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Poisson Bracket

The algebraic operation controlling time evolution in Hamiltonian mechanics.

公開 2026年4月26日 · 1 分

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Definition

For phase-space functions $f(q,p)$ and $g(q,p)$ ,

\{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).

If $H$ is the Hamiltonian,

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

Momentum defined by the Lagrangian, not always equal to mechanical momentum.

The noncommutativity measure that organizes quantum observables.

Dynamics on phase space using coordinates and conjugate momenta.