Thismonographdealswiththeexistenceofperiodicmotionsof Lagrangiansystemswith ndegreesoffreedom ij + V'(q) =0, where Visasingularpotential.Aprototypeofsuchaproblem, evenifitisnottheonlyphysicallyinterestingone, istheKepler problem .. q 0 q+yqr= . This, jointlywiththemoregeneralN-bodyproblem, hasalways beentheobjectofagreatdealofresearch.Mostofthoseresults arebasedonperturbationmethods, andmakeuseofthespecific featuresoftheKeplerpotential. OurapproachismoreonthelinesofNonlinearFunctional Analysis: ourmainpurposeistogiveafunctionalframefor systemswithsingularpotentials, includingtheKeplerandthe...