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Publ. Natl. Astron. Obs. Japan, Vol. 9, 55{112 (2007) Trends in 20th Century Celestial Mechanics Takashi Ito and Kiyotaka Tanikawa (Received April 28, 2006; Accepted March 14, 2007) Abstract We review some of the major achievements of celestial mechanics in the twentieth century, and discuss some unsolved problems that are left to the twenty-¯rst century. The four major research ¯elds in celestial mechanics are treated: dynamical systems, three-body problems, solar system dynamics, and numerical methods. Over the past decades, celestial mechanics has become extremely specialized and categorized along many trajectories, from purely mathematical to quite practical points of view, as well as in what has involved digital computer innovation. These various ends of celestial mechanics sometimes exist in isolation, losing their connections with each other. The purpose of this manuscript is to ¯nd clues for rediscovering connections between each of these ¯elds by listing some of major achievements in hoping to ¯nd new paths for celestial mechanics in the new century. Key words: celestial mechanics, dynamical system, three-body problem, planetary dynamics, numerical method Contents recently ended twentieth century. Not only celestial me- 1 Introduction 55 chanics but the entire ¯eld of astronomy has evolved and 2 Dynamical systems|from Poincar¶eto chaos 56 drastically changed over the past one hundred years. To 2.1 The KAM theory 58 commemorate the beginning of the new century, we ven- 2.2 Chaos 60 2.3 Twist maps 63 ture this review of the history of celestial mechanics in the 2.4 Integrability 65 past century in order to identify new paths to explore in the 3 The three-body problem 68 forthcoming years. It goes without saying that we possess 3.1 Analytical and qualitative studies 68 neither su±cient knowledge nor experience to thoroughly 3.2 Numerical studies 69 review the celestial mechanics of the past century. Neither 3.3 Hierarchical triple systems 70 3.4 New frontiers of the three-body problems 71 can we predict its way in the new century. What is written 3.5 Applications to actual planetary and stellar systems 73 in this manuscript is our recognition and anticipation of 4 Solar system dynamics 73 the past and the future of several speci¯c areas of celestial 4.1 Progress of perturbation theories 74 mechanics in which we are interested. In this sense, our 4.2 Long-term dynamical stability of planetary motion 76 description may be biased toward the authors' favor. Long 4.3 Dynamics in planet formation study 78 before our attempt, V. Szebehely (1997) stated Open prob- 4.4 Dynamics of small bodies 79 4.4.1 Asteroids and resonant dynamics 79 lems on the eve of the next millennium. Though the scope 4.4.2 The Yarkovsky e®ect and asteroid families 83 of our manuscript is not as large as the next millennium, 4.4.3 Kuiper Belt objects 84 we hope our attempt serves as a milestone for getting clues 4.4.4 The Oort Cloud 85 and opening new frontiers in celestial mechanics. 4.5 Planetary rotation 86 Major flows in nineteenth-century celestial mechanics 4.5.1 Nutation and precession of the Earth 86 converged at H. Poincar¶e.And major flows in twentieth cen- 4.5.2 The Earth{Moon system 87 4.5.3 Evolution of planetary rotations 87 tury celestial mechanics diverged out from G. D. Birkho®. 5 Numerical methods 88 The stability analysis by A. M. Lyapunov, which he started 5.1 Regularization 88 in the late nineteenth century, also grew into a large flow 5.2 Symplectic integrators 89 in the early twentieth century. These three great masters 5.3 Other promising numerical techniques 91 established many of the fundamental concepts of modern 5.4 GRAPE: A special-purpose computer for N-body dynamics 93 celestial mechanics|integrability, convergence of pertur- 6 Concluding Remarks 93 bative developments, normal forms, stability of dynamical Acknowledgments 95 systems, and so on. In the 1940s, C. L. Siegel worked on the References 95 singularity of the three-body and small denominator prob- lems. Based on what these researchers had established, the 1. Introduction research of dynamical systems in the 1950s and the 1960s produced the greatest achievement of celestial mechanics This manuscript represents our e®ort to summarize in the twentieth century: the Kolmogorov{Arnold{Moser some of the major trends in celestial mechanics in the (KAM) theory. Later this research flow morphed into the investigation of chaos phenomena, thanks to the develop- Corresponding author: [email protected] (Takashi Ito). ment of computer technology. 