# Normalization of Hamiltonian and nonlinear stability of triangular equilibrium points in the photogravitational restricted three body problem with P-R drag in non-resonance case

## TL;DR

This paper normalizes the Hamiltonian of a photogravitational restricted three-body system with P-R drag to analyze the nonlinear stability of triangular equilibrium points, revealing instability within the classical stability range due to perturbations.

## Contribution

It provides a fourth-order Hamiltonian normalization and nonlinear stability analysis of equilibrium points considering radiation pressure and P-R drag effects.

## Key findings

- Existence of a critical mass parameter where stability fails
- Triangular points are unstable under perturbations within the stability range
- Hamiltonian normalization up to fourth order aids stability analysis

## Abstract

Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects. This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries. The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is computed followed by nonlinear stability of the equilibrium points is examined. Similar to the case of classical problem, we have found that in the presence of assumed perturbations, there always exists one value of mass parameter within the stability range at which the discriminant $D_4$ vanish, consequently, Arnold-Moser theorem fails, which infer that triangular equilibrium points are unstable in nonlinear sense within the stability range. Present analysis is limited up to linear effect of the perturbations, which will be helpful to study the more generalized problem.