The Product of Poisson Manifolds
Given two Poisson manifolds (M,{⋅,⋅}M) and (N,{⋅,⋅}N), a Poisson bracket may be defined on the product manifold M × N. Letting f1 and f2 be two smooth functions defined on M × N, one can define a new Poisson bracket {⋅,⋅}M × N in terms of {⋅,⋅}M and {⋅,⋅}N as follows:
- ,
where x ∈ M and y ∈ N are to be held constant. In other words, if
- ,
then both
and
are implied.
Read more about this topic: Poisson Manifold
Famous quotes containing the word product:
“Labor is work that leaves no trace behind it when it is finished, or if it does, as in the case of the tilled field, this product of human activity requires still more labor, incessant, tireless labor, to maintain its identity as a work of man.”
—Mary McCarthy (19121989)