Integrability and Poisson structures of three dimensional dynamical systems and equations of hydrodynamic type

We show that the Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We shall take advantage of this fact and use the theory of foliations 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. We shall show that the Halphen system can be formulated in terms of a flat SL{2, /i)-valued connection and belongs to a non-trivial GodbillonVey class. On the other hand, for the Euler top and a special case of 3- species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable biHamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sb structure is a quadratic unfolding of an integrable 1-form in 3 -f 1 dimensions. We shall show that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and present some new techniques for incorporating arbitrary constants into the Poisson 1- form. This leads to some extensions, analoguous to q-extensions, of Poisson structure. We shall find that 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. In the second part, we complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler’s equation governing the motion of plane sound waves of finite amplitude and another quasi-linear second order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics which degenerate into one, namely the Benney sequence, for shallow water waves. We present further infinite sequences of conserved quantities for these equations. In the case of multi-component equations of hydrodynamic type, we show that Kodama’s generalization of the shallow water equations admits bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. Using dimensional analysis we are led to an Ansatz for both the Hamiltonian operator as well as the conserved quantities in terms of ratios of polynomials. The coefficients of these polynomials are determined from the Jacobi identities. The resulting bi-Hamiltonian structure of Kodama equations consists of generalization of the Cavalcante-McKean’s work for the shallow water waves. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The choice of the Hamiltonian density lor the second Hamiltonian structure is a crucial step and the analysis of recursion relations becomes necessary. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.