91Ó°ÊÓ

In classical mechanics, the Poisson bracket of two functions, F1 and F2, with respect to generalized coordinates (\(q_i\)) and momenta (\(p_i\)) is defined as follows: $$ \{F_1, F_2\} = \sum_{i=1}^n \left( \frac{\partial F_1}{\partial q_i} \frac{\partial F_2}{\partial p_i} - \frac{\partial F_1}{\partial p_i} \frac{\partial F_2}{\partial q_i}\right) $$ Leibniz's rule states that the derivative of the product of two functions is given by: $$ \frac{d(F_1F_2)}{dx} = F_1\frac{dF_2}{dx} + F_2\frac{dF_1}{dx} $$

Let's compute the Poisson bracket of the product (F1F2) and H. We will first differentiate F1F2 with respect to the generalized coordinates \(q_i\) and momenta \(p_i\), and then apply the definition of the Poisson bracket to these derivatives. $$ \begin{aligned} \{F_1F_2, H\} &= \sum_{i=1}^n \left( \frac{\partial (F_1F_2)}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial (F_1F_2)}{\partial p_i} \frac{\partial H}{\partial q_i}\right) \\ &= \sum_{i=1}^n \left( \left(F_1\frac{\partial F_2}{\partial q_i} + F_2\frac{\partial F_1}{\partial q_i}\right) \frac{\partial H}{\partial p_i} - \left(F_1\frac{\partial F_2}{\partial p_i} + F_2\frac{\partial F_1}{\partial p_i}\right) \frac{\partial H}{\partial q_i}\right) \end{aligned} $$ We applied Leibniz's rule for differentiation to obtain these expressions.

Now we apply the definition of the Poisson bracket again to simplify the expression we obtained in Step 2: $$ \begin{aligned} \{F_1F_2, H\} &= \sum_{i=1}^n \left( F_1\frac{\partial F_2}{\partial q_i}\frac{\partial H}{\partial p_i} + F_2\frac{\partial F_1}{\partial q_i}\frac{\partial H}{\partial p_i} - F_1\frac{\partial F_2}{\partial p_i}\frac{\partial H}{\partial q_i} - F_2\frac{\partial F_1}{\partial p_i}\frac{\partial H}{\partial q_i}\right) \\ &= \sum_{i=1}^n F_1\left(\frac{\partial F_2}{\partial q_i}\frac{\partial H}{\partial p_i} - \frac{\partial F_2}{\partial p_i}\frac{\partial H}{\partial q_i}\right) + \sum_{i=1}^n F_2\left(\frac{\partial F_1}{\partial q_i}\frac{\partial H}{\partial p_i} - \frac{\partial F_1}{\partial p_i}\frac{\partial H}{\partial q_i}\right) \\ &= F_1\{F_2, H\} + F_2\{F_1, H\} \end{aligned} $$

We have shown that the Poisson bracket of a product can be calculated using Leibniz's rule: $$ \{F_1F_2, H\} = F_1\{F_2, H\} + F_2\{F_1, H\} $$