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By Barutello V., Ferrario D.L., Terracini S.

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Additional info for On the singularities of generalized solutions to n-body type problems

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Although the potentials in the previous examples do not depend on time, our assumptions allow an effective time–dependence of the potentials. For instance, we can choose positive and bounded C 1 functions mi (t), i = 1, . . , n. Obviously, the simplest example is the class of α-homogeneous n-body problem n Uα (t, x) = i

As we have already noticed, the class of potentials satisfying (U6) and (U7)h is not stable with respect to the sum of potentials. In order to deal with a class of potentials which is closed with respect to the sum, we introduce the following variant of Theorem 7. ˜ has Theorem 8. In addition to (U0), (U1), (U2)h , (U3)h , (U4)h , (U5), assume that U the form N Kν ˜ (x) = U α (dist(x, Vν )) ν=1 where Kν are positive constants and Vν is a family of linear subspaces, with codim(Vν ) ≥ 2, for every ν = 1, .

Inst. H. Poincar´e Anal. Non Lin´eaire, 8(6):561–649, 1991. [4] V. L. Ferrario, and S. Terracini. Symmetry groups of the planar 3body problem and action–minimizing trajectories. DS/0404514, preprint (2004). [5] V. Barutello and S. Secchi. Morse index properties of colliding solutions to the n-body problem. Arxiv:math/0609837, preprint (2006). [6] V. Barutello and S. Terracini. Action minimizing orbits in the n-body problem with simple choreography constraint. Nonlinearity, 17(6):2015–2039, 2004.

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