September  2003, 2(3): 323-353. doi: 10.3934/cpaa.2003.2.323

Regularization of the two-body problem via smoothing the potential

1. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', 00133, Roma, Italy

2. 

Dipartimento di Matematica, Universitµa dell'Aquila , 67100, L'Aquila, Italy

3. 

Dipartimento di Matematica, Universita di Pisa, 56126, Pisa, Italy

Received  March 2002 Revised  February 2003 Published  June 2003

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.
Citation: G. Bellettini, G. Fusco, G. F. Gronchi. Regularization of the two-body problem via smoothing the potential. Communications on Pure & Applied Analysis, 2003, 2 (3) : 323-353. doi: 10.3934/cpaa.2003.2.323
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