Mathematical Physics |
My main interest lies in the area of geometric mechanics and here especially systems with constraints. In contrast to the traditional approaches mainly based on tangent bundles, I analyse such systems from the point of view of the formal theory of partial differential equations. I have shown that the classical Dirac algorithm for Hamiltonian systems with constraints is nothing but a special case of the general Cartan-Kuranishi completion procedure. However, this equivalence holds only for finite dimensional systems; for field theory the often used naive extension of the Dirac algorithm is in general not sufficient for proving the consistency of the system. Already in my Ph.D. thesis an explicit example for this problem was given. The formal theory yields also an interesting and elegant approach to counting the degrees of freedom of very general systems including anholonomic ones (which is much better than Einstein's notion of the `strength' of a system ;-). Most of my results in this area have been published in the series of papers [8, 9, 17] . For some time, I was working with colleagues on a proper geometric unification of Hamiltonian mechanics and the formal theory of differential equations. This requires the consequent use of the rich geometric structure of jet bundles. For the finite-dimensional case we achieved this goal in the article [22] . Making use of cosymplectic geometry (as the differential equations theory automatically includes explicitly time dependent systems, too), we could show in an intrinsic manner that the completion algorithm of the formal theory is equivalent to the mechanical constraint algorithms. The extension of these results to field theories is still a challenge. Another interest of mine is the numerical integration of Hamiltonian systems with constraints. Due to the presence of the constraints there exist many possibilities for setting up the equations of motion of such systems. I studied the numerical properties of several such formulations, especially of the Hamilton-Dirac equations [16] and of the impetus-striction formalism [89] . For all these physically motivated formulations it turned out that the constraint manifold is usually orbitally stable and that thus only a moderate drift off the manifold can be observed in simulations. More on this topic can be found on the page on numerical analysis. |
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