  We consider the following evolutionary Hamilton-Jacobi equation with initial
condition: \begin{equation*} \begin{cases}
\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases}
\end{equation*} where $\phi(x)\in C(M,\mathbb{R})$. Under some assumptions on
the convexity of $H(x,u,p)$ with respect to $p$ and the uniform Lipschitz of
$H(x,u,p)$ with respect to $u$, we establish a variational principle and
provide an intrinsic relation between viscosity solutions and certain minimal
characteristics. By introducing an implicitly defined {\it fundamental
solution}, we obtain a variational representation formula of the viscosity
solution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss the
large time behavior of the viscosity solution of the evolutionary
Hamilton-Jacobi equation and provide a dynamical representation formula of the
viscosity solution of the stationary Hamilton-Jacobi equation with strictly
increasing $H(x,u,p)$ with respect to $u$.
