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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.

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  • $\begingroup$ Of course, $\int_{t}^{t+\tau} \dot{q}\; dt = q(t+\tau) - q(t)$, and similarly for $p$. $\endgroup$ – Robert Israel Jul 28 '17 at 2:53
  • $\begingroup$ Yes @RobertIsrael, but what if you do not directly have access to q, but you have access to $H$? And so you need to use that somehow when getting the derivative of $H$ w.r.t $p$? $\endgroup$ – nundo Jul 28 '17 at 3:02
  • $\begingroup$ @RobertIsrael any comment or any help please? $\endgroup$ – nundo Aug 2 '17 at 14:40

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