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

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Abstract

The present study deals with the normalisation of Hamiltonian for the nonlinear stability analysis in non-resonance case of the triangular equilibrium points in the perturbed restricted three-body problem with perturbation factors as radiation pressure due to first oblate-radiating primary, albedo from second oblate primary, oblateness and a disc. The problem is formulated with these perturbations and Hamiltonian of the problem is normalised up to fourth order by Lie transform technique consequently a Birkhoff’s normal form of the Hamiltonian is obtained. The Arnold–Moser theorem is verified for the nonlinear stability test of the triangular equilibrium points in non-resonance case with the assumed perturbations. It is found that in the presence of radiation pressure, stability range expanded, significantly with respect to the classical range of stability; however, because of albedo, oblateness and the disc, it contracted gradually. Moreover, it is observed that alike to the classical problem, in the perturbed problem under the impact of the assumed perturbations, there always exist one or more values of the mass ratio \(\mu \) within the stability range at which discriminant \(D_4=0\), which means the triangular equilibrium points are unstable in nonlinear sense.

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Acknowledgements

Authors are thankful to UGC, Govt. of India, for providing partial support through the UGC start-up research Grant No.-F.30-356/2017(BSR) and UGC-JRF (Ref. No.- 21/06/2015(i)EU-V). We are also thankful to Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune (India), for sharing some of the references used in this article. Last but not least, we are thankful to all esteemed reviewer(s) for shaping the manuscript in its present form.

Funding

This work was supported by UGC, Govt. of India (Grant numbers [UGC start-up research Grant No.-F.30-356/ 2017(BSR)] and [UGC-JRF (Ref. No.- 21/06/2015(i)EU-V). ]).

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  1. Department of Mathematics, Central University of Rajasthan, Kishangarh, Ajmer, Rajasthan, 305817, India

    Saleem Yousuf & Ram Kishor

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  1. Saleem Yousuf
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SY performed conceptualisation, formal analysis, methodology, software, validation, visualisation, writing— original draft. RK helped in conceptualisation, formal analysis, methodology, resources, software, supervision, validation, visualisation, writing— reviewing and editing.

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Correspondence to Ram Kishor.

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Yousuf, S., Kishor, R. Nonlinear stability of triangular equilibrium points in non-resonance case with perturbations. Nonlinear Dyn 112, 1843–1859 (2024). https://doi.org/10.1007/s11071-023-09142-x

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