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New methods in celestial mechanics and mission design
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by Jerrold E. Marsden and Shane D. Ross PDF
Bull. Amer. Math. Soc. 43 (2006), 43-73 Request permission


The title of this paper is inspired by the work of Poincaré [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, François Lekien, Martin Lo, Sina Ober-Blöbaum, 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 had a positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gómez, George Haller, Charles Jaffé, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Carles Simó, Turgay Uzer, Steve Wiggins, and Roby Wilson. In an upcoming monograph (see Koon, Lo, Marsden, and Ross [2005]), 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.
    [Bate et al.(1971)Bate, Mueller, and White]BaMuWh1971 Bate, R. R., D. D. Mueller, and J. E. White [1971], Fundamentals of Astrodynamics. Dover, New York.
  • Edward Belbruno, Capture dynamics and chaotic motions in celestial mechanics, Princeton University Press, Princeton, NJ, 2004. With applications to the construction of low energy transfers; With a foreword by Jerry Marsden. MR 2029316, DOI 10.1515/9780691186436
  • [Combes et al.(1999)Combes, Leon, and Meylan]CoLeMe1999 Combes, F., S. Leon, and G. Meylan [1999], $N$-body simulations of globular cluster tides, Astron. Astrophys. 352, 149–162.
  • Michael Dellnitz, Gary Froyland, and Oliver Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 145–174, 805–807. MR 1850305
  • [Dellnitz et al.(2005)Dellnitz, Grubits, Marsden, Padberg, and Thiere]DeGrMaPaTh2005 Dellnitz, M., K. Grubits, J. E. Marsden, K. Padberg, and B. Thiere [2005], Set-oriented computation of transport rates in 3-degree of freedom systems: the Rydberg atom in crossed fields, Regular and Chaotic Dynamics 10, 173–192.
  • Michael Dellnitz and Oliver Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 221–264. MR 1900656, DOI 10.1016/S1874-575X(02)80026-1
  • Michael Dellnitz, Oliver Junge, Wang Sang Koon, Francois Lekien, Martin W. Lo, Jerrold E. Marsden, Kathrin Padberg, Robert Preis, Shane D. Ross, and Bianca Thiere, Transport in dynamical astronomy and multibody problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), no. 3, 699–727. MR 2136742, DOI 10.1142/S0218127405012545
  • [Dellnitz et al.(2005a)Dellnitz, Junge, Lo, Marsden, Padberg, Preis, Ross, and Thiere]DeJuLoMa2005 Dellnitz, M., O. Junge, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross, and B. Thiere [2005a], Transport of Mars-crossing asteroids from the quasi-Hilda region, Physical Review Letters 94, 231102-1–231102-4. [Dellnitz et al.(2005b)Dellnitz, Junge, Marsden, Padberg, Preis, Ross, and Thiere]De-astro Dellnitz, M., O. Junge, J. E. Marsden, K. Padberg, R. Preis, S. Ross, and B. Thiere [2005b], Almost invariant sets and celestial mechanics (in preparation). [Dunn(1962)]Dunn1962 Dunn, G. L. [1962], A high-speed data link for farside lunar communications, General Electric Co. Report 62 SPC-5, March 1962. [Farquhar(1966)]Farquhar1966 Farquhar, R. W. [1966], Station-Keeping in the Vicinity of Collinear Libration Points with an Application to a Lunar Communications Problem. In Space Flight Mechanics, Science and Technology Series, volume 11, pages 519–535. American Astronautical Society, New York. [Farquhar and Dunham(1981)]FaDu1981 Farquhar, R. W. and D. W. Dunham [1981], A new trajectory concept for exploring the Earth’s geomagnetic tail, Journal of Guidance and Control, 4, 192–196. [Farquhar et al.(1980)Farquhar, Muhonen, Newman, and Heuberger]FaMuNeHe1980 Farquhar, R. W., D. P. Muhonen, C. Newman, and H. Heuberger [1980], Trajectories and Orbital Maneuvers for the First Libration-Point Satellite, Journal of Guidance and Control, 3, 549–554. [Farquhar et al.(1977)Farquhar, Muhonen, and Richardson]FaMuRi1977 Farquhar, R. W., D. P. Muhonen, and D. L. Richardson [1977], Mission Design for a Halo Orbiter of the Earth, Journal of Spacecraft and Rockets 14, 170–177. [Farquhar(1968)]Farquhar1968 Farquhar, R. [1968], The Control and Use of Libration-Point Satellites, PhD thesis, Stanford University. [Fukushige et al.(2000)Fukushige and Heggie]FuHe2000 Fukushige, T. and D. C. Heggie [2000], The time-scale of escape from star clusters, Mon. Not. R. Astron. Soc. 318, 753–761. [Gabern et al.(2005)Gabern, Koon, Marsden and Ross]GaKoMaRo2005 Gabern, F., W.-S. Koon, J. E. Marsden and S. D. Ross [2005], Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom, Physica D (to appear).
