Skip to main content
SpringerLink
Log in

Numerical Integration of the Equations of Motion

  • Chapter
  • First Online:
Practical Astrodynamics

Part of the book series: Springer Aerospace Technology ((SAT))

  • 2536 Accesses

Abstract

As has been shown in Sect. 1.14, the n-body problem (which consists in determining the motion of an isolated set of n bodies, having masses m 1, m 2, …, m n , and attracting one another with Newtonian gravitational forces) cannot be solved analytically in the general case, when n is greater than two.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J.L. Engvall, An engineering exposé of numerical integration of ordinary differential equations. Technical Note No. TN D-3969 (NASA, Houston, 1966)

    Google Scholar 

  2. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962)

    MATH  Google Scholar 

  3. M.H. Rogers, Ordinary differential equations, in Handbook of Applicable Mathematics, vol. III, ed. by R.F. Churchhouse (Wiley, Chichester, 1981). ISBN 0-471-27947-1

    Google Scholar 

  4. E. Fehlberg, Low-order classical Runge–Kutta formulas with stepsize control and their application to some heat transfer problems. NASA TR R-315, p. 48. Website http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690021375.pdf

  5. L. Collatz, The Numerical Treatment of Differential Equations, 3rd edn. (Springer, Berlin, 1960)

    Book  MATH  Google Scholar 

  6. E. Fehlberg, Classical fifth-, sixth-, seventh-, and eighth-order Runge–Kutta formulas with stepsize control. NASA TR R-287. Oct 1968, p. 89. Website http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680027281.pdf

  7. J.H. Verner, Explicit Runge–Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 15(4), 772–790 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, Chichester, 2003). ISBN 0-471-96758-0

    Book  MATH  Google Scholar 

  9. F.T. Krogh, in An Adams Guy Does the Runge–Kutta, Computing Memorandum, vol. 554 (California Institute of Technology, Jet Propulsion Laboratory, Pasadena, 1997), pp. 1–16

    Google Scholar 

  10. R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA Education Series, New York, 1987). ISBN 0-930403-25-8

    MATH  Google Scholar 

  11. D.G. Bettis, A Runge–Kutta Nyström Algorithm. Celest. Mech. 8(2), 229–233 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Hairer, S.P. Nørsett, G. Wanner, in Solving Ordinary Differential Equations, vol. I, 2nd edn. (Springer, Berlin, 2000). ISBN 3-540-56670-8

    Google Scholar 

  13. A. Marthinsen, Continuous extensions to Nyström methods for second order initial value problems. BIT 36(2), 309–332 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. P.W. Sharp, J.H. Verner, Completely imbedded Runge–Kutta pairs. SIAM J. Numer. Anal. 31(4), 1169–1190 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. P.J. Prince, J.R. Dormand, High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  17. O. Montenbruck, E. Gill, Satellite Orbits (Springer, Berlin, 2005). ISBN 3-540-67280-X

    MATH  Google Scholar 

  18. J.R. Dormand, P.J. Prince, New Runge–Kutta algorithms for numerical simulation in dynamical astronomy. Celest. Mech. 18(3), 223–232 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Filippi, J. Gräf, in RKN Methods by Filippi and Gräf (1986). www.josef-graef.de

  20. M.K. Horn, Fourth- and fifth-order, scaled Runge–Kutta algorithms for treating dense output. SIAM J. Numer. Anal. 20(3), 558–568 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Owren, M. Zennaro, in Continuous Explicit Runge–Kutta Methods. Proceedings of the London 1989 Conference on Computational ODEs (1989), pp. 1–9

    Google Scholar 

  22. B. Owren, Continuous explicit Runge–Kutta methods with applications to ordinary and delay differential equations. Ph.D. thesis (1997)

    Google Scholar 

  23. B. Owren, Private communications, 24 and 27 Feb 2006

    Google Scholar 

  24. J.R. Dormand, Numerical Methods for Differential Equations—A Computational Approach (CRC Press, Boca Raton, 1996). ISBN 0-8493-9433-3

    MATH  Google Scholar 

  25. O. Montenbruck, E. Gill, State interpolation for on-board navigation systems. Aerosp. Sci. Technol. 5, 209–220 (2001)

    Article  MATH  Google Scholar 

  26. W.H. Enright, D.J. Higham, B. Owren, P.W. Sharp, A survey of the explicit Runge–Kutta method, Technical report 291/94 (revised in Apr 1995), Department of Computer Science, University of Toronto (1994), pp. 1–36

    Google Scholar 

  27. W.H. Enright, Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math. 125, 159–170 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. W.H. Enright, K.R. Jackson, S.P. Nørsett, P.G. Thomsen, Interpolants for Runge–Kutta formulas. ACM Trans. Math. Softw. 12(3), 193–218 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Görtz, R. Scherer, Reducibility and characterization of symplectic Runge–Kutta methods. Electron. Trans. Numer. Anal. 2, 194–204 (1994)

    MathSciNet  MATH  Google Scholar 

  30. L.Y. Chou, P.W. Sharp, Order 5 symplectic Runge–Kutta Nyström methods. J. Appl. Math. Decis Sci 4(2) (2000)

    Google Scholar 

  31. D.I. Okunbor, E.J. Lu, Eighth-order explicit symplectic Runge–Kutta–Nyström integrators. Technical report CSC 94-21, Department of Computer Science, University of Missouri-Rolla (1994), pp. 1–13

