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PHY1500F Problem Set 1 (roflcopter)

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 $x^{i}\neq {\bar {x}}^{i}$ then $\delta (x^{i}(x_{0}^{i},t)-{\bar {x}}^{i}({\bar {x}}_{0}^{i},t))=0$

Problem 3

$H={\frac {p^{2}}{2m}}+{\frac {1}{2}}kq^{2}$

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.

${\frac {\partial F}{\partial t}}=-{\dot {\bar {p}}}(t)\partial _{1}F-{\dot {\bar {q}}}(t)\partial _{2}F=-{\dot {\bar {p}}}(t){\frac {\partial F}{\partial p}}-{\dot {\bar {q}}}(t){\frac {\partial F}{\partial q}}$