- #1
sarason
- 2
- 0
Dear all,
Please help me to solve the following problems
about Poisson brackets.
Let M be a 2n-manifold and w is a closed non-degenerate di®eren-
tial 2-form. (Locally we write w = w_ij dx^i ^ dx^j with [w_ij ] being a
non-degenerate anti-symmetric real matrix-valued local function on M)
Let f, g be two smooth functions on M. Define the Poisson bracket
{f, g} as follows: locally,
{f, g} = w^ij \partial_i f \partial_j g
where [w^ij ] is the inverse of [w_ij ].
SHOW that
1) The Poisson bracket is well-defined, i.e., on the intersection of two
coordinate patches, the two definitions, one written in each local co-
ordinate system, actually always agree.
2) {f, g} = -{g, f} and
{f, gh} = {f, g}h + g{f, h}
for any three smooth functions f, g, h.
3) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0
for any three smooth functions f, g, h.
Thank you very much for taking time to consider these problems.
Sarason
Please help me to solve the following problems
about Poisson brackets.
Let M be a 2n-manifold and w is a closed non-degenerate di®eren-
tial 2-form. (Locally we write w = w_ij dx^i ^ dx^j with [w_ij ] being a
non-degenerate anti-symmetric real matrix-valued local function on M)
Let f, g be two smooth functions on M. Define the Poisson bracket
{f, g} as follows: locally,
{f, g} = w^ij \partial_i f \partial_j g
where [w^ij ] is the inverse of [w_ij ].
SHOW that
1) The Poisson bracket is well-defined, i.e., on the intersection of two
coordinate patches, the two definitions, one written in each local co-
ordinate system, actually always agree.
2) {f, g} = -{g, f} and
{f, gh} = {f, g}h + g{f, h}
for any three smooth functions f, g, h.
3) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0
for any three smooth functions f, g, h.
Thank you very much for taking time to consider these problems.
Sarason