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

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

## Definition

Let $M$ be a smooth manifold and let a lie algebra bracket

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

A poisson structure on $M$ is a smooth bivector field $\pi$ on $M$ satisfying $[\pi, \pi]_{s} = 0$, where $[\cdot, \cdot]_{s}$ is the Schouten bracket. Let $U$ be an open subset of $M$ and let $F, G \in C^{\infty}(U)$ be smooth functions, the bilinear operation $\{ \cdot, \cdot\}_{U}$ is defined by $\{F, G\}_{U}(m) = \langle d_{m}F \wedge d_{m}G, \pi_{m} \rangle$ for all $m \in M$, which satisfies the Jacobi Identity.

## Hamiltonian formalism

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

If we take the 2-form $\pi^{2}|_{x}$ at a point $x$ of a manifold $M$ and it’s differentiable, then we say that it’s a 2-form on the tangent space $T_{x}M$ (i.e. A 2-linear skew-symmetric function of 2 vectors $P_{1}, P_{2}$ tangent to $M$ at x).

Let $A$ be a $\mathbb{R}$-algebra, then $\Delta: A \rightarrow A$ is called a linear differential operator. Let Diff$_{k}$ the set of all differential operator of order $\leq k$ acting on $A$ from $A$. Let $S_{k}(A) = \frac{\text{Diff}_{k}(A)}{\text{Diff}_{k-1}(A)}$ be a quotient module which is called the $k$-symbols, now let us define the algebra of symbols for the algebra $A$ in the following way:

Let $\mathfrak{s} = \text{smbl}_{i}\Delta$, where $\text{smbl}_{i}\Delta \in S_{k}(A)$ for $i = 1, \dots, k$ and thanks to \ref{a}, we can say that

since $[f, g] = [g, f] = 0$ for $f, g \in C^{\infty}(U)$ where $U \subset M$. Applying \ref{c} to $F = f(x)p^{\alpha}, G = g(x)p^{\beta}$ where $\{p^{\alpha}, p^{\beta}\} = 0$, we have

by the equality

After some computations, we get

which is the standard Poisson bracket on $T^{*}M$, making $X_{F} = \{F, G\}$, then $X_{F}$ is the Hamiltonian vector field on $T^{*}M$ with the Hamiltonian $F$.

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.