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Regularization of the two-body problem via smoothing the potential

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  • We investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form $\frac{1}{r^\alpha}$, for $\alpha >0$. Our solutions are pointwise limits of approximate solutions $u_\alpha(\epsilon_k,\nu_k)$ which solve the equation of motion with the regularized potential $\frac{1}{(r^2+\epsilon_k^2)^{\alpha/2}}$, and with an initial condition $\nu_k$; $(\epsilon_k,\nu_k)_k$ is a sequence converging to $(0,\overline \nu)$ as $k\to +\infty$, where $\overline \nu$ is an initial condition leading to collision in the non-regularized problem. We classify all the possible limits and we compare them with the already known solutions, in particular with those obtained in the paper [9] by McGehee using branch regularization and block regularization. It turns out that when $\alpha > 2$ the double limit exist, therefore in this case the problem can be regularized according to a suitable definition.
    Mathematics Subject Classification: 70F05, 70F16.


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