Solution to Hamilton-Jacobi differential equations 
Let $H(x,y)$ be a $C^2$ function on $\mathbb{R}^2$ and let $(x(t),y(t))$ be a solution of the Hamilton-Jacobi equations $$\frac{dx}{dt}=\frac{\partial}{\partial y}H(x(t),y(t))$$$$\frac{dy}{dt}=-\frac{\partial}{\partial x}H(x(t),y(t))$$
Show that the function $H$ is constant along the integral curves of the equations above.

I'm not so sure what "$H$ constant along the integral curves" means. Does it mean $H(x(t),y(t))$ is a constant value for any $t$? If so, how to proceed in proving that?
 A: By integral curve they mean a solution to the system.
Suppose $t \mapsto (x(t),y(t))$ is a solution, then let $\phi(t) = H(x(t),y(t))$.
Then
\begin{eqnarray}
\dot{\phi}(t) &=& \frac{\partial H(x(t),y(t))}{\partial x} \dot{x}(t) + \frac{\partial H(x(t),y(t))}{\partial y} \dot{y}(t) \\
&=& (-\dot{y}(t)) \dot{x}(t) + (\dot{x}(t)) \dot{y}(t) \\
&=& 0
\end{eqnarray}
It follows that $\phi$ is constant.
A: $$
\dot{H} = H_{x}\dot{x} + H_{y}\dot{y} = H_{x}H_{y} + H_{y}\left(-H_{x}\right) = 0
$$
A: The equations in the question are Hamilton's equations; the Hamilton-Jacobi equation is related, though different, see this Widipedia page.  In any event, we have
$\frac{dH}{dt} = \frac{\partial H}{\partial x} \dot x + \frac{\partial H}{\partial y} \dot y = \frac{\partial H}{\partial x}\frac{\partial H}{\partial y} + \frac{\partial H}{\partial y} (-\frac{\partial H}{\partial x}) = 0, \tag{1}$
which from an elementary perspective says the the integral curves $(x(t), y(t))$ of
$\frac{dx}{dt}=\frac{\partial}{\partial y}H(x(t),y(t)), \tag{2}$
$\frac{dy}{dt}=-\frac{\partial}{\partial x}H(x(t),y(t)), \tag{3}$
run orthogonal to $\nabla H$, so they lie in surfaces of constant $H$.  A more advanced treatment requires the use of symplectic forms and symplectic mechanics, information about which is easily found via google, widipedia, etc.
Hope this helps.  Cheers,
and as always, 
Fiat Lux!!!
