We prove the existence of Cantor families of small amplitude periodic solutions for wave and Schrodinger equations on compact Lie groups and homogeneous spaces with merely differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the highly degenerate eigenvalues of the Laplace Beltrami operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions. We need to develop a detailed harmonic analysis on Lie groups and homogeneous spaces to control the multiple eigenvalues of the Laplace operator, as well as the properties of multiplication of eigenfunctions.