KAM theory without action-angle variables

KAM Theory Without A tion-Angle Coordinates

A KAM theorem without action-angle variables for elliptic lower dimensional tori

We study elliptic lower dimensional invariant tori of Hamiltonian systems via parameterizations. The method is based in solving iteratively the functional equations that stand for invariance and reducibility. In contrast with classical methods, we do not assume that the system is close to integrable nor that is written in action-angle variables. We only require an approximation of an invariant ...

Integrability and Action-Angle Variables

We prove that the dynamical system charaterized by the Hamiltonian H = λN ∑N j pj+μ ∑N j,k (pjpk) 1 2{cos[ν(qj−qk)]} proposed and studied by Calogero [1,2] is equivalent to a system of non-interacting harmonic oscillators. We find the explicit form of the conserved currents which are in involution. We also find the action-angle variables and solve the initial value problem in simple form. 2

Treating Small Denominators without Kam

where v : R/Z → R is the potential, α ∈ R \ Q is the frequency, and θ ∈ R is the phase. The most important example is given by the almost Mathieu operator, when v(x) = 2 cos(2πx). There are two classical small denominator problems associated with this model. First, in the regime of small λ , the system is close to constant coefficients and thus, it is natural to attempt to prove reducibility to...

Liouville theory without an action

We show that the conformal bootstrap allows to solve the Liouville conformal field theory without any perturbative computation. In particular, the bootstrap provides the special structure constants that were previously computed perturbatively, up to rescalings of the vertex operators. We also show that there are constraints on the allowed normalization and rescalings of the vertex operators. ar...