On the integrability of the n-centre problem

It is known that for $n \geq 3$ centres and positive energies the $n$-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary $g >1$ independent integrals of Gevrey class $g$ exist, no real-analytic (that is, Gevrey class 1) independent integral exists.Comment: 22 pages, a short announcement see in math.DS/031242

Symbolic dynamics for the NN-centre problem at negative energies

We consider the planar $N$-centre problem, with homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets

Cluster Approximation for the Farey Fraction Spin Chain

We consider the Farey fraction spin chain in an external field $h$. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At $h=0$ we reproduce the rigorously known asymptotic temperature dependence of the free energy. For $h \ne 0$, our results are largely consistent with those found previously using mean field theory and renormalization group arguments.Comment: 17 pages, 3 figure

Modeling the CO2-effects of forest management and wood usage on a regional basis

BACKGROUND: At the 15(th) Conference of Parties of the UN Framework Convention on Climate Change, Copenhagen, 2009, harvested wood products were identified as an additional carbon pool. This modification eliminates inconsistencies in greenhouse gas reporting by recognizing the role of the forest and timber sector in the global carbon cycle. Any additional CO(2)-effects related to wood usage are not considered by this modification. This results in a downward bias when the contribution of the forest and timber sector to climate change mitigation is assessed. The following article analyses the overall contribution to climate protection made by the forest management and wood utilization through CO(2)-emissions reduction using an example from the German state of North Rhine-Westphalia. Based on long term study periods (2011 to 2050 and 2100, respectively). Various alternative scenarios for forest management and wood usage are presented. RESULTS: In the mid- to long-term (2050 and 2100, respectively) the net climate protection function of scenarios with varying levels of wood usage is higher than in scenarios without any wood usage. This is not observed for all scenarios on short and mid term evaluations. The advantages of wood usage are evident although the simulations resulted in high values for forest storage in the C pools. Even the carbon sink effect due to temporal accumulation of deadwood during the period from 2011 to 2100 is outbalanced by the potential of wood usage effects. CONCLUSIONS: A full assessment of the CO(2)-effects of the forest management requires an assessment of the forest supplemented with an assessment of the effects of wood usage. CO(2)-emission reductions through both fuel and material substitution as well as CO(2) sink in wood products need to be considered. An integrated assessment of the climate protection function based on the analysis of the study’s scenarios provides decision parameters for a strategic approach to climate protection with regard to forest management and wood use at regional and national levels. The short-term evaluation of subsystems can be misleading, rendering long-term evaluations (until 2100, or even longer) more effective. This is also consistent with the inherently long-term perspective of forest management decisions and measures

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre on the real plane, we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.Comment: 22 pages, 6 figure