Calculus of Variations and Geometric Measure Theory

G. Bellettini - G. Fusco - G. F. Gronchi

Regularization of the two-body problem via smoothing the potential

created on 14 Jun 2002
modified on 17 Dec 2003


Published Paper

Inserted: 14 jun 2002
Last Updated: 17 dec 2003

Journal: Comm. Pure Appl. Anal.
Volume: 2
Pages: 323-353
Year: 2003


We investigate the existence of global solutions for the two-body problem, when the particles interact with a potential of the form ${1/r^\alpha}$, 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 ${1/(r^2+\epsilon_k^2)^{\alpha/2}}$, and with an initial condition $\nu_k$; ${(\epsilon_k,\nu_k)}_k$ is a sequence converging to $(0,\nu ^*)$ as $k$ tends to infinity, where $\nu^*$ 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 a paper 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.

Keywords: two-body problem, binary collisions, regularization