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February  2012, 32(2): 679-701. doi: 10.3934/dcds.2012.32.679

Closed trajectories on symmetric convex Hamiltonian energy surfaces

Wei Wang 1,

1. 

Key Laboratory of Pure and Applied Mathematics, School of Mathematical Science, Peking University, Beijing, 100871, China

Received  August 2010 Revised  May 2011 Published  September 2011

In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin, where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sigma$.
Keywords: Hamiltonian systems., closed characteristics, Compact convex hypersurfaces.
Mathematics Subject Classification: Primary: 58E05, 37J45; Secondary: 37C7.
Citation: Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679
References:
[1]

A. Borel, "Seminar on Transformation Groups,", Annals of Mathematics Studies, (1960).   Google Scholar

[2]

I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 1 (1984), 19.   Google Scholar

[3]

I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19 (1990).   Google Scholar

[4]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories,, Comm. Math. Phys., 113 (1987), 419.  doi: 10.1007/BF01221255.  Google Scholar

[5]

I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 4 (1987), 307.   Google Scholar

[6]

E. Fadell and P. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Invent. Math., 45 (1978), 139.  doi: 10.1007/BF01390270.  Google Scholar

[7]

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

[8]

Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.   Google Scholar

[9]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics,, Advances in Math., 154 (2000), 76.  doi: 10.1006/aima.2000.1914.  Google Scholar

[10]

Y. Long, "Index Theory for Symplectic Paths with Applications,", Progress in Math., 207 (2002).   Google Scholar

[11]

C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$,, Math. Ann., 323 (2002), 201.  doi: 10.1007/s002089100257.  Google Scholar

[12]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$,, Ann. of Math., 155 (2002), 317.  doi: 10.2307/3062120.  Google Scholar

[13]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[14]

A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171.   Google Scholar

[15]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems,, Trans. Amer. Math. Soc., 311 (1989), 621.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[16]

W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces,, Duke Math. J., 139 (2007), 411.  doi: 10.1215/S0012-7094-07-13931-0.  Google Scholar

[17]

W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$,, J. Eur. Math. Soc., 11 (2009), 575.  doi: 10.4171/JEMS/161.  Google Scholar

[18]

W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$,, J. Diff. Equa., 246 (2009), 4322.  doi: 10.1016/j.jde.2008.10.003.  Google Scholar

[19]

A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math., 108 (1978), 507.  doi: 10.2307/1971185.  Google Scholar

show all references

References:
[1]

A. Borel, "Seminar on Transformation Groups,", Annals of Mathematics Studies, (1960).   Google Scholar

[2]

I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 1 (1984), 19.   Google Scholar

[3]

I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19 (1990).   Google Scholar

[4]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories,, Comm. Math. Phys., 113 (1987), 419.  doi: 10.1007/BF01221255.  Google Scholar

[5]

I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 4 (1987), 307.   Google Scholar

[6]

E. Fadell and P. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Invent. Math., 45 (1978), 139.  doi: 10.1007/BF01390270.  Google Scholar

[7]

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

[8]

Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.   Google Scholar

[9]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics,, Advances in Math., 154 (2000), 76.  doi: 10.1006/aima.2000.1914.  Google Scholar

[10]

Y. Long, "Index Theory for Symplectic Paths with Applications,", Progress in Math., 207 (2002).   Google Scholar

[11]

C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$,, Math. Ann., 323 (2002), 201.  doi: 10.1007/s002089100257.  Google Scholar

[12]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$,, Ann. of Math., 155 (2002), 317.  doi: 10.2307/3062120.  Google Scholar

[13]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157.  doi: 10.1002/cpa.3160310203.  Google Scholar

[14]

A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171.   Google Scholar

[15]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems,, Trans. Amer. Math. Soc., 311 (1989), 621.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[16]

W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces,, Duke Math. J., 139 (2007), 411.  doi: 10.1215/S0012-7094-07-13931-0.  Google Scholar

[17]

W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$,, J. Eur. Math. Soc., 11 (2009), 575.  doi: 10.4171/JEMS/161.  Google Scholar

[18]

W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$,, J. Diff. Equa., 246 (2009), 4322.  doi: 10.1016/j.jde.2008.10.003.  Google Scholar

[19]

A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math., 108 (1978), 507.  doi: 10.2307/1971185.  Google Scholar

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