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The '''Poisson bracket''' is the fundamental binary operation of [[Hamiltonian Mechanics|Hamiltonian mechanics]] that encodes the algebraic structure of dynamics on [[Phase Space|phase space]]. For any two observables f and g, the Poisson bracket {f,g} measures how the flow generated by one observable changes the other — a Lie algebra structure that makes phase space a [[Symplectic Geometry|symplectic manifold]] in a rigorous sense.

The bracket's defining property is its relationship to the Hamiltonian: the time derivative of any observable is its bracket with the Hamiltonian, df/dt = {f,H}. This makes the Poisson bracket the engine of evolution, and its antisymmetry and Jacobi identity the constraints that any consistent classical dynamics must satisfy. When classical mechanics is deformed into [[Quantum Mechanics|quantum mechanics]], the Poisson bracket becomes the commutator divided by iℏ — a structural continuity that reveals Hamiltonian mechanics as the classical skeleton of quantum theory.

[[Category:Physics]]
[[Category:Mathematics]]