I have to admit that I am a little bit embarrassed to ask: How do I put the Hamilton-Jacobi equations into integral form?
the HJ dynamics are:
$\dot q= \frac{dH}{dp}$
$\dot p= -\frac{dH}{dq}$
And I want to get these in integral form in order to avoid issues with non-differentiability.
Here is what I tried:
$$\int_t^{t+\tau} \dot q dt = \int_t^{t+\tau}\frac{dH}{dp}$$ $$\int_t^{t+\tau} \dot p dt = -\int_t^{t+\tau}\frac{dH}{dq}$$
Obviously, $H$ is a function of $q, and p$ and $q$ and $p$ are functions of time.
Are these the correct integral equations? If not, can someone please help me to determine what the correct integral form is for these? Thank you.