**Authors**:
G. Gallavotti

**Title**:
*
A Criterion of Integrability for Perturbed Nonresonant Harmonic
Oscillators. "Wick Ordering" of the Perturbations in Classical Mechanics and
Invariance of the Frequency Spectrum
*

Communication in Mathematical Physics, ** 87**, 365-383, 1982

**Abstract:**
We introduce an analogue to the renormalization theory (of quantum
fields) into classical mechanics. We also find an integrability criterion
guaranteeing the convergence of Birkhoff s series and an algorithm for
modifying the hamiltonian to fix the frequency spectrum of the quasi-periodic
motions. We point out its possible relevance to the transition to chaos.

**Key words: **
Classical Mechanics, Integrability, Counterterms, Perturbation theory