ar X iv : h ep - t h / 95 07 09 2 v 1 1 9 Ju l 1 99 5 An elegant solution of the n - body Toda problem

The solution of the classical open-chain n-body Toda problem is derived from an ansatz and is found to have a highly symmetric form. The proof requires an unusual identity involving Vandermonde determinants. The explicit transformation to action-angle variables is exhibited. [email protected] Permanent address 1 The Toda chain is one of the paradigmatic examples of an integrable many-body system of interacting particles. The discovery of its conserved integrals of motion[1, 2] and its subsequent solution[3, 4, 5] were important steps in the development of the theory of integrable systems[6]. An almost universal feature of analytical studies of the Toda system is the use of the Lax pair formalism. In this paper, an alternative derivation of the solution of the classical open-chain n-body Toda system is given. The derivation proceeds essentially from an ansatz about the form of the solution and therefore lacks the power and generality of the Lax pair treatment. The solution however has an elegant structure which is not evident in previous representations. More, it can be interpreted as the classical canonical transformation from the Toda system to a free theory. This is an important clue to constructing the classical and quantum solutions by a sequence of elementary canonical transformations[7]. Following the successful solution of the three-body Toda problem with this approach[8], work is in progress on the classical and quantum openchain n-body problems. The Hamiltonian for the (n+1)-body open chain Toda system is H = 1 2 n+1 ∑ k=1 pk + n ∑ k=1 ekk+1. (1) The arguments of the exponential potentials can be interpreted as expressions for the root vectors of An in the Cartan basis[5]. A coordinate transformation will put the root vectors into the Chevalley basis and separate out the motion of the