  Let $P_N$ be a trigonometric polynomial of degree $N$ and satisfy
$\|P_N\|_{C^r}\leq \epsilon$. If $P_N$ destroys the Lagrangian torus with the
rotation vector $\omega$ of an integrable Hamiltonian system, then {what are
the relation among $\epsilon$, $N$, $r$ and the arithmetic property of
$\omega$}? By addressing this, we provide quantitatively higher dimensional
generalizations of the remarkable results on destruction of invariant circles
for the area-preserving twist map by Herman [22] and Mather [30] in the 1980's.
As an application, we give a partial answer to an open problem by Mather [30,
p.212] in 1988.
