Classical mechanics
6. Poisson brackets
One of the most important tools at our disposal in Hamiltonian mechanics are Poisson brackets. These help us determine if transformations are canonical and, on a deeper level, act as a sort of bridge between classical and quantum mechanics^{1}.
Defining Poisson brackets
Consider two functions $f$ and $g$ of the generalised variables $p$ and $q$, where the variables can refer to absolutely any set of quantities that fully describe a system. The notations $p$ and $q$ are usually termed generalised momenta and generalised coördinates respectively.
The Poisson bracket of $f$ and $g$ is then defined as $$ [f, g]_{p,q} \equiv {\partial f \over \partial q} {\partial g \over \partial p} - {\partial f \over \partial p} {\partial g \over \partial q} $$
and may be generalised to $$ \begin{equation} [f, g] \equiv {\partial f \over \partial q_i} {\partial g \over \partial p_i} - {\partial f \over \partial p_i} {\partial g \over \partial q_i} \label{eq:pbrackets} \end{equation} $$ for sets of $n$ variables, $(q_1, q_2, \ldots, q_n)$ and $(p_1, p_2, \ldots, p_n)$, with summation implied as per the Einstein notation which tells us that otherwise undefined, doubly repeating indices in a term must be summed over as contextually relevant.
This immediately gives rise to two important properties of the Poisson bracket: $$ \begin{align} [q_i,q_j] &= [p_i,p_j] = 0 \\[.5em] \textrm{and}\; [q_i,p_j] &= -[p_i,q_j] = \delta_{ij} \label{eq:antisymmetry} \end{align} $$
where $\delta_{ij}$ is the Kronecker delta function. The property in eq. $\eqref{eq:antisymmetry}$ is known as antisymmetry.
More properties
The property that makes Poisson brackets particularly useful for canonical transformations is that during a transformation Poisson brackets remain invariant: $$ \begin{equation} [f,g]_{p,q} = [f,g]_{P,Q} \label{eq:invariance} \end{equation} $$
Poisson brackets, like any mathematical tool, adhere to certain rules that help us manipulate them. Think of this as analogous to the basic arithmetic operations for Poisson brackets. These extend to the quantum realm^{2} as well. $$ \begin{align} [f,f] &= 0 \nonumber \\[.5em] [f,g] &= -[g,f] \nonumber \\[.5em] \left[\alpha f + \beta g, h\right] &= \alpha [f, h] + \beta [g,h] \label{eq:linearity} \\[.5em] [fg,h] &= [f,h]g + f[g,h] \label{eq:last-rule} \end{align} $$
We saw the first two of these already; and all four can be proven by simply expanding the brackets as in eq. \eqref{eq:pbrackets} using partial derivatives. Equation \eqref{eq:linearity} is known as linearity which is the same idea as in algebra^{3}.
There is a fifth, extremely important cyclic relationship known as Jacobi's identity that all Poisson brackets obey: $$ \begin{equation} [f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0 \label{eq:jacobi-identity} \end{equation} $$
We prove this in chapter 7.
- The name Poisson is pronounced ‘pwa-son’, nasalised, referring to the eighteenth century French physicist Siméon-Denis Poisson. ↩
- If you are wondering where Poisson brackets appear in quantum mechanics, we use the operator-led definition of the Poisson bracket between $f$ and $g$ as $ (\hat{f}\hat{g} - \hat{g}\hat{f})/i\hbar $ for commutators. ↩
- To help you recall, linearity in algebra (at least additive linearity) is defined as $\alpha (m + n) = \alpha m + \alpha n$ ↩