About the basics of Poisson bracket

In summary, the conversation is about solving problems related to Poisson brackets on a 2n-manifold with a closed non-degenerate differential 2-form. The Poisson bracket is defined using the inverse of the local function matrix and has three properties: well-definedness, skew-symmetry, and the Leibniz rule. The person asking for help is advised to consult a book on differential geometry or manifolds for information about smooth functions before attempting to solve the problems.
  • #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
 
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  • #2
Erm, the answer is to "just do it".

calculate {f,g} on the overlap of two elements in the atlas and use the transformation rules for f and g (it's a smooth function) to show they are equal.

2 and 3, it now suffices to work locally, so do so.

it's not pleasant but that's the way it works I'm afraid.
 
  • #3
Thanks for your reply. Would you please cite some
references so that I may consult the transformation rules
or the information related to my questions.

Sarason
 
  • #4
Eh? any book on differential geometry or differential manifolds, or even differentiable manifolds will tell you what it means for a function to be smooth on the manifold. If you've not seen this then trying to do poisson brackets is a little adventurous.
 

1. What is a Poisson bracket?

A Poisson bracket is a mathematical operation used in classical mechanics to calculate the time evolution of a system. It is denoted by {f, g} and is defined as the partial derivative of f with respect to the coordinates and momenta of the system, multiplied by the partial derivative of g with respect to the same coordinates and momenta, and then subtracting the same product but with the order of the functions reversed.

2. What is the purpose of using a Poisson bracket?

The Poisson bracket is used to calculate the equations of motion for a classical mechanical system. It helps determine how the coordinates and momenta of the system change over time, allowing for the prediction of the future behavior of the system.

3. How is a Poisson bracket different from other mathematical operations?

The Poisson bracket is different from other mathematical operations because it involves both the coordinates and momenta of a system, whereas most other operations only involve one or the other. It also takes into account the time evolution of the system, making it a dynamic operation rather than a static one.

4. Can a Poisson bracket be used for any type of system?

No, a Poisson bracket is specifically designed for classical mechanical systems. It is not applicable to quantum mechanics or other types of systems.

5. Are there any practical applications of Poisson brackets?

Yes, Poisson brackets have several practical applications in physics and engineering. They are used in the study of celestial mechanics, fluid dynamics, and electromagnetism, among other fields. They also have applications in control theory and optimal control, such as in robotics and aerospace engineering.

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