The title of this paper is inspired by the work of Poincare [1890, 1892], who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three-body problem. Since then, many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and, more recently, with the design of space missions.
This paper will give a survey of some of these exciting ideas, and we would especially like to acknowledge the work of Michael Dellnitz, Frederic Gabern, Katalin Grubits, Oliver Junge, Wang-Sang Koon, Francois Lekien, Martin Lo, Sina Ober-Blobaum, Kathrin Padberg, Robert Preis, and Bianca Thiere.
One of the purposes of the AMS Current Events session is to discuss work of others. Even though we were involved in the research reported on here, this short paper is intended to survey many ideas due to our collaborators and others.
This survey is by no means complete and we apologize for not having time or space to do justice to many important and fundamental works. In fact, the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics, mission design and in dynamical systems. We mention just a few whose work has had positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gomez, George Haller, Charles Jaffe, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Carles Simo, Turgay Uzer, Steve Wiggins, and Roby Wilson.
In an upcoming monograph (see Koon, Lo, Marsden, and Ross ), the dynamical systems and computational approach and its application to mission design are discussed in detail.
One of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the Sun, planets and moons. The boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three-body problems. In particular, tube-like structures form an interplanetary transport network which will facilitate the exploration of Mercury, the Moon, the asteroids, and the outer solar system, including a mission to assess the possibility of life on Jupiter's icy moons. The use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed.