4. Fourier-Space Hamilton's Equations

Apply the Fourier transform to the Hamilton's equations:

	$$\dot{\tilde{x}}_k$$	$$= \frac{1}{\sqrt{N}} \sum_j \dot{x}_j e^{-i k j} = \frac{1}{\sqrt... ...rtial H}{\partial p_j} e^{-i k j} = \frac{\partial H}{\partial \tilde{p}_{-k}},$$	(39)
	$$\dot{\tilde{p}}_k$$	$$= \frac{1}{\sqrt{N}} \sum_j \dot{p}_j e^{-i k j} = - \frac{1}{\sq... ...ial H}{\partial x_j} e^{-i k j} = - \frac{\partial H}{\partial \tilde{x}_{-k}}.$$	(40)

Hence, the Fourier-space Hamilton's equations are

$$\boxed{ \dot{\tilde{x}}_k = \frac{\partial H}{\partial \tilde{p}_{-k}}, \quad \dot{\tilde{p}}_k = - \frac{\partial H}{\partial \tilde{x}_{-k}}. }$$ (41)