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We consider a restricted $(N+1)$-body problem, with $N \geq 3$ andhomogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove theexistence of infinitely many collision-free periodic solutions with negativeand small Jacobi constant and small values of the angular velocity, for anyinitial configuration of the centres. We will introduce a Maupertuis' typevariational principle in order to apply the broken geodesics techniquedeveloped in the paper "N. Soave and S. Terracini. Symbolic dynamics for the$N$-centre problem at negative energies. Discrete and Cont. Dynamical SystemsA, 32 (2012)". Major difficulties arise from the fact that, contrary to theclassical Jacobi length, the related functional does not come from a Riemaniannstructure but from a Finslerian one. Our existence result allows us tocharacterize the associated dynamical system with a symbolic dynamics, wherethe symbols are given partitions of the centres in two non-empty sets.

Symbolic dynamics: from the $N$-centre to the $(N+1)$-body problem, a preliminary study