An Introduction to Differential Algebra and the Differential Algebra Manifold Representation

Conference paper
Part of the Astrophysics and Space Science Proceedings book series (ASSSP, volume 44)


Differential Algebra techniques have been used extensively in the past decade to treat various problems in astrodynamics. In this paper we review the Differential Algebra technique and present four different views of the method. We begin with the introduction of the mathematical definition of the technique as a particular algebra of polynomials. We then give an interpretation of the computer implementation of the method as a way to represent function spaces on a computer, which naturally leads to a view of the method as an automatic differentiation technique. We then proceed to the set theoretical view of Differential Algebra for representing sets of points efficiently on a computer, which is of particular value in astrodynamics. After this introduction to the well known classical DA techniques, we introduce the concept of a DA manifold and show how they naturally arise as an extension of classical DA set propagation. A manifold propagator that allows the accurate propagation of large sets of initial conditions by means of automatic domain splitting (ADS) is described. Its function is illustrated by applying it to the propagation of a set of initial conditions in the two-body problem.


Truncation Error Cartesian Space Differential Algebra Float Point Number Expansion Point 
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  1. Alessi, E.M., Farres, A., Vieiro, A., Jorba, À., Simó, C.: Jet transport and applications to neos. In: Proceedings of the 1st IAA Planetary Defense Conference, Granada (2009)Google Scholar
  2. Armellin, R., Di Lizia, P., Bernelli-Zazzera, F., Berz, M.: Asteroid close encounters characterization using diffferential algebra: the case of apophis. Celest. Mech. Dyn. Astron. 107 (4), 451–470 (2010)ADSCrossRefzbMATHGoogle Scholar
  3. Berz, M.: The method of power series tracking for the mathematical description of beam dynamics. Nucl. Instrum. Methods A258 (3), 431–436 (1987)ADSCrossRefGoogle Scholar
  4. Berz, M.: Modern Map Methods in Particle Beam Physics. Academic, New York (1999)Google Scholar
  5. Bignon, E., Pinède, R., Azzopardi, V., Mercier, P.: Jack: an accurat numerical orbit propagator using taylor differential algebra. In: Presentation at KePASSA Workshop, Logroño, 23–25 April 2014Google Scholar
  6. Di Lizia, P., Armellin, R., Lavagna, M.: Application of high order expansions of two-point boundary value problems to astrodynamics. Celest. Mech. Dyn. Astron. 102, 355–375 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. Di Lizia, P., Armellin, R., Zazzera, F.B., Jagasia, R., Makino, K., Berz, M.: Validated integration of solar system dynamics. In: Proceedings of the 1st IAA Planetary Defense Conference, Granada (2009)Google Scholar
  8. Lee, J.: Manifolds and Differential Geometry. Graduate Studies in Mathematics, vol. 107. American Mathematical Society, Providence (2009)Google Scholar
  9. Makino, K.: Rigorous analysis of nonlinear motion in particle accelerators. PhD thesis, Michigan State University (1998)Google Scholar
  10. Makino, K., Berz, M.: Cosy infinity version 9. Nuclear Instrum. Methods A558, 346–350 (2005)ADSGoogle Scholar
  11. Topputo, F., Zhang, R., Zazzera, F.B.: Numerical approximation of invariant manifolds in the restricted three-body problem. In: Proceedings of the 64th International Astronautical Congress. International Astronautical Federation, Paris (2013). IAC-13,C1,9,11,x18153Google Scholar
  12. Valli, M., Armellin, R., Di Lizia, P., Lavagna, M.R.: Nonlinear mapping of uncertainties in celestial mechanics. J. Guid. Control Dyn. 36 (1), 48–63 (2013)ADSCrossRefGoogle Scholar
  13. Wittig, A., Di Lizia, P., Armellin, R., Makino, K., Bernelli-Zazzera, F., Berz, M.: Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting. Celest. Mech. Dyn. Astron. 122 (3), 239–261 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.ESA Advanced Concepts TeamNoordwijkThe Netherlands

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