eikonal equation - 程函方程

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2018/08/08

eikonal equation如下：$$|\nabla_x \tau (x)| = n(x).$$

定义Hamiltonian：$H(p,x) = \tfrac 1 2 n^{-2}(x)|p|^2 - \tfrac 1 2$，于是可得$$0 = \textrm{d}H = \sum_j \frac{\partial H}{\partial p_j} \textrm{d}p_j + \sum_j \frac{\partial H}{\partial x_j} \textrm{d}x_j.$$ 若我们参数化x和p，令$\frac{\textrm{d}x_j(t)}{\textrm{d}t} = \frac{\partial H}{\partial p_j},\quad \frac{\textrm{d}p_j(t)}{\textrm{d}t} = - \sum_j \frac{\partial H}{\partial x_j}$，则此时$x(t)$和$p(t)$满足$\textrm{d}H(p(t),x(t)) = 0$。若我们同时再要求$x(t)$与$p(t)$满足$H(p(t),x(t)) = 0$，则我们得到了原eikonal equation的characteristics。令$\tau(t)$满足$\frac{\textrm{d}\tau(t)}{\textrm{d}t} = \sum_j p_j \frac{\partial H}{\partial p_j} = 1.$

ODE的characteristics的性质，可参见Evans的Partial Differential Equations (v2)的section 3.2。