Regularization of the two-body problem via smoothing the potential

G. Bellettini; G. Fusco; G. F. Gronchi

doi:10.3934/cpaa.2003.2.323

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2003, Volume 2, Issue 3: 323-353. Doi: 10.3934/cpaa.2003.2.323

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Regularization of the two-body problem via smoothing the potential

1.
Dipartimento di Matematica, Università di Roma 'Tor Vergata', 00133, Roma
2.
Dipartimento di Matematica, Universitµa dell'Aquila , 67100, L'Aquila
3.
Dipartimento di Matematica, Universita di Pisa, 56126, Pisa

Received: February 28, 2002

Revised: January 31, 2003

Published: August 2003

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Abstract

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.

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Mathematics Subject Classification: 70F05, 70F16.

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