The Poisson brackets for u,v can be written as,
∂u∂q∂v∂p−∂u∂p∂v∂q.
We can write this as determinant of this matrix
[∂u∂q∂u∂p∂v∂q∂v∂p]
Which is the Jacobian matrix.
Is there any relation between them?
Answer
For a 2-dimensional phase space, they are the same.
More generally, for a 2n-dimensional symplectic manifold (M;ω) with symplectic two-form ω = 12dzI ωIJ∧dzJ,ωIJ = −ωJI,
the Poisson bracket is given by {f,g}PB = ∂f∂zIπIJ∂g∂zJ,πIJ := (ω−1)IJ.See also e.g. this and this Phys.SE posts.The canonical volume form on (M;ω) Ω = ωn = ρ dz1∧…∧dz2n,
is the n'th exterior power of the symplectic two-form ω, with volume density ρ = Pf(ωIJ)given by the Pfaffian, which is a square root of the determinant. This is closely related to Liouville's theorem.
No comments:
Post a Comment