Integrability of One Particle in a Perturbed Central Quartic Potential

Abstract

We show here that the Hamiltonian (1.1) has n functionally independent integrals of motion in involution which are rational both in phase space variables and in parameters. Moreover these integrals are quadratic in momenta and the Hamilton-Jacobi equation of the system (1.1) is separable in generalized elliptic coordinates. A Lax representation for (1.1) and for higher flows is found. The system (1.1) constrained to an eUipsoid remains integrable.

Quasi–periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type

“…The integrability of this case and separability in ellipsoidal coordinates was proved by Wojciechowski [48] (see also [27,44]). We employ this result to integrate the system in terms of ultraelliptic functions (hyperelliptic functions of the genus two curve) and then execute reduction of hyperelliptic functions to elliptic ones by imposing additional constrains on the parameters of the system.…”

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

“…We remark that once a single anharmonic contribution with parameter δ k is added to the first harmonic term the maximal superintegrability is lost, but the resulting system (for any number of arbitrary δ k 's) always keeps the (2N − 3) integrals of motion (1.4). In particular, the latter are just the integrals of the motion for the radial Garnier system [4,5], which is recovered by taking ω and δ 1 as the only non-vanishing parameters. We also recall that the integrability properties of some quartic oscillators can be generalized to the Calogero-Moser systems defined with such nonlinear oscillators as external potentials (see [6] and references therein).…”

“…Furthermore, a geometric analysis shows [20] that the potential 5) can be interpreted as the intrinsic harmonic oscillator on this curved space, that turns out to be MS, despite of the introduction of a non-constant curvature. We remark that, as expected, for N = 2 this model is listed in the classification of MS potentials for the Darboux space of type III given in [9].…”

Explicit integration of the Hénon-Heiles Hamiltonians 1

“…Quite similarly the canonical transformation to elliptic coordinates [49], 12) maps the Hamilton's equations to the hyperelliptic system (7.8) with…”