I have the following example of the Hamilton-Jacobi Equation (and first some notations):
$$ \partial_t u+H(x, \nabla u(t, x))=0 $$
The associated Hamiltonian equations are
$$ \dot{\gamma}=H_p(\gamma, p)$$ $$\dot{z}=p \cdot H_p-H $$
$$\dot{p}=-H_x(\gamma, p) $$ where $z(t) = u(x, \gamma(t))$, $p(t) = \nabla u(t, \gamma(t))$
The example uses the Hamiltonian $H(x, p)=\frac{1}{2} |p|^2$
I have two questions regarding the interpretation of the solution and this might help me understand the idea behind the method of characteristics in general:
- The notes tell me that "the characteristic emerging from some $\bar{x}$ at time $t=0$ has constant velocity $p(0)=$ $\nabla u(0 ; \bar{x})$. Consequently, unless $u$ is linear in $x$, there must exist crossing characteristics.
What I need clarification is the consequently part: I am completely missing the picture of why non-linearity in $x$ of $u$ implies that characteristics need to cross. Why is that a consequence of the constant velocities?
- I also need clarification on the idea of using characteristics. I think by considering solutions $z$ (and gradient of the solution $p$) only along the curve $\gamma(t)$ for which we can sometimes solve, we are able to recover parts of the general solution $u(x, t)$. But how do I see this recovery at work? Explicitly, if we take this same example, I will have the solutions for $\gamma, z, p$ as
$$p(t) = p(0), \gamma(t) = \gamma(0) + tp(0), z(t) = z(0) + \frac{1}{2}|p(0)|^2$$.
But how do I interpret this? What does this tell me about a general solution $u$?