Variational attraction of the KAM torus for the conformally symplectic system

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 presence of the KAM torus dominates the $C^1-$convergence speed of the Lax-Oleinik semigroup.