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We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.
Schrödinger equations in Einsteinïs theory of gravity with time derivative of the wave function
(1995)
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.
General Relativity and Gravitation is a journal of studies in general relativity and related topics, published under the auspices of the International Committee on General Relativity and Gravitation. The journal publishes original, high-quality research papers on the theoretical and experimental aspects of general relativity and related topics; surveys and review articles on current research in general relativity and gravitation; news regarding conferences and other enterprises of interest to scientists in this field; and book reviews. All manuscripts and editorial correspondence, as well as books for review, should be submitted to the Editor, and authors may propose who among the Associate Editors will deal with their paper. All submitted articles are acknowledged and refereed.
Fourier transformation of Hilbert C*-systems, with compact groups, by their regular representation
(1995)
Edge-solid varieties
(1995)