Non-integrability of the 1:1:2–resonance

Abstract

A Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

Perturbations of Superintegrable Systems

“…In the definite case 1 : 1 : 2 the cubic normal form is known to be non-integrable [12,18] while for m = ±1 : 2 : 2 the cubic normal form may serve as intermediate system [1,25,31]; in the additional cases of genuine second order the cubic normal form is again trivial. In indefinite cases with double eigenvalues of opposite symplectic sign these may be involved in a Krein collision and leave the imaginary axis during a subordinate Hamiltonian Hopf bifurcation.…”

Hans Duistermaat’s contributions to Poisson geometry

“…I will give a slightly anachronistic account of Duistermaat's paper [24] on monodromy in integrable systems, which takes into consideration later work of Dazord and Delzant [22].…”

“…Johannes Jisse (Hans) Duistermaat (1942Duistermaat ( -2010 earned his doctorate in 1968 at the University of Utrecht under the direction of Hans Freudenthal. After holding a postdoctoral position at the University of Lund, a professorship at the University of Nijmegen, and a visiting position at the Courant Institute, he returned to Utrecht in 1975 to take over Freudenthal's chair after the latter's retirement.…”

“…Let me now discuss in a bit more detail four of Hans Duistermaat's papers that are of obvious relevance to the topics of this conference, namely those on the spectrum of elliptic operators [28], global action-angle variables [24], the quantum-mechanical spherical pendulum [21], and the Duistermaat-Heckman theorem [29].…”

The Quantum Birkhoff Normal Form and Spectral Asymptotics

“…It may lead to very accurate numerics, much more accurate than could be achieved by a simple perturbation analysis. Many works show that resonances (integer (1) For instance when ν = (1, 1, 2) Duistermaat proved that in general H cannot be completely integrable [5].…”