Hamilton-Jacobi equations

Question

Consider the Hamilton-Jacobi equation:

Using the method of Characteristics, one can define a curve $x(t)$ along which the PDE can be transformed into an ODE. Let

so that

Differentiating the equation w.r.t. $x_i$, one can get

So let

then

Differentiating $z(t)$ w.r.t. time results in the characteristic system of ODEs

Personally, I find $$\dot z(t)= D_pH \cdot p - H.$$

What am I missing?

PS: D denotes here the gradient

Answer 1

You are right, and the given formula for $\dot z$ is incorrect. Let's check on a simple example, inviscid Burgers: $u_t+uu_x=0$. Here $H=up$, hence $\dot x=D_pH=u$. Along the characteristic $u$ is constant, thus $\dot z\equiv 0$. This is in agreement with your formula: $$\dot z = (D_p H) p-H=up-up=0$$ but not with the formula for $\dot z$ in the quoted source.

You can also compare with lecture notes by Xinwei Yu, where $H$ is not as general, but the key points are the same.