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Stabilization by Making Use of a Generalized Hamiltonian Variational Formalism

Published online by Cambridge University Press:  12 April 2016

Joachim W. Baumgarte
Joachim W. Baumgarte*
Affiliation:
Mechanik-Zentrum, Technische Universität Braunschweig, Federal Republic of Germany, Swiss Federal Institute of Technology, Zürich, Switzerland

Abstract

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A generalized Hamiltonian formalism is established which is invariant not only under canonical transformations but under arbitrary transformations. Moreover the dependent variables, coordinates and momenta, as well as the independent variable are allowed to be transformed. This is to say that instead of the physical time t another independent variable s is used, such that t becomes a dependent variable or, more precisely, an additional coordinate. The formalism under consideration permits also to include nonconservative forces.

In case of Keplerian motion we propose to use the eccentric anomaly as the independent variable. By virtue of our generalized point of view a Lyapunov-stable differential system is obtained, such that all coordinates, including the time t, are computed by stable procedures. This stabilization is performed by control terms. As a new result a stabilizing control term also for the time integration is established, such that no longer any kind of time element is needed. This holds true for the usual coordinates as well as for the KS-coordinates.

Type
Part III. Numerical and Other Techniques
Information
International Astronomical Union Colloquium , Volume 41: Dynamics of Planets and Satellites and Theories of their Motion , 1978 , pp. 149 - 156
Copyright
Copyright © Reidel 1978

References

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3. Baumgarte, J.: Stabilization, Manipulation and Analytic Step Adaption, Long-Time Predictions in Dynamics, edited by Szebehely, V. and Tapley, B. D., Nato Advanced Study Institutes Series, Series C -Mathematical and Physical Sciences, Vol. 26, D. Reidel Publishing Company, Dordrecht-Holland/Bostǫn-U.S.A., (1976).Google Scholar
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