Solving Poisson Brackets: Expanding & Showing

This confirms that the Poisson brackets are indeed valid. In summary, the correct expansion using Poisson brackets shows that the double terms are obtained by combining the terms involving \frac{\partial}{\partial t} together, confirming the validity of the Poisson brackets.
  • #1
UrbanXrisis
1,196
1
I need to show using Poisson brackets that:

[tex]\left( \frac{\partial}{\partial t} \right) {f,g} = \left( \frac{\partial f}{\partial t} , g} \right)+ \left( {f, \frac{\partial g}{\partial t} \right)[/tex]

I know that:

[tex] (f,g) = \left( \frac{\partial f}{\partial q} \frac{\partial g}{\partial p}} \right)- \left( {\frac{\partial f}{\partial p} \frac{\partial g}{\partial q} \right) [/tex]

To show the above statement, I will expand using Poisson Brackets:

[tex] \left( \frac{\partial f}{\partial f} , g \right) [/tex] and [tex] \left( f, \frac{\partial g}{\partial t} \right) [/tex]

[tex] \left( {\frac{\partial f}{\partial f} , g} \right)= \frac{\partial }{\partial q} \frac{\partial f}{\partial t} \frac{\partial g}{\partial p} - \frac{\partial }{\partial p} \frac{\partial f}{\partial t} \frac{\partial g}{\partial q} [/tex]

[tex] \left( {f, \frac{\partial g}{\partial t} } \right)= \frac{\partial f}{\partial q} \frac{\partial }{\partial p} \frac{\partial g}{\partial t} - \frac{\partial f}{\partial p} \frac{\partial }{\partial q} \frac{\partial g}{\partial t} [/tex]

[tex] \frac{\partial}{\partial t} \left( {f,g} \right)= \frac{\partial }{\partial q} \frac{\partial f}{\partial t} \frac{\partial g}{\partial p} - \frac{\partial }{\partial p} \frac{\partial f}{\partial t} \frac{\partial g}{\partial q} + \frac{\partial f}{\partial q} \frac{\partial }{\partial p} \frac{\partial g}{\partial t} - \frac{\partial f}{\partial p} \frac{\partial }{\partial q} \frac{\partial g}{\partial t} [/tex]

[tex] \frac{\partial}{\partial t} \left( f,g \right)= \frac{\partial}{\partial t} \left( 2 \frac{\partial f}{\partial q} \frac{\partial g}{\partial p}} - 2 \frac{\partial f}{\partial p} \frac{\partial g}{\partial q} \right)[/tex]

Am I using Poisson brackets correctly? Not sure how I got double the terms I wanted.
 
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  • #2


Yes, you are using Poisson brackets correctly. However, it seems like you have made a mistake in your expansion. The correct expansion should be:

\frac{\partial}{\partial t} \left( f,g \right) = \frac{\partial}{\partial q} \frac{\partial f}{\partial t} \frac{\partial g}{\partial p} - \frac{\partial}{\partial p} \frac{\partial f}{\partial t} \frac{\partial g}{\partial q} + \frac{\partial f}{\partial q} \frac{\partial}{\partial p} \frac{\partial g}{\partial t} - \frac{\partial f}{\partial p} \frac{\partial}{\partial q} \frac{\partial g}{\partial t}

= \left( \frac{\partial f}{\partial t}, g \right) + \left( f, \frac{\partial g}{\partial t} \right)

Notice that the terms involving \frac{\partial}{\partial t} have been combined together to give the desired result.
 

1. What is a Poisson bracket?

A Poisson bracket is a mathematical concept used in classical mechanics to describe the relationship between two physical quantities. It is represented by the symbol {A, B} and is defined as the product of the partial derivatives of A and B with respect to their respective variables, multiplied by the canonical variable of momentum, denoted by p.

2. How do you expand a Poisson bracket?

To expand a Poisson bracket, you first need to identify the quantities A and B, and their corresponding variables. Then, you can use the formula {A, B} = ∂A/∂q * ∂B/∂p - ∂A/∂p * ∂B/∂q to calculate the expanded bracket. This process is also known as applying the Poisson bracket operator to the two quantities.

3. What is the significance of solving Poisson brackets?

Solving Poisson brackets is important because it allows us to analyze the behavior and dynamics of a physical system in classical mechanics. By calculating the Poisson bracket of two quantities, we can determine whether they are in a state of equilibrium or if there is a change in their values over time. This information is crucial in understanding the behavior of systems such as celestial bodies and particles.

4. Can you show an example of solving a Poisson bracket?

Sure, let's say we have the quantities A = x and B = py, where x and y are position variables and p is the canonical momentum variable. The expanded Poisson bracket would be {A, B} = ∂x/∂x * ∂py/∂p - ∂x/∂p * ∂py/∂x = y. This result tells us that the two quantities are not in equilibrium and their values will change over time.

5. How is solving Poisson brackets related to Hamilton's equations?

Solving Poisson brackets is closely related to Hamilton's equations, which describe the evolution of a physical system in classical mechanics. In fact, Hamilton's equations can be derived from the Poisson bracket by setting A = H (Hamiltonian) and B = q (generalized coordinate). This shows that the Poisson bracket is a fundamental concept in classical mechanics and its solutions can be used to determine the behavior of a system over time.

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