4474
1990
eng
postprint
0
2010-09-13
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On a geometrical interpretation of differential-algebraic equations
The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.
urn:nbn:de:kobv:517-opus-46683
4668
Circuits, Systems, and Signal Processing 9 (1990), 4, S. 367-382
<hr>
first published in:<br><a href="http://www.springerlink.com/content/109373">Circuits, Systems, and Signal Processing </a> 9 (1990), 4, p. 367-382 <br>
doi: <a href="http://dx.doi.org/10.1007/BF01189332">10.1007/BF01189332</a>
egger@...
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Sebastian Reich
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
paper 157
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.uni-potsdam.de/files/4474/on_a_geometrical.pdf
1549
1994
eng
postprint
0
2008-03-19
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Momentum conserving symplectic integrators
In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).
urn:nbn:de:kobv:517-opus-16824
1682
Physica D: Nonlinear Phenomena. - 76 (1994), 4, p. 375 - 383. - ISSN 0167-2789
<hr>
first published in:<br><a href="http://www.sciencedirect.com/science/journal/01672789"> Physica D: Nonlinear Phenomena </a> - 76 (1994), 4, p. 375 - 383
<br>
ISSN: 0167-2789 <a href="http://dx.doi.org/doi:10.1016/0167-2789(94)90046-9 ">doi:10.1016/0167-2789(94)90046-9 </a>
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Sebastian Reich
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
paper 044
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.uni-potsdam.de/files/1549/reich_1994.pdf
4480
1995
eng
postprint
0
2010-09-13
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On the local qualitative behavior of differential-algebraic equations
A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.
urn:nbn:de:kobv:517-opus-46739
4673
Circuits, Systems, and Signal Processing 14 (1995), 4, p. 427-443
<hr>
first published in:<br><a href="http://www.springerlink.com/content/109373">Circuits, Systems, and Signal Processing </a> 14 (1995), 4, p. 427-443 <br>
doi: <a href="http://dx.doi.org/10.1007/BF01260330">10.1007/BF01260330</a>
egger@...
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Sebastian Reich
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
paper 159
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.uni-potsdam.de/files/4480/local.pdf
4477
1991
eng
postprint
0
2010-09-13
--
--
On an existence and uniqueness theory for nonlinear differential-algebraic equations
An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.
urn:nbn:de:kobv:517-opus-46706
4670
Circuits, Systems, and Signal Processing 10 (1991), 3, S. 343-359
<hr>
first published in:<br><a href="http://www.springerlink.com/content/109373">Circuits, Systems, and Signal Processing</a>10 (1991), 3, p. 343-359<br>doi: <a href="http://dx.doi.org/10.1007/BF01187550">10.1007/BF01187550</a>
egger@...
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Sebastian Reich
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
paper 158
Mathematik
open_access
Institut für Mathematik
Universität Potsdam
https://publishup.uni-potsdam.de/files/4477/existence.pdf