  Over the space of K\"ahler metrics associated to a fixed K\"ahler class, we
first prove the lower bound of the energy functional $\tilde E^\beta$, then we
provide the criterions of the geodesics rays to detect the lower bound of
$\tilde {\mathfrak J}^\beta$-functional. They are used to obtain the properness
of Mabuchi's $K$-energy. The criterions are examined by showing the convergence
of the negative gradient flow of $\tilde {\mathfrak J}^\beta$-functional.
