An Introduction to Symplectic Maps and Generalizations of the Toda Lattice
نویسندگان
چکیده
We construct particular Hamiltonian systems associated to graphs embedded in Euclidean n−dimensional space and apply the dual transformation to this system. This result extends the one-dimensional dual transformation associated to the Toda lattice. We also discuss some applications to solid state lattice theory.
منابع مشابه
A discrete time relativistic Toda lattice
Four integrable symplectic maps approximating two Hamiltonian flows from the relativistic Toda hierarchy are introduced. They are demostrated to belong to the same hierarchy and to examplify the general scheme for symplectic maps on groups equiped with quadratic Poisson brackets. The initial value problem for the difference equations is solved in terms of a factorization problem in a group. Int...
متن کاملToda versus Pfaff Lattice and Related Polynomials
The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of gl (∞) into sp (∞) and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of “τ -functions”. The latter satisfy their own set of bilinear identities, which moreover characterize them. In the semi...
متن کاملThe Pfaff lattice on symplectic matrices
Abstract. The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [Intern. Math. Res. Notices, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of ...
متن کاملNon-commutative and Commutative Integrability of Generic Toda Flows in Simple Lie Algebras
1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s variables [F] the finite non-periodic Toda lattic...
متن کاملUniversity of Cambridge Conservative Methods for the Toda Lattice Equations Conservative Methods for the Toda Lattice Equations
We are concerned with the numerical integration of the Toda lattice equations by using diierent conservative methods. Numerical experiments suggest that the global error for isospectral schemes decreases exponentially with time but it is almost constant for either symplectic or more general integrators. We provide a theoretical explanation for these experimental ndings.
متن کامل