218 - Österreichische Mathematische Gesellschaft

Here H j is a term proportional to the product of j amplitudes A k and the interaction
Hamiltonian H int describes nonlinear dynamics, while the quadratic Hamiltonian
∞
H 2 = ∑ ω k |A k | 2 (21)
n=1
produces a linear equation of motion,
i dA k
dt
= ω k A k , (22)
and thus describes linear waves with the dispersion function ω k ≡ ω(k). Accordingly,
the term H s in (19) has the order ε s and corresponding resonance conditions
in its most general form read
ω 1 ± ··· ± ω s = 0,, (23)
k 1 ± ··· ± k s = 0, (24)
which are time and space synchronization conditions and can be interpreted as the
conservation laws for energy and momentum.
The problem of persistence, under perturbation, of quasi-periodic motions in
Hamiltonian dynamical systems is studied in the framework of the renowned
KAM theory, named after its creators Kolmogorov, Arnold and Moser in 1960th
(for a comprehensive contemporary review see [16]).
An important example is given by the dynamics of nearly-integrable Hamiltonian
systems. In general, the phase space of a completely integrable Hamiltonian system
of n degrees of freedom is foliated by invariant n-dimensional tori. KAM theory
shows that, under suitable regularity and non-degeneracy assumptions, most
(in measure theoretic sense) of such tori persist (slightly deformed) under small
Hamiltonian perturbations.
This means that if the system is subjected to a weak nonlinear perturbation, some
of the invariant tori are deformed and survive, while others are destroyed. The
ones that survive are those that have ‘sufficiently irrational’ frequencies, i.e. do
not satisfy resonance condition (23). An important consequence of this is that for
a large set of initial conditions the motion remains perpetually quasiperiodic (cf.
to the Theorem on vector-field linearization discussed above).
In KAM theory all results are formulated in terms of resonant and non-resonant
tori but no indication is given about how to compute explicitly solutions of resonance
conditions. Accordingly, the major technical problem arising in this context
is due to the appearance of resonances and, consequently, of small divisors in the
associated formal perturbation series.
Another important issue which is not paid much attention in mathematical studies
is, that for a physical system to be resonant both conditions (23), (24) should
21