Regularization and topological entropy for the spatial \(n\)-center problem (Q2716648)

Let \(P=\{p_1,\dots,p_n\}\) be a finite set in \(\mathbb R^3\). The \(n\)-center problem is a Lagrangian system NEWLINE\[NEWLINE\ddot q=-\nabla V(q),\;q\in\mathbb R^3 \setminus\{p\}\tag{1}NEWLINE\]NEWLINE with the Lagrangian NEWLINE\[NEWLINEL(q,\dot q)=\frac 12 |\dot q |^2-V(q),\quad V(q)=-\sum^n_{i=1} {\mu_i\over|q-p_i|}+ \varphi(q),\;\mu_i >0,NEWLINE\]NEWLINE where \(\varphi\in C^\infty (\mathbb R^3)\). The authors show that (1) has positive topological entropy for \(n\geq 3\). The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. Moreover the \(n\)-center problem in \(S^3\) is also studied.