|
This article is cited in 9 scientific papers (total in 9 papers)
Stability of a Hamiltonian System in a Limiting Case
Kenneth R. Meyera, Jesús F. Palaciánb, Patricia Yanguasb a Department of Mathematical Sciences, University of Cincinnati, 839, Cincinnati, 45221-0025 Ohio, USA
b Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain
Abstract:
We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the $1: -1$ resonance case where the linearized system has double pure imaginary eigenvalues $\pm i\omega, \omega \ne 0$ and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.
This result implies the stability of the Lagrange equilateral triangle libration points, $\mathcal{L}_4$ and $\mathcal{L}_5$, in the planar circular restricted three-body problem when the mass ratio parameter is equal to $\mu_R$, the critical value of Routh.
Keywords:
stability, Lagrange equilateral triangle, KAM tori, Liapunov stable, planar circular restricted three-body problems, Routh’s critical mass ratio.
Received: 23.09.2011 Accepted: 09.11.2011
Citation:
Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas, “Stability of a Hamiltonian System in a Limiting Case”, Regul. Chaotic Dyn., 17:1 (2012), 24–35
Linking options:
https://www.mathnet.ru/eng/rcd380 https://www.mathnet.ru/eng/rcd/v17/i1/p24
|
Statistics & downloads: |
Abstract page: | 88 |
|