In [11], the authors provided an implicit variational principle for the contact Hamilton's equations \begin{align*} \begin{array}{l} \dot{x}=\frac{\partial H}{\partial p}(x,u,p), \dot{p}=-\frac{\partial H}{\partial x}(x,u,p)-\frac{\partial H}{\partial u}(x,u,p)p, \dot{u}=\frac{\partial H}{\partial p}(x,u,p)⋅ p-H(x,u,p), \end{array} \end{align*} where $M$ is a closed, connected and smooth manifold and $H=H(x,u,p)$ is strictly convex, superlinear in $p$ and Lipschitz in $u$. In the present paper, we focus on two applications of the variational principle: 1. A representation formula for the solution semigroup of the evolutionary partial differential equation \[ w_t(x,t)+H(x,w(x,t),w_x(x,t))=0; \] 2. Weak KAM theorem for the contact Hamiltonian $H$. More precisely, we find pairs $(u,c)$ with $u𝟄 C(M,𝐑)$ and $c𝟄𝐑$ which, in the viscosity sense, satisfy the stationary partial differential equation \[ H(x,u(x),u_x(x))=c. \]