Destruction of Lagrangian torus for positive definite Hamiltonian systems

For an integrable Hamiltonian $H_0=1/2\sum_{i=1}^dy_i^2$ $(d\geq 2)$, we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily $C^{2d-\delta}$-small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under $C^{2d+\delta}$-small perturbations.