On resonant classical Hamiltonians with n frequencies (Q923189)

The author considers Hamiltonians of the quadratic forms \(H=\sum^{n}_{k=1}\omega_ kN_ k+V(Z,\bar Z),\) where \(\omega_ k\) \((k=1,2,...,n)\) are non-zero numbers, \(N_ k=| z_ k|^ 2\), \(Z=(z_ k)^ n_{k=1}\), \(\bar Z=(\bar z_ k)^ n_{k=1}\) and \(\overline{V(\bar Z,Z)}=V(Z,\bar Z)\). Here \(\bar z_ k\) denotes the complex conjugate to \(z_ k\) and \(V(Z,\bar Z)\) is a convergent power series starting with a term of order three. By studying the integrable approximation to the Hamiltonian he obtains rigorous results too complicate to be presented here, about the flow induced near the origin of the phase space.