# Normalization of Hamiltonian and nonlinear stability of the triangular equilibrium points in non-resonance case with perturbations

Normalization of Hamiltonian and nonlinear stability of the triangular equilibrium points in... For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information as regards a realistic solution of the problem. In the present study, normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc is performed. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed by computing the fourth order normalized Hamiltonian with the help of Lie transforms. In non-resonance case, nonlinear stability of the system is discussed using the Arnold–Moser theorem. Again, the effects of radiation pressure, oblateness and the presence of the disc are analyzed separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in the nonlinear sense within the stability range 0 < μ < μ 1 = μ c ¯ $0<\mu<\mu_{1}=\bar{\mu_{c}}$ due to failure of the Arnold–Moser theorem. However, perturbation parameters affect the values of μ $\mu$ at which D 4 = 0 $D_{4}=0$ , significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but it can be extended in the future. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Astrophysics and Space Science Springer Journals

# Normalization of Hamiltonian and nonlinear stability of the triangular equilibrium points in non-resonance case with perturbations

, Volume 362 (9) – Aug 3, 2017
18 pages

/lp/springer_journal/normalization-of-hamiltonian-and-nonlinear-stability-of-the-triangular-Ww5Hs3LZbs
Publisher
Springer Netherlands
Subject
Physics; Astrophysics and Astroparticles; Astronomy, Observations and Techniques; Cosmology; Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics) ; Astrobiology
ISSN
0004-640X
eISSN
1572-946X
D.O.I.
10.1007/s10509-017-3132-x
Publisher site
See Article on Publisher Site

### Abstract

For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information as regards a realistic solution of the problem. In the present study, normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc is performed. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed by computing the fourth order normalized Hamiltonian with the help of Lie transforms. In non-resonance case, nonlinear stability of the system is discussed using the Arnold–Moser theorem. Again, the effects of radiation pressure, oblateness and the presence of the disc are analyzed separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in the nonlinear sense within the stability range 0 < μ < μ 1 = μ c ¯ $0<\mu<\mu_{1}=\bar{\mu_{c}}$ due to failure of the Arnold–Moser theorem. However, perturbation parameters affect the values of μ $\mu$ at which D 4 = 0 $D_{4}=0$ , significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but it can be extended in the future.

### Journal

Astrophysics and Space ScienceSpringer Journals

Published: Aug 3, 2017

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