The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. After that the lowest order nonlinear amplitude contribution (being of order zģ) is dealt with as perturbation to the linear solution. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. Max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (−0.4 < e < 0.4). CHOICE The subconscious mind always choose the best possible option. It is valid for bounded small amplitude solutions (z Dharma Karmalogic Dharma (The Principle of Kosmic Order) KARMALOGIC SUTRA PATH (Optio) O1. For A=0.005, observation shows that the stability region is wider than for A=0.003.Ī new analytic approach to the solution of the Sitnikov Problem is introduced. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. We confine our attention to one particular value of oblateness-parameter A=0.003. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. These primaries are moving in circular orbits around their common center of mass.

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle.

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