Poisson Brackets

Abstract

Let F = F(q _i , p _i , t) be any dynamical variable of a system represented by the conjugate variables q _i , p _i . Then:

$$\overset{\cdot }{\mathop{F}}\,\equiv \frac{dF}{dt}=\sum\limits_{i}{\frac{\partial F}{\partial {{q}_{i}}}}\overset{\cdot }{\mathop{{{q}_{i}}}}\,+\sum\limits_{i}{\frac{\partial F}{\partial {{p}_{i}}}}{{\overset{\cdot }{\mathop{p}}\,}_{i}}+\frac{\partial F}{\partial t}$$

(8.1)

From Hamilton’s canonical equations (5.16) this becomes:

$$\overset{\cdot }{\mathop{F}}\,=\sum\limits_{i}{\left( \frac{\partial F}{\partial {{q}_{i}}} \right.}\frac{\partial H}{\partial {{p}_{i}}}-\frac{\partial F}{\partial {{p}_{i}}}\left. \frac{\partial H}{\partial {{q}_{i}}} \right)+\frac{\partial F}{\partial t}$$

(8.2)

The quantity$\sum\limits_{i}{\left( \frac{\partial F}{\partial {{q}_{i}}} \right.}\frac{\partial F}{\partial {{p}_{i}}}-\frac{\partial F}{\partial {{p}_{i}}}\left. \frac{\partial H}{\partial {{q}_{i}}} \right)$ turns out to be a very significant one in the formal development of mechanics and is called the Poisson bracket of F and H. In general, the Poisson bracket of any two dynamical variables X and Y is defined as:

$${{\left[ X,Y \right]}_{q,p}}=\sum\limits_{i}{\left( \frac{\partial X}{\partial {{q}_{i}}} \right.}\frac{\partial Y}{\partial {{p}_{i}}}-\frac{\partial X}{\partial {{p}_{i}}}\left. \frac{\partial Y}{\partial {{q}_{i}}} \right)$$

(8.3)

The concept does not assist materially in the complete solution of the equations of motion of a system, but is of use in discussing the constants of motion, as will be seen. It leads to a formalism which, when re-interpreted according to a simple recipe, forms a convenient way of introducing quantum rules in the Heisenberg development of quantum mechanics.