부산대학교 도서관

The central theme of this paper is the study of the perturbed Kepler problemdescribed 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 respectto the central body.\parThe author uses the variables that were introduced by Levi-Civita(1920) in order to regularize the collision singularity in the Keplerproblem, in which the planar Kepler problem appears as a harmonicoscillator in two dimensions, and the KS variables (Kustaanheimo andStiefel, 1965), in which the spatial Kepler problem appears as aharmonic oscillator in four dimensions.\parIn the first part of the paper, the author gives the three stepsnecessary for regularizing the perturbed planar Kepler problem byLevi-Civita's transformation. He provides a nice overview of howquaternion numbers lead to a representation of the perturbedthree-dimensional Kepler problem as a perturbed harmonicoscillator. The use of quaternions to regularize theKepler 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 wayof handling the three-dimensional case by using anunconventional conjugation of quaternions which he had introduced in a previouswork [in {\it Chaoticworlds}, 231--251, Springer, Dordrecht,2006].\parFinally, in the last section of the paper the author outlines theBirkhoff regularizing transformation in the plane with an emphasison 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 throughquaternions.\parThis paper is organized in a convenient way. The results are presentedwith clarity.