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Poisson Brackets

A mathematical operation that quantifies the relationship between two variables in Hamiltonian mechanics, denoted by \[ {x, y} ] = \frac{\partial{x}}{\partial{y}} - \frac{\partial{y}}{\partial{x}} \[.
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The statement of the theorem

Define the Poisson bracket {⋅,⋅}\left\{ \cdot , \cdot \right\} for two smooth functions f(q,p,t)f(q, p, t) and g(q,p,t)g(q, p, t) on the phase space M\mathcal{M} as:\n\n{f,g}=∑i=1N(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi)\left\{ f , g \right\} = \sum_{i=1}^{N} \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) \n\nFor the specific case of canonical coordinates xx and yy, the bracket simplifies to {x,y}=∂x∂y−∂y∂x\left\{ x , y \right\} = \frac{\partial x}{\partial y} - \frac{\partial y}{\partial x}. The time evolution of an observable ff is given by dfdt={f,H}+∂f∂t\frac{d f}{d t} = \left\{ f , H \right\} + \frac{\partial f}{\partial t}.
Source: Wikipedia