Poisson bracket

The Poisson bracket measures how two quantities on Phase space fit together dynamically. If $f(q,p,t)$ and $g(q,p,t)$ are functions of coordinates and momenta, then

$$\{f,g\}=\sum _i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right).$$

With the Hamiltonian $H$, the time evolution of any phase-space quantity is

$$\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}.$$

So $H$ is not merely an energy formula; in Hamiltonian mechanics it is the generator of time translations.

Important cases

$$\{q_i,p_j\}={\delta }_{ij},\{q_i,q_j\}=0,\{p_i,p_j\}=0.$$

A quantity $f$ with $\{f,H\}=0$ and no explicit time dependence is conserved.

Quantum bridge

In quantum mechanics, Poisson brackets are replaced by commutators in a precise analogy. Roughly,

$$\{\cdot ,\cdot \}⟶\frac{1}{i\hslash }[\cdot ,\cdot ].$$

This is one reason the Susskind classical book is good preparation for observables.

Common pitfalls

• The order matters: $\{f,g\}=-\{g,f\}$.
• The bracket is defined on phase-space functions, not directly on ordinary position-space curves.
• A zero bracket with $H$ signals conservation only when there is no explicit time dependence.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.