Any one help me the target is to determine sets of initial conditions for a system of differential equations modelling the interaction of two electrons that start from a distant point and move to form a bound state.
In the Mathematica notebook attached, the interaction of two electrons is modeled with a system of differential equations.
In the bookmarks of the Manipulate the initial conditions of several possible interactions are given as examples: 1.- Electron Repulsion: Electrons far apart repel each other [Attached image: Repulsion.jpg]
2.- Pass Through: Electrons with enough speed pass through each other [Attached image: PassThrough.jpg]
3.- Paired Electrons: Electrons close enough remain in a bound state orbiting each other [Attached image: Paired.jpg]
The task is to find initial conditions that take the electrons from being separated by a distance dx>=3.5 (like in the Repulsion or Pass Through examples) to being bound in an stable orbit with the center of masses (CM) turning around each other (like in the Paired Electrons example) for more than 5 CM turns.
One possibility could be that the electrons, starting at dx=3.5 must have the correct phases (ψ) and enough velocity (v) with the right direction (λ) to "start" passing through each other but not too much so that they stay paired instead of completing the pass through.
We are looking for the initial conditions to generate something like the following plot (which is only for example purposes, visualizing only CM1 & CM2): [Attached image: Target.jpg]
The system is highly non-linear and sometimes produces erroneous numerical results (not always detected by the checks in the notebook): complex numbers, distances between center of charge 'r' and center of mass 'q' >1.5, speeds of the center of charge 'u'<>1, erratic paths for the center of mass etc. Only realistic results should be considered and any improvement in the algorithm or solving method should not change the underlying differential equations.
Note 1: Please refrain from asking to award the contest to you unless you can prove with an image that you have a solution.
Note 2: You can test the notebook online without the desktop Wolfram Mathematica or any other software in: https://www.wolframcloud.com/obj/jonbarandiaran/Published/PoincareInteractionDiracParticles_Bounty2D.nb
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