Normally stable hamiltonian systems

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

Normally stable hamiltonian systems

Kenneth R. Meyer ^1, , Jesús F. Palacián ^2, and Patricia Yanguas ^2,

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

Abstract
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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.

Keywords: formal stability, Hamiltonian systems, Liapunov stability, normal form., normal stability, parametric stability.

Mathematics Subject Classification: Primary: 34C20, 34C25, 37J40; Secondary: 70F10, 70K6.

Citation: Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete and 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.
[2]	A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219.
[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.
[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.
[5]	N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531.
[6]	G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88. doi: 10.1515/crll.1846.32.85.
[7]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292. doi: 10.1007/PL00004245.
[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.
[9]	J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526.
[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.
[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.
[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.
[13]	K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522.
[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.
[15]	J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120.
[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.
[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.
[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.
[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.
[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.
[21]	V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967.
[22]	C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294.
[23]	A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.
[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.
[25]	A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248. doi: 10.1007/BF01214573.
[26]	V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975.

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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.
[2]	A. D. Bryuno, Formal stability of Hamiltonian systems, Mat. Zametki, 1 (1967), 325-330; English translation, Math. Notes, 1 (1967), 216-219.
[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.
[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.
[5]	N. G. Chetaev, Un théorème sur l'instabilité, Dokl. Akad. Nauk SSSR, 1 (1934), 529-531.
[6]	G. L. Dirichlet, Über die stabilität des Gleichgewichts, J. Reine Angew. Math., 32 (1846), 85-88. doi: 10.1515/crll.1846.32.85.
[7]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Arch. Ration. Mech. Anal., 158 (2001), 259-292. doi: 10.1007/PL00004245.
[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.
[9]	J. Glimm, Formal stability of Hamiltonian systems, Comm. Pure Appl. Math., 16 (1963), 509-526.
[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.
[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.
[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.
[13]	K. R. Meyer, Normal forms for Hamiltonian systems, Celestial Mech., 9 (1974), 517-522.
[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.
[15]	J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87-120.
[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.
[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.
[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.
[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.
[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.
[21]	V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, New York, 1967.
[22]	C. Vidal, Stability of equilibrium positions of Hamiltonian systems, Qual. Theory Dyn. Syst., 7 (2008), 253-294.
[23]	A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.
[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.
[25]	A. Weinstein, Bifurcations and Hamilton's principle, Math. Z., 159 (1978), 235-248. doi: 10.1007/BF01214573.
[26]	V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," Vol. 1 and 2, John Wiley & Sons, New York, 1975.

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