Download On the singularities of generalized solutions to n-body type by Barutello V., Ferrario D.L., Terracini S. PDF
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
Example text
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.