Papers by Ali Abdulhussein
arXiv (Cornell University), Dec 21, 2021
It is proven that second order vectorial nonlinear differential systems y ′′ = f (y) , possess a ... more It is proven that second order vectorial nonlinear differential systems y ′′ = f (y) , possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If f (y) is an odd function of y , then y ′′ = f (y) is shown also to possess a continuum of odd solutions. The results apply to a significant family of second order vectorial nonlinear differential systems that are not dissipative. This family of differential equations includes the celebrated N body problem of celestial mechanics and other central force problems.
arXiv (Cornell University), Dec 20, 2021
Newton's equations of celestial mechanics are shown to possess a continuum of solutions in which ... more Newton's equations of celestial mechanics are shown to possess a continuum of solutions in which the future trajectories of the N bodies are a perfect reflection of their past. These solutions evolve from zero initial velocities of the N bodies. Consequently, the future gravitational forces acting on the N bodies are also a perfect reflection of their past. The proof is carried out via Taylor series expansions. A perturbed system of equations of the N body problem is also considered. All real valued solutions of this perturbed system have no singularities on the real line. The perturbed system is shown to have a continuum of solutions that possess symmetry where the future velocities of the N bodies are a perfect reflection of their past. The positions and accelerations of the N bodies are then odd functions of the time. All N bodies then evolve from one location in space.
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Papers by Ali Abdulhussein