Loop Algebras Toda Mechanics Systems Research

Many body mechanics systems have received particular attention after two body systems were exactly solved. Because it is connected with many fields such as long-range correlation, nonlinear wave propagation and inverse scattering method, theoretical physicists and mathematicians have payed much attention to it for a long time and make some important breakthrough. Since the late 1960's and the early 1970's, when a number of exactly solvable many body systems in one dimension were discovered and solved by means of the inverse scattering method, among which the classical Toda chains, the Calogero-Moser systems and the Ruijsenaars-Schneider models are the most famous and important examples.In this paper, we give the generalization of Toda chain based on the loop algebra. the many body mechanics systems based on the loop algebras are considered to be more realistic than the case for classical Lie algebras because the spectral curves of the Lax matrices of the Toda theory on loop algebras are identified with the elliptic curves generated by the moduli parameters of the four dimensional supersymmetry gauge theory in the context of Seiberg-Witten theory, and then a method was given to get the moduli parameters of the four dimensional supersymmetry gauge theory and the pre-power. This is the reason we generalize Toda chains based on loop algebra in our paper.This paper is organized as follows. Chapter one is the introduction. Chapter two is the basic knowledge. In chapter three, we generalize the Toda mechanics system with long range interaction to the case of loop algebra L(D_r) . By using a pair of ordered positive integer (X,Y) to describe Toda chains, we extract the equation of motion and the Hamiltonian structure of this system for X, Y ≤ 3 . It turns out that both extra coordinates and standard Toda variables are poisson non-commutative in the case of nontrivial generalization, and for some case, extra variables appear linearly on the right hand side of the poisson brackets. Then we check the fact that the poisson brackets assumed from Hamiltonian equations are identified with the result receivedfrom f matrices inand the above poisson brackets are satisfied with the Jacobi identification, too. In chapter four, we prove the Lax equationis exactly solvable if we transform this problem to a regular Riemann-Hilbert problem by means of letting M in Lax pair is an antisymmetrical matrix. In order to prove our result, we exactly solve a R-H problem under a given initial condition.