First order normalization in the generalized photo gravitational non-planar restricted three body problems
Keywords:Normalisation, Photogravitational, Non-Planar, RTBP.
In this paper, we normalised the second-order part of the Hamiltonian of the problem. The problem is generalised in the sense that fewer massive primary is supposed to be an oblate spheroid. By photogravitational we mean that both primaries are radiating. With the help of Mathematica, H2 is normalised to H2 = a1b1w1 + a2b2w2. The resulting motion is composed of elliptic motion with a short period (2p/w1), completed by an oscillation along the z-axis with a short period (2p/w2).
 Benettin, G.; Fasso, F.; Guzzo, M. Nekhoroshev, 1998, Stability of L4 and L5 in the spatial restricted three-body problem. Regular and chaotic dynamics V. 3, No. 3. http://dx.doi.org/10.1070/rd1998v003n03ABEH000080.
 Celleti, A. and Giorgilli, A., 1991, on the stability of the Lagrangian points in the spatial Restricted Three Body Problem. Celest. Mech. Dynam. Astrr. 50(1): 31-58. http://dx.doi.org/10.1007/BF00048985.
 Duskos, C. N. and Markellos, V.V. 2006, Out-of-plane equilibrium points in the restricted three body problem with oblateness, A & a 446, 357-360.
 Ito, H., 1992, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Math. Ann. 292, 411-444. http://dx.doi.org/10.1007/BF01444629.
 Jorba, A., 2001, Numerical computation of the normal behaviour of invariant curves of n-dimensional maps. Nonlinearity, 14(5): 943-976. http://dx.doi.org/10.1088/0951-7715/14/5/303.
 Kusvah, B. S., Sharma J. P. and Ishwar, B. 2007, Non-linear stability in generalisedphotogravitational restricted three body problems with P-R drag. Ap& SS, 312, 279-293. http://dx.doi.org/10.1007/s10509-007-9688-0.
 Meyer, K. R.; Hall, G. R., 1992, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer-Verlag. New York. http://dx.doi.org/10.1007/978-1-4757-4073-8.
 Oberti, P. andVienne, A., 2003, An upgraded theory forHelene, Telesto and Calypso. A&A 397, 353-359.
 Poschel, J., 1982, the concept of integrability on Cantor sets for Hamiltonian systems, Celestial Mechanics 28, 133-139. http://dx.doi.org/10.1007/BF01230665.
 Sharma, R. K. and Subba Rao, P.V., 1976, Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid. Celes. Mech., 13,137. http://dx.doi.org/10.1007/BF01232721.
 Sokol'skii,A.G., 1978, Proof of the stability of Lagrangian solutions for a critical mass ratio. Sov. Astronom. Lett. 79-81.
 Szebehely, V., 1967, Theory of orbits. Academic Press.New York.