Soliton Methods in the Theory of Integrable Mechanical Systems

Abstract: This dissertation is devoted to the study of integrable sets of mechanical equations with the use of soliton techniques. A Lax representation of the coupled Kortevegde Vries (cKdV) hierarchy with sow-ces is derived. It is shown that all stationary and restricted flows of the cKdV and the Harry Dym (HD) hierarchies admit a parametrization in the form of Newton equations. New infinite families of Newton equations are found and their integrable structures (such as Lax representation and bi-Hamiltonian formulation) are derived from the underlying soliton hierarchy. Next, an application of the r-matrix formalism to the theory of lattice integrable systems is presented. New families of lattice integrable systems are derived and several Miuralike maps between them are established. One of the offsprings of the obtained results is a completely new theory - theory of quasi-Lagrangian Newton (qLN) equations with two quadratic integrals of motion which contains as its special realization the classical separability theory for natural two-dimensional Hamiltonian systems. The theory of qLN systems goes however far beyond the separability theory - all qLN equations with two quadratic integrals of motion are characterized by a second order linear partial differential equation with polynomial coefficients of third order which generalizes the Bertrand-Darboux equation describing separable potentials in the plane. Hamiltonian formulation of the obtained qLN systems is presented and their complete integrability is proved. Finally, two families of Poisson structures are investigated. It is shown how these families naturally lead to separable systems or to quasi-Lagrangian Newton systems.