    Google Scholar 

  32. S. Blanes, P.C. Moan, Practical Symplectic Partitioned Runge–Kutta and Runge–Kutta–Nyström Methods. University of Cambridge, Numerical Analysis Report No. DAMTP 2000/NA13, Nov 2000

    Google Scholar 

  33. M.P. Calvo, J.M. Sanz-Serna, High-order symplectic Runge–Kutta–Nyström methods. SIAM J. Sci. Comput. 14(5), 1237–1252 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. D.I. Okunbor, R.D. Skeel, Canonical Runge–Kutta–Nyström methods of orders five and six. J. Comput. Appl. Math. 51, 375–382 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. P.W. Sharp, R. Vaillancourt, The error growth of some symplectic explicit Runge–Kutta Nyström methods on long N-body simulations, 24 Aug 2001

    Google Scholar 

  36. O. Montenbruck, Numerical integration methods for orbital motion. Celest. Mech. Dyn. Astron. 53, 59–69 (1992)

    Article  Google Scholar 

  37. E. Fehlberg, Klassische Runge–Kutta–Nyström-Formeln mit Schrittweiten-Kontrolle für Differentialgleichungen x″= f(t, x, x′). Computing 14, 371–387 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)

    Google Scholar 

  39. J.R. Dormand, M.E. El-Mikkawy, J.P. Prince, High-order embedded Runge–Kutta–Nyström formulae, IMA J. Numer. Anal. 7(4), 423–430

    Google Scholar 

  40. W.B. Gragg, On extrapolation algorithms for ordinary initial value problems. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal. 2(3), 384–403 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  41. L.F. Richardson, The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philos. Trans. R. Soc. Ser. A 210, 307–357 (1910)

    Article  Google Scholar 

  42. L.F. Richardson, J.A. Gaunt, The deferred approach to the limit. Philos. Trans. R. Soc. Ser. A 226, 299–361 (1927)

    Article  MATH  Google Scholar 

  43. R. Bulirsch, J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8(1), 1–13 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  44. B. Démidovitch, I. Maron, Eléments de Calcul Numérique (Editions MIR, Moscou, 1979)

    Google Scholar 

  45. P. Lynch, Richardson extrapolation: the power of the 2-gon. Math. Today 159–160 (2003)

    Google Scholar 

  46. C.F. Gerald, P.O. Wheatley, Applied Numerical Analysis (Addison-Wesley, Reading, 1984). ISBN 0-201-11577-8

    Google Scholar 

  47. P. Deuflhard, Order and stepsize control in extrapolation methods. Numer. Math. 41(3), 399–422 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Kirpekar, Implementation of the Bulirsch Stoer extrapolation method, non-refereed technical report (2003)

    Google Scholar 

  49. P.E. Chase, Stability properties of predictor-corrector methods for ordinary differential equations. J. Assoc. Comput. Mach. 9(4), 457–468 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  50. R.W. Hamming, Stable predictor-corrector methods for ordinary differential equations. J. Assoc. Comput. Mach. 3(1), 37–47 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  51. J.B. Scarborough, Numerical Mathematical Analysis (Oxford University Press, London, 1962)

    MATH  Google Scholar 

  52. M.M. Berry, A variable-step double-integration multi-step integrator. Doctoral dissertation (Blacksburg, 2004)

    Google Scholar 

  53. J. Jackson, Note on the numerical integration of d2 x/dt 2 = f(x, t). Mon. Not. R. Astron. Soc. 84, 602–606 (1924)

    Article  Google Scholar 

  54. R.H. Merson, Numerical integration of the differential equations of celestial mechanics. Royal Aircraft Establishment Technical Report 74184 (1974)

    Google Scholar 

  55. S. Herrick, Astrodynamics, vol. 1 (Van Nostrand Reinhold, London, 1971). ISBN 0-442-03370-2

    MATH  Google Scholar 

  56. S. Herrick, Astrodynamics, vol. 2 (Van Nostrand Reinhold, London, 1972). ISBN 0-442-03371-0

    MATH  Google Scholar 

  57. A. Ralston, Numerical integration methods for the solution of ordinary differential equations, in Mathematical Methods for Digital Computers, ed. by A. Ralston, H.S. Wilf (Wiley, New York, 1960)

    Google Scholar 

  58. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1965). ISBN 0-486-61272-4

    MATH  Google Scholar 

  59. K. Fox, Numerical integration of the equations of motion in celestial mechanics. Celest. Mech. 33, 127–142 (1984)

    Article  MATH  Google Scholar 

  60. K.F. Sundman, Mémoire sur le problème des trois corps. Acta Math. 36, 105–179 (1912)

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Nacozy, The intermediate anomaly. Celest. Mech. 16, 309–313 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  62. M.M. Berry, L. Healy, The generalized Sundman transformation for propagation of high-eccentricity elliptical orbits. Paper AAS 02-109, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, 27–30 Jan 2002

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rome, Italy

    Alessandro de Iaco Veris

Authors
  1. Alessandro de Iaco Veris
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro de Iaco Veris .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

de Iaco Veris, A. (2018). Numerical Integration of the Equations of Motion. In: Practical Astrodynamics. Springer Aerospace Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-62220-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62220-0_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62219-4

  • Online ISBN: 978-3-319-62220-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation