# Normalization of Hamiltonian and Nonlinear Stability of the Triangular Equilibrium Points in Non-resonance Case with Perturbations

## TL;DR

This paper normalizes the Hamiltonian and analyzes nonlinear stability of the triangular equilibrium points in a perturbed non-resonance case, revealing instability within the stability range and effects of perturbations on stability parameters.

## Contribution

It provides a detailed normalization of the Hamiltonian up to fourth order and applies Arnold-Moser theorem to analyze nonlinear stability under perturbations in a Chermnykh-like problem.

## Key findings

- Triangular equilibrium points are unstable in the nonlinear sense within the stability range.
- Perturbation parameters significantly influence the stability boundary values.
- The study extends understanding of stability under radiation pressure, oblateness, and disc effects.

## 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 about realistic solution of the problem. Present paper reflects about 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. 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. Due to the presence of perturbations and a tedious huge algebraic computation for intermediate terms, we have computed only up to the fourth order normalized Hamiltonian using Lie transforms. In non-resonance case, nonlinear stability of the system is discussed with the help of Arnold-Moser theorem. Again, the effects of radiation pressure, oblateness and 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 nonlinear sense within the stability range $0<\mu<\mu_1=\bar{\mu_c}$ due to failure of Arnold-Moser theorem. However, perturbation parameters affect the values of $\mu$ at which $D_4=0$, 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 in future it can be extended.