In 1st January 2016 I submitted a paper on Arxiv about how to describe a Poisson structure on a manifold with corners, I’ll write about it in a later post, now I’m going to talk about an introduction to understand what a poisson structure is.

Introduction

Suppose that we want to describe a system evolution, How can we do that? First, let’s list elements of a system.

  • Let be a smooth manifold (physical system).
  • Let be a point on (state of the system).
  • Let be a smooth function on (measuring device).

So, we can say that any state on can be measured by . How the evolution works in this case?, we need to get a collection of measuring devices (an -algebra homomorphism) and then the evolution depends about how is the the behaviour in this collection, thus can be viewed as a hamiltonian function, more precisely a Poisson structure on a smooth manifold associates to every smooth function on , a vector field on . After this short intuition, let’s define formally some concepts.

Definition

Let be a smooth manifold and let a lie algebra bracket

which is called a Poisson bracket on the vector space of smooth functions on and it’s skew-symmetry

A poisson structure on is a smooth bivector field on satisfying , where is the Schouten bracket. Let be an open subset of and let be smooth functions, the bilinear operation is defined by for all , which satisfies the Jacobi Identity.

Hamiltonian formalism

To talk about Hamiltonian formalism, let’s recall firstly the definition of a differential -form.

If we take the 2-form at a point of a manifold and it’s differentiable, then we say that it’s a 2-form on the tangent space (i.e. A 2-linear skew-symmetric function of 2 vectors tangent to at x).

Let be a -algebra, then is called a linear differential operator. Let Diff the set of all differential operator of order acting on from . Let be a quotient module which is called the -symbols, now let us define the algebra of symbols for the algebra in the following way:

Let , where for and thanks to \ref{a}, we can say that

since for where . Applying \ref{c} to where , we have

by the equality

After some computations, we get

which is the standard Poisson bracket on , making , then is the Hamiltonian vector field on with the Hamiltonian .

So, there a lot of work to do and study about Poisson structure, this is just a short introduction, however depending of the space over we work, it’s not always easy to define a Poisson structure, next time we’ll talk about how can we describe a Poisson structure on a manifold with singularities, this is thanks to the work of Miss Maria Sorokina: Poisson structure on manifolds with singularities.