# American Institute of Mathematical Sciences

March  2013, 33(3): 1201-1214. doi: 10.3934/dcds.2013.33.1201

## Normally stable hamiltonian systems

 1 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, United States 2 Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain, Spain

Received  May 2011 Revised  November 2011 Published  October 2012

We study the stability of an equilibrium point of a Hamiltonian system with $n$ degrees of freedom. A new concept of stability called normal stability is given which applies to a system in normal form and relies on the existence of a formal integral whose quadratic part is positive definite. We give a necessary and sufficient condition for normal stability. This condition depends only on the quadratic terms of the Hamiltonian. We relate normal stability with formal stability and Liapunov stability. An application to the stability of the $L_4$ and $L_5$ equilibrium points of the spatial circular restricted three body problem is given.
Citation: Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
##### References:
 [1] G. Benettin, F. Fassò and M. Guzzo, Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regul. Chaotic Dyn., 3 (1998), 56-71.  Google Scholar [2] A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219.  Google Scholar [3] H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems, Nonlinearity, 12 (1999), 1351-1362. doi: 10.1088/0951-7715/12/5/309.  Google Scholar [4] T. M. Cherry, On periodic solutions of Hamiltonian systems of differential equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 227 (1928), 137-221. doi: 10.1098/rsta.1928.0005.  Google Scholar [5] N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531. Google Scholar [6] G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88. doi: 10.1515/crll.1846.32.85.  Google Scholar [7] F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292. doi: 10.1007/PL00004245.  Google Scholar [8] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7.  Google Scholar [9] J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526.  Google Scholar [10] L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh., 35 (1971), 423-431; English translation, J. Appl. Math. Mech., 35 (1971), 384-391.  Google Scholar [11] A. L. Kunitsyn and A. A. Tuyakbayev, The stability of Hamiltonian systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 56 (1992), 672-675; English translation, J. Appl. Math. Mech., 56 (1992), 572-576.  Google Scholar [12] A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; French translation of the 1892 Russian article.  Google Scholar [13] K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522.  Google Scholar [14] K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009.  Google Scholar [15] J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120.  Google Scholar [16] J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), 81-114. doi: 10.1002/cpa.3160110105.  Google Scholar [17] J. Moser, On the elimination of the irrationality condition and Birkhoff's concept of complete stability, Bol. Soc. Mat. Mexicana (2), 5 (1960), 167-175.  Google Scholar [18] J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747; addendum in: Comm. Pure Appl. Math., 31 (1978), 529-530. doi: 10.1002/cpa.3160290613.  Google Scholar [19] F. dos Santos, J. E. Mansilla and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems with $n$-degrees of freedom in the case of single resonance, J. Dynam. Differential Equations, 22 (2010), 805-821. doi: 10.1007/s10884-010-9176-z.  Google Scholar [20] C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., 128 (1954), 144-170. doi: 10.1007/BF01360131.  Google Scholar [21] V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967. Google Scholar [22] C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294.  Google Scholar [23] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.  Google Scholar [24] A. Weinstein, Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265-271. doi: 10.1002/cpa.3160300207.  Google Scholar [25] A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248. doi: 10.1007/BF01214573.  Google Scholar [26] V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975.  Google Scholar

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##### References:
 [1] G. Benettin, F. Fassò and M. Guzzo, Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regul. Chaotic Dyn., 3 (1998), 56-71.  Google Scholar [2] A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219.  Google Scholar [3] H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems, Nonlinearity, 12 (1999), 1351-1362. doi: 10.1088/0951-7715/12/5/309.  Google Scholar [4] T. M. Cherry, On periodic solutions of Hamiltonian systems of differential equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 227 (1928), 137-221. doi: 10.1098/rsta.1928.0005.  Google Scholar [5] N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531. Google Scholar [6] G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88. doi: 10.1515/crll.1846.32.85.  Google Scholar [7] F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292. doi: 10.1007/PL00004245.  Google Scholar [8] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7.  Google Scholar [9] J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526.  Google Scholar [10] L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances, Prikl. Mat. Mekh., 35 (1971), 423-431; English translation, J. Appl. Math. Mech., 35 (1971), 384-391.  Google Scholar [11] A. L. Kunitsyn and A. A. Tuyakbayev, The stability of Hamiltonian systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 56 (1992), 672-675; English translation, J. Appl. Math. Mech., 56 (1992), 572-576.  Google Scholar [12] A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; French translation of the 1892 Russian article.  Google Scholar [13] K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522.  Google Scholar [14] K. R. Meyer, G. R. Hall and D. Offin, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," $2^{nd}$ edition, Springer-Verlag, New York, 2009.  Google Scholar [15] J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120.  Google Scholar [16] J. Moser, New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math., 11 (1958), 81-114. doi: 10.1002/cpa.3160110105.  Google Scholar [17] J. Moser, On the elimination of the irrationality condition and Birkhoff's concept of complete stability, Bol. Soc. Mat. Mexicana (2), 5 (1960), 167-175.  Google Scholar [18] J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747; addendum in: Comm. Pure Appl. Math., 31 (1978), 529-530. doi: 10.1002/cpa.3160290613.  Google Scholar [19] F. dos Santos, J. E. Mansilla and C. Vidal, Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems with $n$-degrees of freedom in the case of single resonance, J. Dynam. Differential Equations, 22 (2010), 805-821. doi: 10.1007/s10884-010-9176-z.  Google Scholar [20] C. L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differential gleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., 128 (1954), 144-170. doi: 10.1007/BF01360131.  Google Scholar [21] V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967. Google Scholar [22] C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294.  Google Scholar [23] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.  Google Scholar [24] A. Weinstein, Symplectic $V$-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265-271. doi: 10.1002/cpa.3160300207.  Google Scholar [25] A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248. doi: 10.1007/BF01214573.  Google Scholar [26] V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975.  Google Scholar
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