### Abstract

We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions (L_{4} and L_{5}) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named μ_{0}, where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.

Original language | English (US) |
---|---|

Pages (from-to) | 243-253 |

Number of pages | 11 |

Journal | Qualitative Theory of Dynamical Systems |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Apr 15 2013 |

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### Keywords

- KAM theory
- Restricted Three Body Problem
- Stability

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

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*Qualitative Theory of Dynamical Systems*, vol. 12, no. 1, pp. 243-253. https://doi.org/10.1007/s12346-012-0093-x

**Stability of Elliptic Periodic Points with an Application to Lagrangian Equilibrium Solutions.** / Hua, Yongxia; Xia, Zhihong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stability of Elliptic Periodic Points with an Application to Lagrangian Equilibrium Solutions

AU - Hua, Yongxia

AU - Xia, Zhihong

PY - 2013/4/15

Y1 - 2013/4/15

N2 - We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions (L4 and L5) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named μ0, where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.

AB - We study the stability of elliptic rest points and periodic points of Hamiltonian systems of two degrees of freedom. We try to understand to what extent the linear stability would imply nonlinear stability. In case of one and one half degrees of freedom, linear stability, most of the times, do imply nonlinear stability provided that certain conditions on the eigenvalues are satisfied. This is a result of Herman. However, this is no longer the case for two degrees of freedom. We will analyse this case and it turns out that we can still say a lot about the nonlinear stability even in this case. As an example, we consider the Lagrange solutions (L4 and L5) of circular restricted three body problem. For certain mass ratios of the primaries, the Lagrangian solution is elliptic and the high order term in the normal form is non-degenerate, and therefore the Lagrangian point is stable from the standard KAM theory. However, there is one particular value, famously named μ0, where the fourth order term in the normal form happens to be degenerate. We will apply our result to this case.

KW - KAM theory

KW - Restricted Three Body Problem

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=84875975638&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875975638&partnerID=8YFLogxK

U2 - 10.1007/s12346-012-0093-x

DO - 10.1007/s12346-012-0093-x

M3 - Article

AN - SCOPUS:84875975638

VL - 12

SP - 243

EP - 253

JO - Qualitative Theory of Dynamical Systems

JF - Qualitative Theory of Dynamical Systems

SN - 1575-5460

IS - 1

ER -