792804133941. |
[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. |
[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. |
[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. |
[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. |
[19] |
W. Klingenberg, Lectures on Closed Geodesics,, Die Grundlehren der Math. Wissenschaften, 230 (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.
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.
doi: 10.1016/j.aim.2005.05.005. |
[23] |
W. J. Merry, Lagrangian Rabinowitz Floer homology and twisted cotangent bundle,, , 42 (2011), 355.
doi: 10.1007/s00526-011-0391-1. |
[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. |
[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.
|
[27] |
R. S. Palais, Homotopy theory of infinite dimensional manifolds,, Topology, 5 (1966), 1.
doi: 10.1016/0040-9383(66)90002-4. |
[28] |
J. Robbin and D. Salamon, The Maslov index for paths,, Topology, 32 (1993), 827.
doi: 10.1016/0040-9383(93)90052-W. |
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