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 - A, 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. Google Scholar

[2]

A. D. Bryuno, Formal stability of Hamiltonian systems,, Mat. Zametki, 1 (1967), 325. Google Scholar

[3]

H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems,, Nonlinearity, 12 (1999), 1351. 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. doi: 10.1098/rsta.1928.0005. Google Scholar

[5]

N. G. Chetaev, Un théorème sur l'instabilité,, Dokl. Akad. Nauk SSSR, 1 (1934), 529. Google Scholar

[6]

G. L. Dirichlet, Über die stabilität des Gleichgewichts,, J. Reine Angew. Math., 32 (1846), 85. 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. 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. 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. Google Scholar

[10]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances,, Prikl. Mat. Mekh., 35 (1971), 423. 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. Google Scholar

[12]

A. Liapounoff, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203. Google Scholar

[13]

K. R. Meyer, Normal forms for Hamiltonian systems,, Celestial Mech., 9 (1974), 517. 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, (2009). Google Scholar

[15]

J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87. Google Scholar

[16]

J. Moser, New aspects in the theory of stability of Hamiltonian systems,, Comm. Pure Appl. Math., 11 (1958), 81. 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. Google Scholar

[18]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein,, Comm. Pure Appl. Math., 29 (1976), 727. 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. 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. doi: 10.1007/BF01360131. Google Scholar

[21]

V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies,", Academic Press, (1967). Google Scholar

[22]

C. Vidal, Stability of equilibrium positions of Hamiltonian systems,, Qual. Theory Dyn. Syst., 7 (2008), 253. Google Scholar

[23]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems,, Invent. Math., 20 (1973), 47. 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. doi: 10.1002/cpa.3160300207. Google Scholar

[25]

A. Weinstein, Bifurcations and Hamilton's principle,, Math. Z., 159 (1978), 235. doi: 10.1007/BF01214573. Google Scholar

[26]

V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients,", Vol. 1 and 2, (1975). Google Scholar

show all references

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. Google Scholar

[2]

A. D. Bryuno, Formal stability of Hamiltonian systems,, Mat. Zametki, 1 (1967), 325. Google Scholar

[3]

H. E. Cabral and K. R. Meyer, Stability of equilibria and fixed points of conservative systems,, Nonlinearity, 12 (1999), 1351. 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. doi: 10.1098/rsta.1928.0005. Google Scholar

[5]

N. G. Chetaev, Un théorème sur l'instabilité,, Dokl. Akad. Nauk SSSR, 1 (1934), 529. Google Scholar

[6]

G. L. Dirichlet, Über die stabilität des Gleichgewichts,, J. Reine Angew. Math., 32 (1846), 85. 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. 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. 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. Google Scholar

[10]

L. G. Khazin, On the stability of Hamiltonian systems in the presence of resonances,, Prikl. Mat. Mekh., 35 (1971), 423. 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. Google Scholar

[12]

A. Liapounoff, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203. Google Scholar

[13]

K. R. Meyer, Normal forms for Hamiltonian systems,, Celestial Mech., 9 (1974), 517. 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, (2009). Google Scholar

[15]

J. Moser, Stabilitätsverhalten kanonischer Differential gleichungs systeme,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa, 6 (1955), 87. Google Scholar

[16]

J. Moser, New aspects in the theory of stability of Hamiltonian systems,, Comm. Pure Appl. Math., 11 (1958), 81. 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. Google Scholar

[18]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein,, Comm. Pure Appl. Math., 29 (1976), 727. 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. 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. doi: 10.1007/BF01360131. Google Scholar

[21]

V. G. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies,", Academic Press, (1967). Google Scholar

[22]

C. Vidal, Stability of equilibrium positions of Hamiltonian systems,, Qual. Theory Dyn. Syst., 7 (2008), 253. Google Scholar

[23]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems,, Invent. Math., 20 (1973), 47. 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. doi: 10.1002/cpa.3160300207. Google Scholar

[25]

A. Weinstein, Bifurcations and Hamilton's principle,, Math. Z., 159 (1978), 235. doi: 10.1007/BF01214573. Google Scholar

[26]

V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients,", Vol. 1 and 2, (1975). Google Scholar

[1]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[2]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612

[3]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317

[4]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

[5]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703

[6]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[7]

Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243

[8]

Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734

[9]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67

[10]

Luca Biasco, Luigi Chierchia. On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 233-262. doi: 10.3934/dcds.2003.9.233

[11]

Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279

[12]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[13]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661

[14]

Ugo Boscain, Grégoire Charlot, Mario Sigalotti. Stability of planar nonlinear switched systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 415-432. doi: 10.3934/dcds.2006.15.415

[15]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

[16]

J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625

[17]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1

[18]

Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373

[19]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[20]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (3)

[Back to Top]