##
**A new analytic approach to the Sitnikov problem.**
*(English)*
Zbl 0757.70006

J. Hagel publishes here a very interesting paper which represents a step towards in one of the particular case of the difficult \(N\)-body problem. The paper is a very complex and however a clear one. The author uses a new analytical method for the solution of the Sitnikov problem. It is valid in a regime of \(-0.4<e<0.4\) and \(0<z(0)<0.2\).

In the beginning there is an introduction which presents the Sitnikov problem and the most important steps obtained to resolve this problem in some particular cases. Five chapters follow the introduction. They start with the equation of motion written in terms of the time \(t\), given by Moser (1973): \(\ddot z+z/[r(t)^ 2+z^ 2]^{3/2}=0\). The author presents the hypothesis and the equation of motion obtaining in this case — the Hill equation. The nonlinear ordinary second order differential equation is solved using Floquet’s theory. The approximate method leads to a recursive system. A numerical solution is obtained using the Runge- Kutta method. There are presented two figures that show approximate results related to the Hill equation. A case is presented which brings the Hill equation to a simple harmonic oscillator form. For finding an appropriate ansatz to two important integrals that must be calculated one uses other new variables and functions. There is a lot of expressions and equations used for the estimate of these two integrals. Finally the author obtains an analytical expression for \(z(t)\) — the distance of the massless body from the barycenter along the system symmetry axis.

The fifth chapter “Evaluation of the results” presents the comparison with the results on numerical integration obtained by usual 4-th order Runge-Kutta procedure. We can observe that the analytical results represent very well the numerical integration, the agreement between analytical and numerical results is very good for the entire range of parameters.

Every significant result is particularized in those cases, being studied in this paper and enumerated in the introduction. Thus Hagel’s paper appears to us like a generalization of related results obtained earlier.

In the beginning there is an introduction which presents the Sitnikov problem and the most important steps obtained to resolve this problem in some particular cases. Five chapters follow the introduction. They start with the equation of motion written in terms of the time \(t\), given by Moser (1973): \(\ddot z+z/[r(t)^ 2+z^ 2]^{3/2}=0\). The author presents the hypothesis and the equation of motion obtaining in this case — the Hill equation. The nonlinear ordinary second order differential equation is solved using Floquet’s theory. The approximate method leads to a recursive system. A numerical solution is obtained using the Runge- Kutta method. There are presented two figures that show approximate results related to the Hill equation. A case is presented which brings the Hill equation to a simple harmonic oscillator form. For finding an appropriate ansatz to two important integrals that must be calculated one uses other new variables and functions. There is a lot of expressions and equations used for the estimate of these two integrals. Finally the author obtains an analytical expression for \(z(t)\) — the distance of the massless body from the barycenter along the system symmetry axis.

The fifth chapter “Evaluation of the results” presents the comparison with the results on numerical integration obtained by usual 4-th order Runge-Kutta procedure. We can observe that the analytical results represent very well the numerical integration, the agreement between analytical and numerical results is very good for the entire range of parameters.

Every significant result is particularized in those cases, being studied in this paper and enumerated in the introduction. Thus Hagel’s paper appears to us like a generalization of related results obtained earlier.

Reviewer: A.Muntean (Constanza)

### MSC:

70F07 | Three-body problems |

### Keywords:

perturbation theory; equation of motion; Hill equation; Floquet’s theory; Runge-Kutta method; harmonic oscillator; numerical integration
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\textit{J. Hagel}, Celest. Mech. Dyn. Astron. 53, No. 3, 267--292 (1992; Zbl 0757.70006)

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### References:

[1] | Courant, E.D., Livingston, M.S. and Snyder, H.S.: 1952, Phys. Rev. 88. |

[2] | Goldstein, H.: 1951, Classical Mechanics, Addison-Wesley, Reading. · Zbl 0043.18001 |

[3] | Lichtenberg, A.J. and Liebermann, M.A.: 1983, Regular and Stochastic Motion, Springer Verlag. |

[4] | MacMillan, W.D.: 1913, ?An Integrable Case in the Restricted Problem of Three Bodies?, AJ. 27, 11. |

[5] | Moser, J.: 1973, ?Stable and Random Motion in Dynamical Systems?, Annals ofMathematics Studies Number 77. · Zbl 0271.70009 |

[6] | Moser J.: 1878, Mathematical Intellegencer 1, 65. |

[7] | Sitnikov, K.: 1960, ?Existence of Oscillating Motion for the Three-Body Problem?, Dokl. Akad. Nauk. USSR 133, no. 2, 303-306. · Zbl 0108.18603 |

[8] | Stumpff K.: 1965, Himmelsmechanik, Band II, VEB, Berlin. |

[9] | Jie Liu and Yi-Sui Sun: 1990, ?On the Sitnikov Problem?, Cel. Mech. 49, 285-302. · Zbl 0718.70005 |

[10] | Wodnar, K.: ?New Formulations of the Sitnikov Problem?, Preprint, Institute of Astronomy, Vienna, Austria (1990), to be submitted to Cel. Mech. |

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