  • G. GĂłmez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont, and S. D. Ross, Connecting orbits and invariant manifolds in the spatial restricted three-body problem, Nonlinearity 17 (2004), no. 5, 1571–1606. MR 2086140, DOI 10.1088/0951-7715/17/5/002
  • K. C. Howell, B. T. Barden, and M. W. Lo, Application of dynamical systems theory to trajectory design for a libration point mission, J. Astronaut. Sci. 45 (1997), no. 2, 161–178. MR 1604793, DOI 10.1007/BF03546374
  • [JaffĂ© et al.(2002)JaffĂ©, Ross, Lo, Marsden, Farrelly, and Uzer]JaRoLoMa2002 JaffĂ©, C., S. D. Ross, M. W. Lo, J. E. Marsden, D. Farrelly, and T. Uzer [2002], Statistical Theory of Asteroid Escape Rates, Phys. Rev. Lett. 89, 011101–1. [Junge et al.(2005)Junge, Marsden, and Ober-Blöbaum]JuMaOb2005 Junge, O., J. E. Marsden, and S. Ober-Blöbaum [2005], Discrete mechanics and optimal control, IFAC Proceedings (to appear). [Koon, Lo, Marsden, and Ross(1999)]KoLoMaRo1999 Koon, W. S., M. W. Lo, J. E. Marsden, and S. D. Ross [1999], Constructing a Low Energy Transfer between Jovian Moons. In Celestial Mechanics : an international conference on celestial mechanics, Evanston, Illinois.
  • Wang Sang Koon, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos 10 (2000), no. 2, 427–469. MR 1765636, DOI 10.1063/1.166509
  • [Koon, Lo, Marsden, and Ross(2005)]book Koon, W. S., M. Lo, J. E. Marsden, and S. Ross [2005], Dynamical Systems, the Three-Body Problem and Space Mission Design (to be published). [Kresák(1979)]Kresak1979 Kresák, L. [1979], Dynamical interrelations among comets and asteroids, in Asteroids, Univ. of Arizona Press, Tucson, 289–309. [Lekien(2003)]Lekien2003 Lekien, F. [2003], Time-Dependent Dynamical Systems and Geophysical Flows, PhD thesis, California Institute of Technology.
  • A. Lew, J. E. Marsden, M. Ortiz, and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg. 60 (2004), no. 1, 153–212. MR 2073073, DOI 10.1002/nme.958
  • [Lo et al.(2001)Lo, Williams, Bollman, Han, Hahn, Bell, Hirst, Corwin, Hong, Howell, Barden, and Wilson]LoWi2001 Lo, M., B. G. Williams, W. E. Bollman, D. Han, Y. Hahn, J. L. Bell, E. A. Hirst, R. A. Corwin, P. E. Hong, K. C. Howell, B. Barden, and R. Wilson [2001], Genesis Mission Design, The Journal of the Astronautical Sciences 49, 169–184. [Lo and Ross(1998)]LoRo1998 Lo, M. W. and S. D. Ross [1998], Low energy interplanetary transfers using invariant manifolds of L1, L2 and halo orbits. In AAS/AIAA Space Flight Mechanics Meeting, Monterey, California. [Lo and Ross(2001)]LoRo2001 Lo, M. W. and S. D. Ross [2001], The Lunar L1 Gateway: Portal to the stars and beyond. In AIAA Space 2001 Conference, Albuquerque, New Mexico.
