The algebra of Poisson brackets

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)

arXiv (Cornell University), 2022

In the phase space R 2d , let us denote {A, B} the Poisson bracket of two smooth classical observables and {A, B} ⊛ their Moyal bracket, defined as the Weyl symbol of i[Â,B], where A is the Weyl quantization of A and [ A, B] = A B − B A (commutator). In this note we prove that if a given smooth Hamiltonian H on the phase space R 2d , with derivatives of moderate growth, satisfies {A, H} = {A, H} ⊛ for any observable A in the Schwartz space S(R 2d), then, as it is expected, H must be a polynomial of degree at most 2 in R 2d. A related answer to this question is given in the Groenewold-van Hove Theorem [4, 5, 8] concerning quantization of polynomial observables. We consider here more general classes of Hamiltonians.

Lecture Notes, available at http://toknotes. …, 2006

Archive for Rational Mechanics and Analysis

In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by A. Weinstein,, in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from , but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity tranformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results give the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.

Journal of Mathematical Physics, 1997

The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket A unified setting for generalized Poisson and Nambu-Poisson brackets is discussed. It is proved that a Nambu-Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively. © 1997 American Institute of Physics. ͓S0022-2488͑97͒04305-3͔

Communications in Mathematical Physics, 2005

We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnković-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.

HAL (Le Centre pour la Communication Scientifique Directe), 2023

In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector fields involving the cofactors of the Jacobian are divergence free. We illustrate the fundamental impact of these properties on research, from the times of Jacobi to our days.

Journal of Mathematical Physics, 1993

The ordinary Poisson brackets in field theory do not fulfill the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. It is shown that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery, and Ratiu for the free boundary problem in hydrodynamics. The definition of Poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets.

1999

On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.

1995

Classi cation of a Poisson bivectors on some Lie algebras 3.1 Poisson bivectors on a compact Lie algebra 3.2 Poisson bivectors and closed 2-forms on elementary (K ahler) Lie algebras 4 Poisson bivectors subordinated to a closed 2-form and groups acting simply transitively on orbits 4.1 Relations between Poisson bivectors and closed 2-forms on a Lie algebra 4.2 Poisson bivectors subordinate d to a closed 2-form 4.3 Case of exact 2-form. Frobenius Lie algebras 4.4 Case of semi-simple Lie algebra. Problem of classi cation of almost simply transitive Lie subgroups of G 4.5 Iwasawa decomposition for SO(p; 2) and the canonical symplectic form on the Iwasawa subalgebra 5 Poisson bivectors associated to a Hermitian symmetric domains for the function X(;) taking the values in G G, where G is a Lie algebra. In order to de ne the quantity X 12 (1 ; 2), following BD 1982], we x an associative algebra A with unit, which contains G and the linear maps ' 12 ; ' 23 and ' 13 , so that ' 12 : G G ! A A A; ' 12 (a b) = a b 1 (0:2) and analogously for maps ' 23 and ' 13. Note that if X(;) is a solution of eq. (0.1) and '(u) is a function with values in G, thenX(;) = ('() '())X(;) is also a solution of (0.1) and we will consider the solutions X andX as equivalent. Let us introduce the following de nition. De nition 0.1. The function X(;) is invariant relative to g 2 Aut G, if (g g)X(;) = X(;): The set of all such g forms the group that is called the invariance group of X(;). The function X(;) is said to be invariant with respect to h 2 G if h 2 1 + 1 h; X(;)] = 0; i.e. if it is invariant relative to expfadhg for any t. Note that if X(;) is a solution of (0.1), which is invariant relative to the subalgebra