3. If xk = (1)/(√(2))(qk + ipk) and x̅k = (1)/(√(2))(qk - ipk), show that the Hamilton's equatio

Step 1: Rewrite the Hamilton's equations of motion using the given expressions for x_k and bar(x

3. If xk = (1)/(√(2))(qk + ipk) and x̅k = (1)/(√(2))(qk -...

3. If $x_k = \frac{1}{\sqrt{2}}(q_k + ip_k)$ and $\bar{x}_k = \frac{1}{\sqrt{2}}(q_k - ip_k)$, show that the Hamilton's equations of motion can be expressed in the form \frac{dx_k}{dt} + i\frac{\partial H}{\partial \bar{x}_k} = 0, where $i = \sqrt{-1}$.

Transcript

00:01 Hi there, so for this problem we know that the function k is just simply x to the y times z to the 3.

00:19 And for this we are given that x is equal to r times x to the square, then y is s plus r times t, and z is just the product between r, s, and t, and this plus 2.

00:38 So we are asked to calculate the partial derivative of this function with respect to s.

00:49 Okay, so this partial derivative can be written as follows.

00:55 The partial derivative of the function k with respect to x times the partial derivative of x with respect to s, then this plus the partial derivative of k with respect to y times the partial derivative of y with respect to, in this case with respect to s.

01:16 Okay, and finally the partial derivative of k with respect to z this times the partial derivative of z with respect to s.

01:27 So let's do those partial derivatives separately.

01:30 So let's start with the partial derivative of k with respect to x.

01:33 Now as you can see the only term that depends on x is this one right here.

01:38 So we treat everything else as a constant.

01:42 So then that will be, we take down the y so that will be then x elevated to y minus one.

01:53 Okay, so that's the derivative with respect to x and then this times z to the three.

02:04 Let's do the partial derivative of k with respect to, well in here we can already substitute the values that we are given for this.

02:18 So we know that y is just s plus r times t, then this times s and that will be elevated to s plus r times t minus one, then c to the three so it'll be r s t plus two and that to this to the three.

02:43 Now let's do the partial derivative of k with respect to y...

Question

3. If $x_k = \frac{1}{\sqrt{2}}(q_k + ip_k)$ and $\bar{x}_k = \frac{1}{\sqrt{2}}(q_k - ip_k)$, show that the Hamilton's equations of motion can be expressed in the form \frac{dx_k}{dt} + i\frac{\partial H}{\partial \bar{x}_k} = 0, where $i = \sqrt{-1}$.

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