How do I construct an explicit diffeomorphism between $TS^1$ and $S^1\times\Bbb{R}$?
It will be something like $\phi:TS^1\to S^1\times\Bbb{R}, (x,v)\to(x,...)$.
Also we know that for $x=(x_1,x_2)$ and $v=(v_1,v_2)$, $x_1^2+x_2^2=1$ and $v_1x_2+v_2x_1=0$. From these two equations we have $$\begin{align}v_1x_2&=-v_2x_1 \\v_1^2x_2^2&=v_2^2x_1^2\\v_1^2x_2^2+v_1^2x_1^2&=v_2^2x_1^2+v_1^2x_1^2\\v_1^2&=x_1^2(v_2^2+v_1^2)\end{align}$$ And similarly $v_2^2=x_2^2(v_2^2+v_1^2)$. But I don't know where should I send $v$.