The Euler equations on Lie algebra so(4): An elementary approach to integrability condition

Abstract

Articles you may be interested inTwo soliton hierarchies associated with SO(4) and the applications of SU(2) SU(2) SO(4) J. Math. Phys. 55, 093510 (2014); 10.1063/1.4895914 SO(4) algebraic approach to the three-body bound state problem in two dimensionsWe give a new derivation of the Manakov and the product case integrability conditions of the Euler equations on Lie algebra so͑4͒. Fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach.

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

“…Here we explicitely write down the fourth integral for the Manakov case and product case in form obtained in [9].…”

“…In both cases the fourth integral can be found among the polynomials of degree at most 2 (see [1,9]). As in [9] the table of these first integrals was not correctly printed, for the sake of completeness we reproduce its correct form in Appendix.…”

“…As in [9] the table of these first integrals was not correctly printed, for the sake of completeness we reproduce its correct form in Appendix.…”

Euler Equations on $$\mathfrak{so}(4)$$ as a Nearly Integrable Hamiltonian System

“…For some aspects of the theory of Euler equations on Lie algebras including results on the integrability and physical applications, we refer to [2,3,5,6,[9][10][11][12]. Here, in the context of the Hamiltonian perturbation theory, we study the Euler dynamics around a singular level set (a singular adjoint orbit) { ξ = const > 0, x = 0} of the two Casimir functions.…”

“…The singular level Δ × S 1 × {0} of K is, viewed as an orbit cylinder, trivially foliated by closed orbits of system (4)-(6) carrying a periodic motion with frequencies ω(s). To be completely integrable, Hamiltonian system (4)-(6) needs an additional integral of motion, functionally independent of H and K. Finding the integrability conditions for systems of such a type is a difficult task even in the case of the "quadratic" Hamiltonian H [2,5,11]. We are interested in the qualitative behavior of the Euler dynamics around the orbit cylinder in the context of the KAM results on the persistence of quasi-periodic tori and the excitation of normal modes [3,4,7,10,14,15].…”