Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of mass invariant, possesses at least 3 fixed points. The proof of this global fixed point theorem does not make use of the generating function technique. The theorem is a consequence of the statement that a Hamiltonian vector-field on a torus T2n, which depends periodically on time, possesses at least (2n+1) forced oscillations. These periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows. A second conjecture of V. I. Arnold proved concerns a Birkhoff-Lewis type fixed point theorem for symplectic maps. Additional keywords; Periodic functions; Variational principles; Equations; Reprints. (Author)