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2018/08/08
eikonal equation如下：$$| abla_x au (x)| = n(x).$$
定义Hamiltonian：$H(p,x) = frac 1 2 n^{-2}(x)|p|^2 - frac 1 2$，于是可得$$0 = extrm{d}H = sum_j frac{partial H}{partial p_j} extrm{d}p_j + sum_j frac{partial H}{partial x_j} extrm{d}x_j.$$ 若我们参数化x和p，令$frac{ extrm{d}x_j(t)}{ extrm{d}t} = frac{partial H}{partial p_j},quad frac{ extrm{d}p_j(t)}{ extrm{d}t} = - sum_j frac{partial H}{partial x_j}$，则此时$x(t)$和$p(t)$满足$ extrm{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$，则我们得到了原eikonal equation的characteristics。令$ au(t)$满足$frac{ extrm{d} au(t)}{ extrm{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。