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

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.

Non-linear stability of triangular equilibrium points in non-resonance case with perturbations

“…2 ) with A, B, C as constants to be determined. This form can be found by imposing non-resonance condition on the frequencies ω 1 , ω 2 described as in [30,31,45] and which is stated as if frequencies of infinitesimal mass in linear dynamics are ω 1 , ω 2 and s ∈ Z is such that s ≥ 2, then…”

“…Since, at least fourth order Birkhoff's normal form of the normalised Hamiltonian is required to verify the Arnold-Moser theorem, which can be obtained from second order normalized Hamiltonian (44) by using the Lie transform method described in [18,22,30,31,46]. Suppose, higher order normalized Hamiltonian [46,47] is given as…”

“…Alvarez-Ramírez et al [25] have obtained fourth order normal form of the Hamiltonian in presence of radiation pressure and have applied Arnold-Moser theorem to examine the non-linear stability of L 4 point. In literature, Several researchers [26][27][28][29][30][31][32] have studied either RTBP or its generalised form with different kinds of perturbations for non-linear stability of equilibrium points and have analysed the effect of perturbations in the context of the nature of the stability property. Further, Zepeda Ramírez et al [33,34] have presented the non-linear stability analysis of equilibrium points in the planar equilateral restricted mass-unequal fourbody problem.…”