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

Noticing that the equation \frac d ^2 O_ n d t^2=\H_ c ,\H_ c , O_ n \\=\lambda O_ 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.