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 $C^{2d-δ}$-small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under $C^{2d+δ}$-small perturbations.

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