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We show that the minimization of the Lagrangian action functional on suitableclasses of symmetric loops yields collisionless periodic orbits of the n-bodyproblem, provided that some simple conditions on the symmetry group aresatisfied. More precisely, we give a fairly general condition on symmetrygroups G of the loop space for the n-body problem (with potential ofhomogeneous degree alpha, with alpha>0) which ensures that the restriction ofthe Lagrangian action to the space of G-equivariant loops is coercive and itsminimizers are collisionless, without any strong force assumption. Many of thealready known periodic orbits can be proved to exist by this result, andseveral new orbits are found with some appropriate choice of G.Comment: 58 pages, 10 figures. Second revised versio

On the existence of collisionless equivariant minimizers for the classical n -body problem