Completely Integrable Systems and Singularity in the Complex t — Plane

Abstract

For certain Hamiltonian dynamical systems with several parameters, we always may obtain integrable cases through a postulate that the solution should have Laurent expansion with sufficient number of arbitrary constants near a singularity in the complex t-plane. Famous classical examples are tops of Euler, Lagrange, and Kovalevskaja. For a Hamiltonian system \(H = \frac{1}{2}(p_1^2 + p_2^2) + \frac{1}{2}(q_1^2 + q_2^2) + q_1^2{q_2} + \frac{1}{3}(1 - 2\varepsilon )q_2^3,\) this postulate (procedure) determines the value of parameter as e = 0, which is shown to be integrable. Henon-Heiles case (e= 1) i s rejected. Another interesting example is a series of Generalized Toda Lattice systems, which can be written in a Lax-type differential equation.