Poisson bracket
E559803
Lie bracket
bilinear operation
derivation
mathematical operator
structure in classical mechanics
structure in symplectic geometry
The Poisson bracket is a fundamental mathematical operator in classical mechanics and symplectic geometry that encodes the time evolution and mutual relationships of dynamical variables in Hamiltonian systems.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Lie bracket
ⓘ
bilinear operation ⓘ derivation ⓘ mathematical operator ⓘ structure in classical mechanics ⓘ structure in symplectic geometry ⓘ |
| appearsIn |
Liouville theorem
NERFINISHED
ⓘ
Noether theorem formulations in Hamiltonian mechanics ⓘ |
| classicalLimitOf | quantum commutator divided by iħ ⓘ |
| codomain | smooth functions on phase space ⓘ |
| definedOn |
Poisson manifold
ⓘ
symplectic manifold ⓘ |
| definesLieAlgebraOn | space of smooth functions on a Poisson manifold ⓘ |
| domain | smooth functions on phase space ⓘ |
| field |
Hamiltonian mechanics
ⓘ
classical mechanics ⓘ mathematical physics ⓘ symplectic geometry ⓘ |
| generalizationOf | canonical commutation relations in classical mechanics ⓘ |
| governs | time evolution via df/dt = {f,H} + ∂f/∂t ⓘ |
| inCanonicalCoordinates | {f,g} = Σ_i (∂f/∂q_i ∂g/∂p_i − ∂f/∂p_i ∂g/∂q_i) ⓘ |
| induces | Poisson structure on a manifold ⓘ |
| isAntisymmetric | true ⓘ |
| isBilinear | true ⓘ |
| isDerivationInFirstArgument | true ⓘ |
| isDerivationInSecondArgument | true ⓘ |
| mathematicalNature | skew-symmetric bilinear map ⓘ |
| namedAfter | Siméon Denis Poisson NERFINISHED ⓘ |
| relatedConcept |
Dirac bracket
ⓘ
Hamiltonian function ⓘ Moyal bracket NERFINISHED ⓘ Poisson manifold ⓘ canonical coordinates ⓘ canonical transformation ⓘ commutator ⓘ quantization ⓘ symplectic form ⓘ |
| satisfies |
{af+bg,h} = a{f,h}+b{g,h} for scalars a,b
ⓘ
{f,gh} = {f,g}h + g{f,h} ⓘ {f,g} = -{g,f} ⓘ {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0 ⓘ |
| satisfiesJacobiIdentity | true ⓘ |
| satisfiesLeibnizRule | true ⓘ |
| usedFor |
describing time evolution in Hamiltonian systems
ⓘ
encoding canonical equations of motion ⓘ expressing conservation laws ⓘ formulating integrability conditions ⓘ studying symmetries of dynamical systems ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.