Normal coordinate in harmonic crystal obtained by virtue of the classical correspondence of the invariant eigen-operator

Noticing that the equation frac d²O_ndt²={H_c,{H_c,O_n}} =λO_n with double-Poisson bracket, where O_n is normal coordinate, H_c is classical Hamiltonian, is the classical correspondence of the invariant eigen-operator equation (2004 Phys. Lett. A. 321 75), we can find normal coordinates in harmonic crystal by virtue of the invariant eigen-operator method.