Poisson Brackets - Definition
In order to understand what Poisson Brackets are used for in physics, you need to know a little about the Hamiltonian and partial derivatives. I assume you know what partial derivatives are, but if you haven’t started yet with the Hamiltonian, let’s try to patch things up a bit.
First, the Hamiltonian digression:
The Hamiltonian is a function `H(q,p,t)` that depends specifically on position `q`, a special version of momentum called “canonical momentum” - `p` (for the moment just think of it as classical momentum), and time `t`. Once you know this function for every particle of your system, you can predict the system’s future evolution by solving Hamilton’s Equations with this function:
`dot p=- (partial H) / (partial q)`
`dot q= (partial H) / (partial p)`
You can think of variables `q`, `p` and `t` as being independent coordinates of the system. I know you usually think of p (momentum) as time derivative of position times mass, but this time momentum is seen just as any position coordinate and in this framework there is even the possibility to transform between position and momentum and vice-versa.
Poisson Brackets are defined as:
`[A,B]=((partial A)/(partial q) (partial B)/(partial p)-(partial B)/(partial q) (partial A)/(partial p))`
Where `A` and `B` are functions of `q` and `p` (and possibly of `t` but this is not relevant here).
Sometimes, you’ll also see this notation `{A,B}`.
To me, “Poisson Bracketing” is a new way to work with math formulas used in mechanics, which probably brings some advantages. We’ll see later what these are.
Steaming from the definition, we have the following properties of poisson brackets:
`[A,B]=-[B,A]`
`[A,A]=0`, in particular `[q,q]=0`, `[p,p]=0` and `[H,H]=0`
`[kA,B]=k[A,B]`, `[A,kB]=k[A,B]`
`[A+B,C]=[A,C]+[B,C]`
`[AB,C]=B[A,C]+A[B,C]`
`[q,p]=1` and `[p,q]=-1`
`[q^n,p]=nq^(n-1)=frac(d(q^n))(dq)`
And since any function can be approximated by a polynomial,
`[f(q,p),p]=frac(partial f(q,p))(partial q)`
which may be generalised to the fact that the Poisson Bracket of any function with `p` implies the differentiation of that function with q.
You probably noticed by now that there is a sort of antisymmetry between `q` and `p` and therefore we surely have:
`[f(q,p),q]=-frac(partial f(q,p))(partial p)`
Returning to our Hamiltonian, Hamilton’s equations may be written as:
`dot p=- (partial H) / (partial q)=[p,H]` (notice: no minus sign this time:-))
`dot q= (partial H) / (partial p)=[q,H]`
which this time do not seem antisymmetric anymore.