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Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 1055–1075 | Cite as

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

  • Ram KishorEmail author
  • M. Xavier James Raj
  • Bhola Ishwar
Article

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.

Keywords

Normalization of Hamiltonian Nonlinear stability Non-resonance case Restricted three body problem Poynting–Roberston drag 

Notes

Acknowledgements

We all are thankful to the Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune for providing references through its library and computation facility in addition to local hospitality. First author is also thankful to UGC, New Delhi for providing financial support through UGC Start-up Research Grant No.-F.30-356/2017(BSR).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral University of RajasthanAjmerIndia
  2. 2.Applied Mathematics DivisionVikram Sarabhai Space CentreThiruvananthapuramIndia
  3. 3.Department of MathematicsB. R. A. Bihar UniversityMuzaffarpurIndia

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