Adding potentials to superintegrable systems with symmetry

Abstract

In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

Integrable and superintegrable extensions of the rational Calogero–Moser model in three dimensions

“…with X 2 reducing to J 2 1 + 2k 3 . In the context of [15,16], the corresponding metric has symmetry algebra e 1 , h 4 (where h 4 = J 1 ), so we can adapt our 'universal coordinates' to reduce this Hamiltonian to 2 degrees of freedom. This is discussed in section 7.1.…”

“…In [15,16] we made a systematic study of subalgebras of the full conformal algebra (in three dimensions). For each subalgebra in the classification, we considered the restriction of the general conformal factor ϕ, in order to be invariant under the action of the subalgebra.…”

“…In section 7 we introduce some two and three dimensional symmetry algebras (linear functions of momenta, which commute with the kinetic energy of H), discussed in detail in [15,16]. The existence of such symmetries restricts the conformal factor ϕ.…”

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

“…The wave functions and energy spectra of the systems were obtained by means of the separation of variables. Superintegrable systems in Darboux spaces were also studied in [17,40,41]. It was shown there that free superintegrable systems (i.e.…”

Introduction to special feature dedicated to Prof. Allan Fordy on the occasion of his 70th birthday

“…As well as integrable evolution equations, Allan has an enduring interest in integrable finite-dimensional Hamiltonian systems, whether as stationary flows of PDEs [28], in connection with integrability of hydrodynamic-type PDE systems [29], or in their own right. In the latter vein, the work [30] presents a construction of superintegrable systems with three degrees of freedom, obtained by adding suitable potentials to purely kinetic Hamiltonians associated with conformally flat metrics. It is clear that, even in finite dimensions, the classification of integrable systems is far from being complete, and Allan still has plenty more to say on the subject.…”