Abstract
Methods for transforming differential equations are developed in such a way that the transformed equation is better suited for numerical integration. In order to obtain theoretical insight into such mechanisms we discuss mostly differential equations which are also solvable in closed analytical form. The methods under consideration are aimed at application in cases where small perturbing terms are added to these equations.
Keywords
- Fundamental Frequency
- Celestial Mechanic
- Control Term
- Restricted Problem
- Closed Analytical Form
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V. Szebehely: Theory of orbits. The restricted problem of three bodies, Academic press 1967.
E.L. Stiefel, G. Scheifele: Linear and regular celestial mechanics, Springer 1971.
C.L. Siegel, J.K. Moser: Lectures on celestial mechanics, Springer 1971.
P. Kustaanheimo, E. Stiefel: Perturbation theory of Kepler motion based on spinor regularization, J.reine und angew. Math. 218, 1965, p. 204–219.
E. Stiefel, M. Rössler, J. Waldvogel, C.A. Burdet: Methods of regularization for computing orbits in celestial mechanics, NASA contractor report CR-769, 1967.
H. Sperling: Computation of Keplerian conic sections, ARS journal, 1961, p. 660–661.
C.A. Burdet: Regularization of the two-body problem, ZAMP 18, 1967, p. 434–438.
: Theory of Kepler motion: The general perturbed two-body problem, ZAMP 19, 1968, p. 345–368.
K.F. Sundman: Mémoire sur le problème des trois corps, Acta math. 36, 1913, p. 105–179.
G. Lemaître: Regularization dans le problème des trois corps, Acad. Roy. Belg. Bull.d.Sci. (5) 40, 1954, p. 759–767.
J. Waldvogel: A new regularization of the planar problem of three bodies, Cel. Mech., vol. 6, No. 2, 1972.
R. F. Arenstorf: New regularization of the restricted problem of three bodies, Astr. J. 68, No. 8, 1963., p. 548–555.
J. Waldvogel: Die Verallgemeinerung der Birkhoff-Regularisierung auf das räumliche Dreikörperproblem, Bulletin astr. série 3, t. II, fasc. 2, 1967.
H. J. Sperling: Bibliography on the singularities of the equations of motion of celestial mechanics, Second edition, Internal note, Marshall Space Flight Center, Febr. 1970.
E. Stiefel, D.G. Bettis, Stabilization of Cowell's method. Numer. Math. 13, 1969, p. 154–175.
D.G. Bettis, Stabilization of finite difference methods of numerical integration, Cel. Mech., vol. 2, 1970, p. 282–295.
D.G. Bettis: Numerical integration of products of Fourier and ordinary polynomials, Numer. Math. 14, 1970, p. 421–434.
G. Scheifele: On numerical integration of perturbed linear oscillating systems, ZAMP, vol. 22, fasc. 1, 1971, p. 186–210.
P.E. Nacozy: The use of integrals in numerical integrations of the N-body problem, Astrophysics and space science 14, 1971, p. 40–51.
J. Baumgarte: Numerical stabilization of the differential equations of Keplerian motion, Cel. Mech., vol. 5, No. 4, p. 490–501.
J. Baumgarte: Stabilization of constraints and integrals of motion in dynamical systems, Comp. meth. in applied mech. and engineering 1, 1972, p. 1–16.
G. Scheifele: On nonclassical canonical systems, Cel. Mech. 2, p. 296, 1970.
G. Scheifele, E. Stiefel: Canonical satellite theory based on independent variables different from time, Report to ESRO, ESOC contract 219/70/AR, 1972.
U. Kirchgraber: An analytical perturbation theory based on polar coordinates in the four-dimensional KS-space, Thesis ETH Zürich, 1972.
H.R. Schwarz: Stability of Kepler motion, Comp. meth. in appl. mech. and engineering 1. p. 279–299.
J. Baumgarte: Asymptotische Stabilisierung von Integralen bei gewöhnlichen Differentialgleichungen 1. Ordnung. To appear.
E. Hochfeld (ed): Stabilization of Computer Circuits, U. of Chicago, Nov. 1957. Available from US Govt., Printing Office, Washington D.C. as WADC TR 57-425 (AD 155740).
G.D. Birkhoff: The restricted problem of three bodies, Rendiconti Palermo 39, 1915, p. 314.
B. Gagliardi: Ueber die Dimensionserhöhung bei der Regularisierung des Keplerproblems, Thesis ETH Zürich, 1972
U. Kirchgraber: The transformational behaviour of perturbation theories. To appear in Cel. Mech., vol. 7, No. 4.
K. Kocher: Eine lineare Theorie der Optimierungsprobleme bei Raketen mit kleinem Schub (Low thrust), Thesis ETH Zürich, 1971.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag
About this paper
Cite this paper
Baumgarte, J., Stiefel, E. (1974). Examples of transformations improving the numerical accuracy of the integration of differential equations. In: Bettis, D.G. (eds) Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations. Lecture Notes in Mathematics, vol 362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066593
Download citation
DOI: https://doi.org/10.1007/BFb0066593
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06602-6
Online ISBN: 978-3-540-37911-9
eBook Packages: Springer Book Archive
