Notes on Integrable Systems

Physica A: Statistical Mechanics and its Applications, 1989

For any Lie algebra g equipped with a second Lie product (engendered by a so-called R-matrix) a natural integrable system arises from the invariant functions on the dual g*. if g has the structure of an associative algebra and admits a trace-form, then in addition to the linear Lie-Poisson structure on g* a quadratic as well as a cubic Poisson bracket arises leading to tri-Hamiitonian formulations of the integrablc equations. The results are applied to PDE's (Korteweg-de Vrics and Boussinesq equation) as well as to lattice systems (classical and relativistic Toda lattice).

arXiv (Cornell University), 2018

In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian system on a Poisson manifold (P, Π) is a smooth manifold P equipped with a bivector field Π satisfying [Π, Π ] = 0 (Jacobi identity), inducing the Poisson bracket on C ∞ (P), {f, g} ≡ Π(df, dg) where f, g ∈ C ∞ (P). For any f ∈ C ∞ (P) the Hamiltonian vector field is defined by X f (g) = {g, f }. The Hamiltonian vector fields X f generate an integrable generalized distribution on P and the leaves of this foliation are symplectic. The flow of any hamiltonian vector field preserves the Poisson structure, it fixes each leaf and the hamiltonian itself is a first integral. It is important to characterize numerical methods preserving some of these fundamental properties of the hamiltonian flow on Poisson manifolds (geometric integrators). We discuss the difficulties of deriving these Poisson methods using standard techniques and we present some modern approaches to the problem.

International Mathematics Research Notices, 2006

We establish the Liouville integrability of the differential equationṠ(t) = [N, S 2 (t)], recently considered by Bloch and Iserles. Here, N is a real, fixed, skewsymmetric matrix and S is real symmetric. The equation is realized as a Hamiltonian vector field on a coadjoint orbit of a loop group, and sufficiently many commuting integrals are presented, together with a solution formula for their related flows in terms of a Riemann-Hilbert factorization problem. We also answer a question raised by Bloch and Iserles, by realizing the same system on a coadjoint orbit of a finite dimensional Lie group.

In this paper, the general form of the Noether theorem is used as a systematic procedure for the identification of integrable twedimensional systems. We give some applications for polynomial potentials, including the generalized Hénon-Heiles case.

Journal of Geometry and Physics, 1985

For a symplectic manifold the Poisson bracket on the space of functions is (uniquely) extended to a graded Lie bracket on the space of differential forms modulo exact forms. A large portion of the Hamiltonian formalism is still working. Let (M, w) be a symplectic manifold. Then there is an exact sequence of Lie algebras and Lie algebra homomorphisms 0 -~H°(M)-~C~(M)~=~(M) -+ H'(M) -~0, where H°(M), H 1(M) are the de Rham cohomology spaces,~= 0(M) is the space of all vector fields X with 0 (X)w = 0 (Lie derivative), a Lie subalgebra of the space~((M)of all vector fields. C(M) is equipped with the Poisson bracket }, and H(f) is the Hamiltonian vector field for the generating function f. 'y(x)

Annals of Physics, 2010

The three integrable two-dimensional Hénon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and Ndimensional completely integrable generalizations of all these systems are constructed by making use of sl(2, R) ⊕ h 3 as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form V(q 2 1 , q 2), and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.

Mathematics, 2021

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expressi...