부산대학교 도서관

The central theme of this paper is the study of the perturbed Kepler problem described by nonlinear differential equations for the vector $x\in \Bbb{R}^n$, $n=2,3$, as a function of time, given by $$ \ddot{x}+\mu\frac{x}{r^3}=\varepsilon f(x,t),\quad r=\|x\|, $$ where $x$ is the position vector of the moving particle with respect to the central body. \par The author uses the variables that were introduced by Levi-Civita (1920) in order to regularize the collision singularity in the Kepler problem, in which the planar Kepler problem appears as a harmonic oscillator in two dimensions, and the KS variables (Kustaanheimo and Stiefel, 1965), in which the spatial Kepler problem appears as a harmonic oscillator in four dimensions. \par In the first part of the paper, the author gives the three steps necessary for regularizing the perturbed planar Kepler problem by Levi-Civita's transformation. He provides a nice overview of how quaternion numbers lead to a representation of the perturbed three-dimensional Kepler problem as a perturbed harmonic oscillator. The use of quaternions to regularize the Kepler problem in the spatial case has been contemplated in other works [J. Vrbik, J. Phys. A {\bf 28} (1995), no.~21, 6245--6252; MR1364798 (97a:70015); M. D. Vivarelli, Meccanica {\bf 29} (1994), no. 1, 15--26; Zbl 0847.70014], but here the author presents a new, elegant way of handling the three-dimensional case by using an unconventional conjugation of quaternions which he had introduced in a previous work [in {\it Chaotic worlds}, 231--251, Springer, Dordrecht, 2006]. \par Finally, in the last section of the paper the author outlines the Birkhoff regularizing transformation in the plane with an emphasis on the generalization to the spatial case, in a similar way as in [E. Stiefel and J. Waldvogel, C. R. Acad. Sci. Paris {\bf 260} (1965), 805; MR0174367 (30 \#4572)], but here he develops the Birkhoff regularization through quaternions. \par This paper is organized in a convenient way. The results are presented with clarity.