Jacobian in Hamiltonian dynamics

Question

I was trying to show that for an infinitesimal time evolution in classical Hamiltonian dynamics preserves volume by showing that the following Jacobian $$ \left|\frac{\partial(q(t),p(t))}{\partial(q(t_{0}),p(t_{0}))}\right| = +1 $$ By expanding $q(t)$, $p(t)$ as, $$ q(t)=q(t_{0})+\dot{q(t)}\Big|_{t_{0}} \delta t+O(\delta t^2) $$ $$ p(t)=p(t_{0})+\dot{p(t)}\Big|_{t_{0}} \delta t+O(\delta t^2) $$ using Hamilton's equation we could say $$ q(t)=q(t_{0})+\frac{\partial H}{\partial p}\Bigg|_{t_{0}} \delta t+O(\delta t^2) $$ $$p(t)=p(t_{0})-\frac{\partial H}{\partial q}\Bigg|_{t_{0}} \delta t+O(\delta t^2) $$ From the definition of jacobian matrix I get the following matrix $$\left|\begin{array}{cc} 1+\partial_{q}\partial_{p}H|_{t_{0}}\delta t+O(\delta t^2) & \partial^2 _{p}H|_{t_{0}} \delta t+O(\delta t^2)\\ -\partial^2 _{q}H|_{t_{0}} \delta t+O(\delta t^2) & 1-\partial_{p}\partial_{q}H|_{t_{0}}\delta t + O(\delta t^2) \end{array}\right|$$ Is there any other way of intuitively understanding this?

Answer 1

In view of your expansion, the temporal derivative of the Jacobian determinant, if evaluated exactly at $t=t_0$, is $$\frac{\partial J}{\partial t}|_{t_0} =\lim_{\delta t \to 0 }\frac{J(t_0+ \delta t) - J(t_0)}{\delta t}= \lim_{\delta t \to 0 }\frac{\left(1 + O(\delta t^2)\right)-1}{\delta t} =0$$ where I used Schwarz' theorem on second derivatives in computing the determinant of the Jacobian matrix you expanded, so that terms of order $\delta t$ cancel (I am assuming that the Hamiltonian function is $C^2$). Next, observe that $$J(t) = J(t_0) J(t-t_0)$$ and also $$J(t+\delta t) = J(t_0+ \delta t) J(t-t_0),$$ so that, the derivative at a generic value of time $t$ is $$\frac{\partial J}{\partial t} =\lim_{\delta t \to 0 }\frac{J(t_0+ \delta t) - J(t_0)}{\delta t} J(t-t_0) = \frac{\partial J}{\partial t}|_{t_0}J(t-t_0)= 0 J(t-t_0)=0.$$ Therefore, $J$ is constant in time for fixed canonical coordinates. As for $t=t_0$, again due to your expansion, $J(t_0)= 1$, $J(t)$ takes that value everywhere and for all times.