The Restricted Three-Body Problem and Holomorphic Curves

Description

diiscusses Kepler problem's Runge-Lenz vector, Moser regularization, Levi-Civita regularization, and completely integrable systems

cf.

• Frauenfelder, Urs, and Joa Weber. “The Fine Structure of Weber’s Hydrogen Atom: Bohr–Sommerfeld Approach.” Zeitschrift Für Angewandte Mathematik Und Physik 70, no. 4 (June 21, 2019): 105.

and U. Augsburg's mathematicians' Fraunfelder's and Cieliebak's talks at Symplectic Topology meets Celestial and Quantum Mechanics via Weber Electrodynamics.

The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of four-dimensional symplectizations and symplectic cobordisms, and their applications to celestial mechanics.

The authors study the restricted three-body problem using recent techniques coming from the theory of pseudo-holomorphic curves. The book starts with an introduction to relevant topics in symplectic topology and Hamiltonian dynamics before introducing some well-known systems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. After an overview of different regularizations of these systems, the book continues with a discussion of periodic orbits and global surfaces of section for these and more general systems. The second half of the book is primarily dedicated to developing the theory of holomorphic curves - specifically the theory of fast finite energy planes - to elucidate the proofs of the existence results for global surfaces of section stated earlier. The book closes with a chapter summarizing the results of some numerical experiments related to finding periodic orbits and global surfaces of sections in the restricted three-body problem.

MR3837531

Why another book on celestial mechanics? Aren't there any introductions to the field, such as the recent book by H. Geiges [An introduction to contact topology, Cambridge Stud. Adv. Math., 109, Cambridge Univ. Press, Cambridge, 2008; MR2397738]? Because it's so important? Classical and beautiful?
The authors preemptively answer this question by quoting G. D. Birkhoff [Trans. Amer. Math. Soc. 14 (1913), no. 1, 14–22; MR1500933]: "This state of affairs seems to me to make it probable that the restricted problem of three bodies admit of reduction to the transformation of a discoid into itself as long as there is a closed oval of zero velocity about J(upiter), and that in consequence there always exists at least one direct periodic orbit of simple type.'' (p. 1) Clearly, Birkhoff asks for the existence of a global disc-like surface of sections. Or, in the words of the authors: "Translated into modern language, Birkhoff asks if below the first critical energy value in each bounded component of the restricted three-body problem there exists a disk-like global surface of section.'' (p. 1) And, indeed, a positive answer is very likely in view of the theory of finite energy holomorphic curves due to H. H. W. Hofer, K. Wysocki and E. J. Zehnder [Duke Math. J. 89 (1997), no. 3, 603–617; MR1470344]. To explain this and to motivate research on Birkhoff's question is the aim of the book under review.
But what is this restricted three-body problem about? It is about "the dynamics of a massless body attracted by two massive bodies according to Newton's law of gravitation.'' (p. 2) Typically, what one has in mind is a satellite in the earth-moon system or the moon together with the sun and the earth. "Or one could think of the massless body' as the planet Tatooine attracted by the two stars Tatoo I and Tatoo II as in the Star Wars saga'' (p. 2), as the authors make clear in their introduction. At this moment, it becomes obvious that "[t]he purpose of these notes is to make young ambitious researchers familiar with … Birkhoff 's question'' (p. 1). The starting point of the fulfillment of this programme is an introduction to symplectic geometry guided by the needs of celestial mechanics. And of course, there is an introduction to the classical techniques to understand the dynamics of the restricted three-body problem such as symmetries, regularisations and Lagrange points. In order to make it clear how holomorphic curves come into play, the authors explain the contact geometry of related energy surfaces and their dynamical convexity properties. An introduction to the Fredholm, compactness and intersection theory of holomorphic curves is given in order to add a new technique to the restricted three-body problem: finite energy planes as disc-like global surfaces of section. After reading this well-written and highly inspiring book one will be convinced that "holomorphic curves … confirm Poincaré's fantastic insight [H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Tome I, reprint of the 1892 original, Grands Class. Gauthier-Villars, Lib. Sci. Tech. Albert Blanchard, Paris, 1987; [MR0926906](https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=926906&loc=fromrevtext); Les méthodes nouvelles de la mécanique céleste. Tome II, reprint of the 1893 original, Grands Class. Gauthier-Villars, Lib. Sci. Tech. Albert Blanchard, Paris, 1987; [MR0926907](https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=926907&loc=fromrevtext); Les méthodes nouvelles de la mécanique céleste. Tome III, reprint of the 1899 original, Grands Class. Gauthier-Villars, Lib. Sci. Tech. Albert Blanchard, Paris, 1987; [MR0926908](https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=926908&loc=fromrevtext)] that periodic orbits in some sense are theskeleton' of the dynamics'' (p. 2). Reviewed by Kai Zehmisch