The Nonabelian Liouville-Arnold Integrability by Quadratures Problem: a Symplectic Approach

Abstract

A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces. 0. Introduction 0.1. As is well known [1,4], the integrability by quadratures of a differential equation in space R n is a method of seeking its solutions by means of finite number of algebraic operations (together with inversion of functions) and "quadratures" -calculations of integrals of known functions. Assume that our differential equation is given as a Hamiltonian dynamical system on some appropriate symplectic manifold M 2n , ω (2) , n ∈ Z + , in the formwhere u ∈ M 2n , H : M 2n → R is a sufficiently smooth Hamiltonian function [1,4] with respect to the Poisson bracket {·, ·} on D(M 2n ), dual to the symplectic structure ω (2) ∈ Λ 2 (M 2n ), and t ∈ R is the evolution parameter. More than one hundred and fifty years ago French mathematicians and physicists, first E. Bour and next J. Liouville, proved the first "integrability by quadratures" theorem which in modern terms [33] can be formulated as follows.

Geometry of Lie integrability by quadratures

“…Let us stress that proving just the existence of such constants of motion, sometimes understood as Arnold-Liouville integrability, does not allow for an explicit integration. Other authors have developed a generalization of the Arnold-Liouville approach by using non-Abelian Lie algebras (see [5,6,9,10]) in the framework of Hamiltonian mechanics.…”

“…Sometimes we have additional geometric structures that are compatible with the dynamics [5,6]. For instance, the m 2 -dimensional manifold M is endowed with a symplectic structure, ω.…”

On particular integrability in classical mechanics

“…, f k ∈ C ∞ (M) we can look at the subalgebra generated by the Poisson brackets {f i , f j }. The Bour-Liouville theorem on integrability by quadratures [33][34][35] states that the solutions of the Hamilton's equations can be obtained by quadratures, provided the existence of a sufficient number of functionallyindependent integrals of motion that form a solvable Lie algebra with the Poisson bracket. For this result the structure of a module of C ∞ (M) over itself is not considered, i.e.…”

“…. , q d(N−1) , (34) any set of generalized coordinates. In particular, one possibility is to take the (N − 1) vectorial Jacobi coordinates defined by the linear relations…”