How to solve the following Poisson bracket

[flower321]

anyone can help me how to solve the following poisson bracket?
{U(x,λ), U(y,µ)} = −(1/λµ) {S^{i} (x), S^{j} (y)} σ_{i} ⊗ σ_{j}

where
U(x, λ) = −(i/λ) S(x)

 

[jvicens]

and U(y, µ) = −(i/µ) S(y)

Sure, I'd be happy to help you solve this Poisson bracket! First, let's break down the notation a bit to make it easier to understand. The curly braces { } indicate a Poisson bracket, which is a mathematical operation that involves taking the partial derivatives of two functions with respect to their variables and then multiplying them together. In this case, the functions are U(x, λ) and U(y, µ). The variables for U(x, λ) are x and λ, while the variables for U(y, µ) are y and µ.

Next, we see that U(x, λ) and U(y, µ) are defined in terms of another function, S(x), which has a subscript i and represents the i-th component of the vector S. This means that S(x) is a vector function with multiple components.

To solve the Poisson bracket, we will need to use the definition of the Poisson bracket and the given functions. The general formula for a Poisson bracket is:

{f, g} = ∂f/∂x * ∂g/∂y - ∂f/∂y * ∂g/∂x

where f and g are two functions with variables x and y.

Using this formula, we can solve the Poisson bracket in the given problem. Let's start with the first function, U(x, λ). Taking the partial derivative with respect to x, we get:

∂U(x, λ)/∂x = -i/λ * ∂S(x)/∂x

Similarly, for the second function U(y, µ), we get:

∂U(y, µ)/∂y = -i/µ * ∂S(y)/∂y

Now, we can plug these derivatives into the formula for the Poisson bracket:

{U(x,λ), U(y,µ)} = (-i/λ) * ∂S(x)/∂x * (-i/µ) * ∂S(y)/∂y - (-i/λ) * ∂S(x)/∂y * (-i/µ) * ∂S(y)/∂x

= (i²/λµ) * (∂S(x)/∂x * ∂S(y)/∂y - ∂S(x)/∂y