  • J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001), 357–514. MR 2009697, DOI 10.1017/S096249290100006X
  • J. D. Meiss, Symplectic maps, variational principles, and transport, Rev. Modern Phys. 64 (1992), no. 3, 795–848. MR 1183196, DOI 10.1103/RevModPhys.64.795
  • [Meyer and Hall(1992)]MeHa1992 Meyer, K. R. and R. Hall [1992], Hamiltonian Mechanics and the N-Body Problem. Texts in Applied Mathematics Science. Springer-Verlag, Berlin.
  • Burkhard Monien, Robert Preis, and Ralf Diekmann, Quality matching and local improvement for multilevel graph-partitioning, Parallel Comput. 26 (2000), no. 12, 1609–1634. MR 1786939, DOI 10.1016/S0167-8191(00)00049-1
  • [PoincarĂ©(1890)]Poincare1890 PoincarĂ©, H. [1890], Sur le problème des trois corps et les Ă©quations de la dynamique, Acta Math. 13, 1–27. [PoincarĂ©(1892)]Poincare1892 PoincarĂ©, H. [1892], Les MĂ©thodes Nouvelles de la MĂ©canique Celeste. 3 volumes. English translation, New Methods of Celestial Mechanics, History of Modern Physics and Astronomy 13, Amer. Inst. Phys., 1993. [Porter and Cvitanović(2005)]PorCv2005 Porter, M.A. and P. Cvitanović [2005], Ground control to Niels Bohr: Exploring outerspace with atomic physics, Notices of the AMS 52 (October 2005), 1020–1025.
  • Vered Rom-Kedar, Transport in a class of $n$-d.o.f. systems, Hamiltonian systems with three or more degrees of freedom (S’AgarĂł, 1995) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 533, Kluwer Acad. Publ., Dordrecht, 1999, pp. 538–543. MR 1720944
  • V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Rational Mech. Anal. 109 (1990), no. 3, 239–298. MR 1025172, DOI 10.1007/BF00375090
  • [Ross(2003)]Ross2003 Ross, S. D. [2003], Statistical theory of interior-exterior transition and collision probabilities for minor bodies in the solar system, in Libration Point Orbits and Applications, World Scientific, 637–652. [Ross(2005)]Ross2005 Ross, S. D. [2005], A mechanism for capture, escape, and collision in dynamical astronomy (in preparation). [Ross et al.(2003)Ross, Koon, Lo, and Marsden]RoKoLoMa2003 Ross, S. D., W. S. Koon, M. W. Lo, and J. E. Marsden [2003], Design of a Multi-Moon Orbiter. In 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico. Paper No. AAS 03-143. [Roy(1988)]Roy1988 Roy, A. E. [1988], Orbital Motion. Adam Hilger, Bristol, 3rd edition.
  • Radu Serban, Wang S. Koon, Martin W. Lo, Jerrold E. Marsden, Linda R. Petzold, Shane D. Ross, and Roby S. Wilson, Halo orbit mission correction maneuvers using optimal control, Automatica J. IFAC 38 (2002), no. 4, 571–583. MR 2131468, DOI 10.1016/S0005-1098(01)00279-5
  • Stephen Wiggins, Chaotic transport in dynamical systems, Interdisciplinary Applied Mathematics, vol. 2, Springer-Verlag, New York, 1992. MR 1139113, DOI 10.1007/978-1-4757-3896-4
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Additional Information
  • Jerrold E. Marsden
  • Affiliation: Control and Dynamical Systems, California Institute of Technology 107-81, Pasa- dena, California 91125
  • Email:
  • Shane D. Ross
  • Affiliation: Department of Aerospace and Mechanical Engineering, University of Southern California, RRB 217, Los Angeles, California 90089-1191
  • Email:
  • Received by editor(s): May 3, 2005
  • Received by editor(s) in revised form: July 19, 2005
  • Published electronically: November 22, 2005
  • Additional Notes: The first author’s research was supported in part by a Max Planck Research Award and NSF-ITR Grant ACI-0204932.
    The second author’s research was supported by an NSF Postdoctoral Fellowship, DMS 0402842.
    This article is based on a lecture presented January 7, 2005, at the AMS Special Session on Current Events, Joint Mathematics Meetings, Atlanta, GA

  • Dedicated: To Henri PoincarĂ© on the 150th anniversary of his birth.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 43 (2006), 43-73
  • MSC (2000): Primary 70F07, 70F15; Secondary 37J45, 70H33
  • DOI:
  • MathSciNet review: 2188175