Particle systems with quasihomogeneous interaction

Date

2018-03-16

Authors

Stoica, Cristina

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Abstract

In this dissertation we analyse from a qualitative standpoint motion in a quasihomogeneous potential field: we offer a complete description of the flow associated with the two-body problem in quasihomogeneous field, obtain necessary and sufficient conditions for the block regularization of the flow and we propose an alternative model for the helium atom within the framework of a Manev-type interaction. We call a potential quasihomogeneous if it is of the form A/rᵃ + B/rᵝ, where r is the distance between the two mass points, 0 < α < β are real parameters and A > 0 and B > 0 inertia factors. To obtain the full description of the flow associated with the two-body problem in quasihomogeneous fields, we use diffeomorphic transformations that lead to an equivalent analytic system and at least differentiable integral energy relation. For each level of energy, we introduce the fictional invariant collision manifold and the infinity manifolds. We offer and analyse the global flow picture and we point out the Lebesgue measure of the set of initial conditions that lead to collision for each different case with respect to α and β. The next chapter focuses on the smoothness of the flow in the neighborhood of the collision manifold. In question is the possibility of extending solutions beyond singularities maintaining good properties with respect to initial data. In this case the singularity set for the system is said to be block regularizable. It is proved that the singularity set block regularizable if and only if β = 2 − [special characters omitted], where n is a positive integer, n ≥ 2. Also, the physical interpretation of this result is pointed out, namely that block regularization is in fact the mathematical expression of constrain imposed over the classical scattering angle. The last chapter presents a model for the Helium atom within the framework of classical mechanics. The set up consists of a planar isosceles 3-body problem formed by one neutron and two electrons, whose law of motion is given by a Manev-type potential with charges. We first describe the qualitative features of the local flow near triple collision, find several properties of the global flow and finally we prove the existence of a large open, connected, positive-measure manifold of bounded and collisionless solution.

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Keywords

Singularities (Mathematics), Particles

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