By Markus Klein, Andreas Knauf
ISBN-10: 3540559876
ISBN-13: 9783540559870
This booklet treats scattering of a classical particle in a scalar strength with a number of attracting Coulombic singularities. For greater than facilities this is often an incredible prototype of chaotic scattering, that is analysed intensive the following utilizing equipment of differential geometry and ergodic thought. particularly, the Cantor set constitution of all bounded orbits is defined when it comes to symbolic dynamics, and rigorous power established bounds are derived for amounts resembling the topological entropy of the stream, the Hausdorff measurement of the bounded orbits and the distribution of time hold up. This exhibits that the chaotic behaviour of such structures is common within the excessive power regime. eventually the scattering orbits are categorized via use of a gaggle. lots of the ends up in the booklet are new. the 1st mathematically rigorous and accomplished therapy of chaotic scattering in Coulombic potentials, together with thirteen figures are given. The e-book may be of curiosity to mathematical physicists, mathematicians, and physicists.
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Extra resources for Classical planar scattering by Coulombic potentials
Example text
3). Thus the length function on K x K attains a maximum t~ax. We assume that E' > 2Vmax • Then for all E > E' one has J'l---V-/-E- ~ -I2J1 - V/ E' which implies that t max for £E(c). 4, one may describe the long-time behaviour of all geodesic segments in GE, E large, by iterating a Poincare map defined below. 3) every geodesic leaves G E after a time bounded by t max . Therefore, we assume n ~ 2. Let D k := 11E1 (d k (I)) C EE be the set of points in the energy shell projecting to the geodesic d k , k E {I, ...
If eo, Cl E flpqM are not homotopic, we set dpq (eo, Cl) := 00, and similarly for AM. Clearly, dpq(eo, Cl) 2: dl(eo, Cl), and similarly for AM. The following proposition is a generalization of Thm. 7 of [26]. 1 If a Riemannian manifold (M, g) is complete, then the metric space (H1(I, M), dd of HI-curves is complete. Proof. The inclusion HI (I, M) '----+ C(I, M) is continuous, since for any Cl-path x: I -+ H1(I,M) with end points X(O) = eo, x(1) = Cl we may find a tM E I such that d(eo, cd = dM(eo(tM),Cl(tM)).
By shortening the curves VI E Dsl,aGEM we obtain closed nonintersecting geodesics dl • By the same method as in the last lemma using the covering transformation G we obtain the geodesics d l which, again, must be free of mutual and self-intersections. Furthermore, dl(aI) c aGE and they meet the boundary with a right angle because otherwise they could be shortened. The virial identity implies that d l (]O,I[) C Int(GE)' so that the d l are neat (see Hirsch [18]). By uniqueness of geodesics starting with directions perpendicular to aGE all endpoints are different.
Classical planar scattering by Coulombic potentials by Markus Klein, Andreas Knauf
by Ronald
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