### Sidebar

This is an old revision of the document!

$\left \{F,G \right \} = \sum_{n = 1}^{N} \left( \frac{\partial F}{\partial q_n} \frac{\partial G}{\partial p_n} - \frac{\partial F}{\partial p_n} \frac{\partial G}{\partial q_n} \right)$

# Poisson Bracket

## Intuitive

The Poisson bracket is a mathematical tool that allows us to calculate the time evolution in classical mechanics.

## Concrete

For two sets of canonical coordinates $q_1,q_2, … q_N$ and momenta $p_1, p_2, … p_N$ the Poisson bracket is defined by

$$\left \{F,G \right \} = \sum_{n = 1}^{N} \left( \frac{\partial F}{\partial q_n} \frac{\partial G}{\partial p_n} - \frac{\partial F}{\partial p_n} \frac{\partial G}{\partial q_n} \right)$$ for any two functions $F$ and $G$ of the canonical coordinates and momenta.

The Poisson bracket of two observables can be thought of as the rate of change of the first along the flow given by the second. The most famous example is the time evolution. To get the time evolution of some observable $O$ all we have to do is calculate the Poisson bracket of $O$ with the Hamiltonian function.

Properties of the Poisson Bracket
It is linear for both $F$ and $G$. It is antisymmetric $$\left[ G,F \right] = -\left[ F,G \right]$$ The Posisson bracket also satisfies the Jacobi identity $$\left[\left[F,G \right],H \right] + \left[\left[G,H \right],F \right] + \left[\left[H,F \right],G \right] = 0. \tag{2.2}$$

The Poisson bracket satisfied

$$\left\{ F G, H \right\} = F \left\{ G, H \right\} + G \left\{ F, H \right\} .$$

which looks like the Leibniz rule in calculus. This suggests that we interpret the Poisson bracket as derivative of the first argument with respect to the second argument.

The Poisson Bracket describe time evolution

In general, the time derivative of a function $F$ that is a function of generalized position and momentum coordinates, but not a direct function of time (which is often accurate) is

$$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{dq}{dt}+\frac{\partial F}{\partial p}\frac{dp}{dt}$$

Hamilton's equations of motion are $$\frac{dq}{dt}=\frac{\partial H}{\partial p}$$ and $$\frac{dp}{dt}=-\frac{\partial H}{\partial q}.$$

Putting these equations into our general time derivative from above yields

$$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial q}\equiv \{H,F\},$$

which shows that indeed Poisson brackets describe the time evolution of observables. If the function $F$ explicitly depends on the time we get instead $$\frac{\mathrm{d}F}{\mathrm{d}t} = \{F,H\} + \frac{\partial F}{\partial t}.$$ Take note how similar this equation is to the Heisenberg equation which describes the time evolution in quantum mechanics $$\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t}.$$ The only difference is that the Poisson brackets have been replaced with the commutator.

This is one way to make the difference between quantum and classical mechanics explicit:

$$\text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad \{f,g\}$$

The Poisson bracket describes general transformations/symmetries

As mentioned above the Poisson bracket of two observables can be thought of as the rate of change of the first along the flow given by the second.

So when we use the Hamiltonian $H$ as the generator, we end up with time evolution. Therefore, as shown above for an observable $A$ without explicit time dependence we have

## Why is it interesting?

Poisson brackets are necessary to describe the time evolution of observables in the Hamiltonian formulation of classical mechanics. Formulated differently, the Poisson bracket controls the dynamics in classical mechanics.

Poisson brackets play more or less the same role in classical mechanics that commutators do in quantum mechanics.

## FAQ

What is the Lie group that is generated by Poisson brackets?
The Lie group which integrates the Poisson bracket is called the “quantomorphism group”. For a nice discussion, see e.g. https://www.quora.com/How-is-the-Poisson-bracket-different-from-a-commutator