Definition:Poisson Bracket

Definition

For $n \in \N$, let:

$\map A {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$

$\map B {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$

be real functions, dependent on canonical variables.

The Poisson bracket of functions $A$ and $B$ is defined as:

$\ds \sqbrk {A, B} := \sum_{i \mathop = 1}^n \paren {\frac {\partial A} {\partial y_i} \frac {\partial B} {\partial p_i} - \frac {\partial B} {\partial y_i} \frac {\partial A} {\partial p_i} }$

where the notation $\dfrac {\partial A} {\partial y_i}$ denotes partial differentiation.

Source of Name

This entry was named for Siméon-Denis Poisson.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.16$: The Canonical Form of the Euler's Equations