The Study On Integrable Many-body System With Symmetric Lax Matrix

Classical integrable many-body system is always one of the most interesting field in mathematical and physics, especially in the past several years when the close relationship between these systems and supersymmetric gauge theories was revealed. For example, Donagi and Witten found that the spectral curves of these integrable many body systens are identified with the Seiberg-Witten curves of the Seiberg-Witten theory. More recently, Dijkgraaf and Vafa found the information about the superpotentials of the chiral fields in some N = 1 SUSY gauge theories from the planar diagram of certain matrix integral. At the same time , N.Dorey found the superpotential from the equilibrium position of Hamiltonian phase flow of a certain classical integrable system. While, N. Dorey and T. J. Hollowood found that these two very different methods give basically the same result. However, so far , the classical many body systems known to play certain role in the studies in both the N = 2 and N = 1 gauge theories are restricted within the classical Toda chains, the Calogero-Moser systems and the Ruijsenaars-Schneider model.Toda only considered the nearest neighbor interaction in the Toda chain, but not the chains with longer range couplings. Bo-Yu Hou and Liu Chao studied the Toda field theory where they considered the next to neighbor interactions. In this paper,based on the above work, we will consider the case where there are longer range interactions(the coupling range is [p/2]), and obtain some novel exactly solvable many body systems.In chapter one,we give a brief introduction to the Hamiltonian method of integrable system. In chapter two, we introduce more off-diagonal variables in the symmetric Lax matrix of the Toda chain model, and generalize the Toda chain model to the case of quasi-long range interactions(the coupling range is [p/2]). We also prove its integrability , and give the method of how to obtain the exact solu-tion. We see that the system contains two sets of particles, the first type of particles behave quite like the usual Toda particles in the sense that they interact only exponentially with each other, while the sencond set of particles interact with the rests only through velocity couplings. A particular case arises when the velocities of all the particles from the second set are zero.In this case the system becomes the union of two ordinary Toda mechanics. In chapter three, we add a traceless condition to the Lax matrix of chapter two, and obtain the generalized Toda model for the Lie algebra Ar. We also notice that the Toda Lattices correspond to the Dynkin Lattices of the semi-simple Lie algebra Ar ,while the generalized Toda Lattices correspond to part of the root Lattices of the semi-simple Lie algebra Ar. Therefore,we construct the r matrix and Lax pair for semi-simple Lie algebra Br, Cr, Dr in the same way as that of the semi-simple Lie algebra Ar, and get the generalized Toda model for the semi-simple Lie algebra BT, Cr, Dr. We also provide the explicit equations of motion the Hamiltonian and the canonical poission relations to the case of q = 1> q = 2 and q = 3, where q is the order of the roots . We find that the forms of equations corresponding to the Lie algebras Br, Cr, Dr are different when we consider long range interactions except that they interact exponentially with each other.