56 Takashi Ito and Kiyotaka Tanikawa In the ¯rst half of the twentieth century, after Poincar¶e jects have helped solar system dynamics achieve a huge and Birkho®, celestial mechanics flourished in a variety of progress. A number of spacecrafts with cutting-edge digital forms. The understanding of planetary motion in our solar technology have explored the solar system objects, giving system progressed, led by the development of accurate per- us precious information about the dynamical nature of our turbation theories. The progress of planetary perturbation planetary system. In addition to this, the recent rush of dis- theories stimulated the search for the ninth planet, result- coveries of extrasolar planetary systems, owing to the devel- ing in the discovery of Pluto by C. W. Tombaugh in 1930. opment of new observing technologies, has signi¯cantly ex- Research on major and minor planets alike became one of tended the ¯eld of celestial mechanics since the mid 1990s. the central issues of celestial mechanics in this era as astro- Bearing this background in mind, from the next section metric observation became increasingly precise. Motivated we review some of the major achievements and remain- by the identi¯cation of asteroid families by K. Hirayama in ing problems of twentieth century celestial mechanics 1 . 1918, the quantitative research of asteroid dynamics began. We have categorized the major ¯elds of celestial mechanics The Soviet Union's launch of the world's ¯rst arti¯cial into four parts: dynamical systems (Section 2), the three- satellite, Sputnik in 1957, turned out to be one of the most body problem (Section 3), solar system dynamics (Section signi¯cant stimuli for celestial mechanics. This epochmak- 4), and numerical methods (Section 5). Section 6 presents ing event marked the start of the space age, and owed much some discussion and concluding remarks. We are fully aware to the development of advanced technology. From the view- of our inability to describe several important topics such point of celestial mechanics, Sputnik represented the tri- as relativistic celestial mechanics, boundary area to stellar umph of Newtonian mechanics and perturbation theories. dynamics, current progress of arti¯cial satellite theories, Theoretical studies of the motion of arti¯cial satellites made observation-related research of near-Earth objects, detailed great advances after Sputnik's launch. description about ring-satellite dynamics, and many other As well as to many other ¯elds of science and engineer- ¯elds. This ignorance just reflects the limitation of our abil- ing, the development of fast computers has made an enor- ities and experience in these ¯elds. We have noted major mous contribution to celestial mechanics, replacing the nu- missing topics and their importance at the beginning of merical studies by hands (\human" computers) that had each section. Throughout this manuscript, we would like to already started in the early twentieth century. The use of provide some thoughts and questions for our readers: What electronic digital computers for numerical studies in ce- is celestial mechanics? How should we de¯ne it? Since the lestial mechanics began in the 1950s. At ¯rst, the general progress of celestial mechanics is quite rapid, the answer three-body problem was targeted as an example study us- to these questions might not be straightforward. But the ing electronic computers. Regularization methods that had subtleties of such questions could help us to delineate new been initially devised for theoretical purposes have become academic frontiers in the new century. important since the 1960s. The techniques of regularization partially overcame one of the biggest di±culties in celestial 2. Dynamical systems|from Poincar¶eto chaos mechanics: collisions. Computers are also used as instru- ments for purely theoretical research. For example, devel- Towards the end of the nineteenth century, H. Poincar¶e opments of dynamical theories were enhanced a great deal revolutionized the human view of nature (Poincar¶e,1892). owing to the detailed numerical survey of phase space. Nature is not so simple as the simple form of the New- The progress of numerical study in celestial mechanics tonian equations of motion suggests, something Poincar¶e owes a lot not only to the development of computer hard- realized through his own experience. His ¯rst experience ware, but also to sophisticated numerical algorithms. A va- perhaps was his failure to show the solvability of the re- riety of numerical integration schemes such as the Runge{ stricted three-body problem in the Grand prix paper for Kutta methods, various predictor{correctors, the extrapo- the Swedish prize in 1889. The failure was said to be related lation method, and the family of symmetric integration to the stable and unstable manifolds of periodic points. At schemes has contributed greatly to the progress of numeri- ¯rst, Poincar¶eimplicitly assumed that these manifolds were cal celestial mechanics. In the 1990s, the knowledge of these connected smoothly. But later, he realized that they gener- numerical techniques was combined with that of Hamilto- ally intersect transversely. His second experience came with nian dynamics, resulting in a totally new type of numerical his success in proving the analytic non-integrability of the scheme: symplectic integrators.

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