Diagonalization of Hamiltonian in the photogravitation-al restricted three body problem with P-R drag

  • Authors

    • Xavier James Raj Vikram Sarabhai Space Centre
    • Bhola Ishwar BRA Bihar University
    2017-08-20
    https://doi.org/10.14419/ijaa.v5i2.7931
  • Hamiltonian, Normalization, Photogravitational, P-R drag, RTBP.

  • Abstract

    In this paper, restricted, three-body problem (RTBP) is generalised to study the non-linear stability of equilibrium points in the photogravitational RTBP with P-R drag. In the present study, both primaries are considered as a source of radiation and effect of P-R drag. Hence the problem will contain four parameters q1, q2, W1 and W2. At first, the Lagrangian and the Hamiltonian of the problem were computed, then the Lagrangian function is expanded in power series of the coordinates of the triangular equilibrium points x and y. Lastly, diagonalized the quadratic term of the Hamiltonian of the problem, which is obtained by expanding original Lagrangian or Hamiltonian by Taylor's series about triangular equilibrium point. Finally, the study concluded that the diagonalizable Hamiltonian is H2=ω1I1-ω2I2.

  • References

    1. [1] Avdhesh Kumar, Sharma JP & Ishwar B (2007) Linear Stability of Triangular Equilibrium Points in the Photogravitational Restricted Three Body Problem with Poynting -Robertson Drag when both Primaries are Oblate Spheroid. Journal of Dynamical Systems and Geometric Theories 5(2), 193-202.https://doi.org/10.1080/1726037X.2007.10698538.
      [2] Chernikov YA (1970) the photo gravitational restricted three body problem. Sov. Astr. AJ. 14(1), 176-181.
      [3] Ishwar B (1997) Non-linear stability in the generalized restricted three body problem. Celestial Mechanics and Dynamical Astronomy 65, 253-289.https://doi.org/10.1007/BF00053509.
      [4] Jaiyeola Sefinat B, Abdulrazaq Abdulraheem &Titiloye Emmanue O (2016) Effects of Poynting-Robertson Drag and Oblateness on the Stability of Photogravitational Restricted Three-Body Problem. Advances in Astrophysics 1(1), 36-46.
      [5] Jorba À (1999) A methodology for the numerical computations of Normal Forms, Centre Manifold and First Integrals of Hamiltonian Systems. Experimental Mathematics 8 154-195.https://doi.org/10.1080/10586458.1999.10504397.
      [6] Kushvah BS, Sharma JP &Ishwar B (2007) nonlinear stability in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. Astrophysics and Space Science 312(3), 279–293.https://doi.org/10.1007/s10509-007-9688-0.
      [7] Murray CD (1994) Dynamical effect of drag in the circular restricted three body problem, 1. Location and stability of Lagrangian equilibrium points. Icarus 112, 465-484.https://doi.org/10.1006/icar.1994.1198.
      [8] Poynting JH (1903) Radiation in the solar system: its Effect on Temperature and its Pressure on Small Bodies. Monthly Notices of the Royal Astronomical Society 64(1), 525-552. https://doi.org/10.1093/mnras/64.1.1a.
      [9] Ragos O &Zafiropoulo FA (1995) A numerical study of the influence of the Poynting-Robertson effect on the equilibrium points of the photogravitational restricted three body problem, 1. Coplanar case. Astron. Soc. 97, 423-438.
      [10] Robertson HP (1937) Dynamical effect of radiation in solar system. Monthly Notices of the Royal Astronomical Society 97, 423-438.https://doi.org/10.1093/mnras/97.6.423.
      [11] Schuerman DW (1980) the restricted three body problem including the radiation pressure. Astrophysics Journal 238, 337-342.https://doi.org/10.1086/157989.
      [12] Singh J &Ishwar B (1999) Stability of the triangular points in generalized photogravitational restricted three body problemâ€, Bulletin Astronomical Society of India 27, 415-424.
      [13] Vivek Kumar Mishra, Sharma JP &Ishwar B (2016) Stability of triangular equilibrium points in the Photogravitational elliptic restricted three body problem with Poynting-Robertson drag. International Journal of Advanced Astronomy, 4(1), 33-38.https://doi.org/10.14419/ijaa.v4i1.5834.
      [14] Vivek Kumar Mishra &Ishwar B (2016) Non-Linear Stability in the Photogravitational Elliptic Restricted Three Body Problem with Poynting-Robertson Drag. Advances in Astrophysics, 1(3), 138-160.

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  • How to Cite

    James Raj, X., & Ishwar, B. (2017). Diagonalization of Hamiltonian in the photogravitation-al restricted three body problem with P-R drag. International Journal of Advanced Astronomy, 5(2), 79-82. https://doi.org/10.14419/ijaa.v5i2.7931

    Received date: 2017-06-05

    Accepted date: 2017-07-07

    Published date: 2017-08-20