User talk:Rxc

Problem 2
$$x_0$$ means any possible initial conditions, $$\bar{x}_0$$ is our initial condition.

$$\rho(x(x_0,t),t)=\prod_{i=1}^{6N}\delta(x^i(x_0^i,t)-\bar{x}^i(\bar{x}_0^i,t))$$

$$\frac{d}{dt}\rho(x(x_0,t),t)=\sum_i\left\{\delta'(x^i(x_0^i,t)-\bar{x}^i(\bar{x}_0^i,t)) [\dot{x}^i(x_0^i,t)-\dot{\bar{x}}^i(\bar{x}_0^i,t)] \prod_{j\neq i}\delta(x^j(x_0^j,t)-\bar{x}^j(\bar{x}_0^j,t))\right\}$$

if $$x^i = \bar x^i$$ then $$\dot{x}^i(x_0^i,t)-\dot{\bar{x}}^i(\bar{x}_0^i,t) = 0$$

if $$x^i \ne \bar x^i$$ then $$\delta(x^i(x_0^i,t)-\bar{x}^i(\bar{x}_0^i,t)) = 0$$

Problem 3
$$H = \frac{p^2}{2m} + \frac 1 2 k q^2$$

Let m = 1 and k = 1,

$$\rho_t = kq\rho_p - \frac p m \rho_q$$

Lemming
If $$\bar{p}(t)$$ and $$\bar{q}(t)$$ obays Hamilton's equations, then
 * $$\rho(p,q,t)=F(p-\bar{p}(t),q-\bar{q}(t))$$

is a solution to the Liouville's equation.


 * Proof.

Only $$\bar{p}(t)$$ and $$\bar{q}(t)$$ are explicit functions of t.

$$\frac{\partial F}{\partial t} = -\dot{\bar p}(t)\partial_1F-\dot{\bar q}(t)\partial_2F = -\dot{\bar p}(t)\frac{\partial F}{\partial p}-\dot{\bar q}(t)\frac{\partial F}{\partial q}$$