Skip to content
ψ Miko-pedia
Map 日本語
← Back to map
Mechanics · undergrad

Poisson Bracket

The algebraic operation controlling time evolution in Hamiltonian mechanics.

· 1 min
View PDF PDF LaTeX

PDF View

Poisson Bracket

Open PDF

The PDF file is linked, but it has not been added yet.

/sources/poisson-bracket.pdf

Definition

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

{f,g}=∑i(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi).\{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).

Time Evolution

If HH is the Hamiltonian,

dfdt={f,H}+∂f∂t.\frac{df}{dt} = \{f,H\} + \frac{\partial f}{\partial t}.
Related
Mechanics · Notes · undergrad

Canonical Momentum

Momentum defined by the Lagrangian, not always equal to mechanical momentum.
Quantum · Notes · undergrad

Commutator

The noncommutativity measure that organizes quantum observables.
Mechanics · Notes · undergrad

Hamiltonian Mechanics

Dynamics on phase space using coordinates and conjugate momenta.