  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 Osgood growth of
$H(x,u,p)$ with respect to $u$, we establish an implicitly variational
principle and provide an intrinsic relation between viscosity solutions and
certain minimal characteristics. Moreover, we obtain a representation formula
of the viscosity solution of the evolutionary Hamilton-Jacobi equation.
