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Poisson bracket (Definition)

Let $ M$ be a symplectic manifold with symplectic form $ \Omega$. The Poisson bracket is a bilinear operation on the set of differentiable functions on $ M$. In terms of local Darboux coordinates $ p_1, \ldots, p_n, q_1, \ldots, q_n$, the Poisson bracket of two functions is defined as follows:

$\displaystyle [f,g] = \sum_{i=1}^n {\partial f \over \partial q_i} {\partial g ... ...\partial p_i} - {\partial f \over \partial p_i} {\partial g \over \partial q_i}$
It can be shown that the value of $ [f,g]$ does not depend on the choice of Darboux coordinates. Therefore, the Poisson bracket is a well-defined operation on the symplectic manifold. Also, some authors use a different sign convention -- what they call $ [f,g]$ is what would be referred to as $ -[f,g]$ here.

The Poisson bracket can be defined without reference to a special coordinate system as follows:

$\displaystyle [f,g] = \Omega^{-1} (df, dg) = \sum_{i=1}^{2n} \Omega^{ij} {\partial f \over \partial x_i} {\partial g \over \partial x_j}$
Here $ \Omega^{-1}$ is the inverse of the symplectic form, and its components in an arbitrary coordinate system are denoted $ \Omega^{ij}$.

The Poisson bracket sastisfies several important algebraic identities. It is antisymmetric:

$\displaystyle [f,g] = -[g,f]$
It is a derivation:
$\displaystyle [fg,h] = f[g,h] + g[f,h]$
It satisfies Jacobi's identitity:
$\displaystyle [f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0$

The Hamilton equations can be expressed elegantly in terms of the Poisson bracket. If $ X$ is a smooth function on $ M$, we can describe the time-evolution of $ X$ by the equation

$\displaystyle {dX \over dt} = [X,H]$
If $ X$ is a smooth function on $ \mathbb{R} \times M$, we can describe the time-evolution of $ X$ by the more general equation
$\displaystyle {dX \over dt} = {\partial X \over \partial t} - [X,H]$



"Poisson bracket" is owned by rspuzio. [ full author list (2) ]
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See Also: quantization, canonical quantization


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Poisson ring (Definition) by rspuzio
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Cross-references: equation, Hamilton equations, derivation, antisymmetric, identities, algebraic, components, inverse, coordinate system, operation, well-defined, functions, Darboux coordinates, terms, differentiable functions, bilinear operation, symplectic form, symplectic manifold
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This is version 8 of Poisson bracket, born on 2004-10-24, modified 2006-06-27.
Object id is 6412, canonical name is PoissonBracket.
Accessed 3646 times total.

Classification:
AMS MSC53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

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