Poisson structure of dynamical systems with three degrees of freedom

Abstract

It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one-form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two-monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a nontrivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of three-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one-form in 3 + 1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one-form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack-McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka-Volterra, May-Leonard, and Maxwell-Bloch systems admit globally integrable bi-Hamiltonian structure.

Super-integrable Calogero-type systems admit maximal number of Poisson structures

“…In this paper we shall present a general method for the construction of 2N − 1 compatible Poisson structures for 2N-dimensional super-integrable dynamical systems. It can be regarded as a kind of generalization of Nambu mechanics [5] and its extensions to integrable dynamical systems with three degrees of freedom [6].…”

A time-extended Hamiltonian formalism

“…[16] and the references therein for the recent resurrection of this system in modern theoretical physics. The lifts of the generators on I ×M of the three-parameter family of transformations (24) along the vector v(m) result precisely in the time-dependent basis (−v(m), U t (m), W t (m)) of sl(2, R) at the point m ∈ M. Here, v(m) is given by (23) and we find that u, w have the representatives u(m) = 2(x∂ x + y∂ y + z∂ z ) , w(m) = ∂ x + ∂ y + ∂ z (25) in the adapted coordinate system [17]. Thus, starting from the transformations (24) and reading the above abstract algebraic construction backward, we recover the geometric and algebraic structure on flow space of the autonomous system (23).…”

Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

“…and It is worth recalling that a complete classification of the Lie-Poisson completely integrable biHamiltonian systems on R 3 , which have non-transcendental integrals of motion, may be found in [14] (see also [13]). In fact, the Euler top system is labeled with the number (6) in Table 1 of [14].…”

“…This system is equivalent to a particular case of the so(3) Euler top, which is a well-known three dimensional bi-Hamiltonian system (see [13]) belonging to the realm of classical mechanics [20].…”

“…In [21,22] (see also [13]), the authors consider the conservative limit of the previous equations, which is obtained through the following rescalling…”