  For the conformally symplectic system \[ \left\{ \begin{aligned}
\dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-\lambda p,
\quad \lambda>0 \end{aligned} \right. \] with a positive definite Hamiltonian,
we discuss the variational significance of invariant Lagrangian graphs and
explain how the KAM torus impacts the $W^{1,\infty}-$convergence speed of the
Lax-Oleinik semigroup.
