Variational principle for contact Tonelli Hamiltonian systems

We establish an implicit variational principle for the equations of the contact flow generated by the Hamiltonian $H(x,u,p)$ with respect to the contact 1-form $\alpha=du-pdx$ under Tonelli and Osgood growth assumptions. It is the first step to generalize Mather's global variational method from the Hamiltonian dynamics to the contact Hamiltonian dynamics.