$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$

5.2. Eikonal and Hamilton-Jacobi equations

1. Eikonal and Hamilton-Jacobi equations
2. Solving non-linear first order PDEs

Eikonal and Hamilton-Jacobi equations

Plugging into (5.1.6) solution $u=e^{ih^{-1}S(x)}A(x)$, using (5.1.8) and ignoring all terms with positive pover of $h$ we arrive to equation \begin{equation} P_0(x,\nabla S(x))=0. \label{eq-5.2.1} \end{equation}

Example 1. In particular for wave equation (5.1.1) we get eikonal equation \begin{equation} S_t^2-c(x)^2|\nabla S|^2=0 \label{eq-5.2.2} \end{equation} (here we take off $t$ from $x=(x_1,\ldots,x_d)$, so $S=S(x,t)$ and $\nabla= (\partial_1,\ldots,\partial_d)$. In optics phase $S$ is called eikonal.

Example 2. In particular for Schrödinger equation (5.1.2) we get Hamilton-Jacobi equation from classical mechanics \begin{equation} S_t + H(x,\nabla S)=0 \label{eq-5.2.3} \end{equation} where \begin{equation} H(x,p)=\frac{1}{2}p^2+V(x). \label{eq-5.2.4} \end{equation} (here we take off $t$ from $x=(x_1,\ldots,x_d)$, so $S=S(x,t)$ and $\nabla= (\partial_1,\ldots,\partial_d)$. See equation (10.3.8) from PDE Textbook. In classical mechanics function $S$ is called action.

Solving non-linear first order PDEs

Solution of such PDEs (with initial data $S|_{t=0}=S_0(x)$) is descrbed in Subsection 2.2.2 from PDE Textbook. See equations (2.2.11)--(2.2.13) from PDE Textbook: \begin{align} &\frac{dx_j}{d\tau}= P_0^{(j)}(x,p), \label{eq-5.2.5}\\ &\frac{dp_j}{d\tau}= -P_{0(j)}(x,p), \label{eq-5.2.6}\\ &\frac{dS}{d\tau}=\sum_j p_j P_0^{(j)}(x,p)- P_0(x,p). \label{eq-5.2.7} \end{align}

Example 3. In particular for eikonal equation after we rewrite it as \begin{equation} S_t + c(x)|\nabla S|=0 \label{eq-5.2.8} \end{equation} (for opposite sign we just reverse time $t\mapsto -t$) we get \begin{align} &\frac{dx_j}{dt}= c(x)p_j/|p|, \label{eq-5.2.9}\\ &\frac{dp_j}{dt}=- c_{(j)}|p|, \label{eq-5.2.10}\\ &dS=0. \label{eq-5.2.11} \end{align} Especially interesting case $c=\const$.

Example 4. In particular for Hamilton-Jacobi equation we get \begin{align} &\frac{dx_j}{dt}= H^{(j)}(x,p), \label{eq-5.2.12}\\ &\frac{dp_j}{dt}=- H_{(j)}(x,p), \label{eq-5.2.13}\\ &dS=\sum_j p_j \frac{dx_j}{dt} -H. \label{eq-5.2.14} \end{align} Observe that if we express $p$ via $x$ and $\frac{dx}{dt}$ from (\ref{eq-5.2.12}) and plug into the right-hand of (\ref{eq-5.2.14}) we get a Lagrangian $L(x,\frac{dx}{dt}, t)$.

Definition 1. (\ref{eq-5.2.12})--(\ref{eq-5.2.14}) define a Hamiltonian flow $\Psi_t$, $\Psi_0=I$.

Theorem 1. Consider $S_0(x)$. At each point $x$ define $p(x)=\nabla S_0(x)$. We get $d$-dimensional surface $\Lambda_0=\{(x,p(x))\}$ in $2d$-dimensional space $\bR^{2d}=T^*\bR^d$ parametrized by $x$.

1. Through each point $\lambda\in \Lambda_0$ let us pass a Hamiltonian curve $\Psi_t(\lambda)$ and also along this define $S(\lambda,t)$ by (\ref{eq-5.2.14}) and $S(\lambda,0)=S_0(x)$. For each $t$ we have a $d$-dimensional surface $\Lambda_t=\Psi_t\Lambda_0$ in $2d$-dimensional space.
2. Assume that in some point $\bar{\lambda} \in \Lambda_t$ projector $\pi_x: \Lambda_t\ni \lambda=(x,p)\to x$ is a local diffeomorphism which means exactly that differential (Jacobi matrix) has rank $d$. Then we can define locally $S(x,t)=S(\pi_x^{-1}(x),t)$.
3. This function satisfies Hamilton-Jacobi equation (\ref{eq-5.2.3}) and also \begin{equation} \partial_{x_j} S=p_j. \label{eq-5.2.15} \end{equation}

Definition 2.

1. This surface $\Lambda_t$ we call a Lagrangian manifold.
2. Points where $\pi_x$ is a local diffeomorphism we call regular points and all other points we call singular points.

Remark 1.

1. Under reasonable assumptions $\Lambda_t$ is defined globally, for all $t\ge 0$ ¹.
2. On the other hand, exists $\tau(x): 0<\tau(x)\le +\infty$ such that $\Psi_t(\lambda)$ is a regular points for all $t:0\le t<\tau (x)$ but for $t=\tau(x)$ we get a singular point. Therefore solution of the Cauchy problem $S(x,0)=S_0(x)$ for Hamilton-Jacobi equation (\ref{eq-5.2.3}) may be defined only locally.
3. Still globally defined $\Lambda_t$ will be of prime significance for construction of asymptotics.
4. In symplectic geometry we consider symplectic spaces, that is $2d$-dimensional manifolds equipped with closed $2$-form $\omega$ of full rank. Then Lagrangian submanifolds are $d$-dimensional manifolds $\Lambda$ such that $\omega|_\Lambda=0$. Canonical example \begin{gather} \omega =\sum_{1\le j\le d} dx_j\wedge d\xi_j=d\Bigl(- \sum \xi_j dx_j\Bigr). \end{gather}

$\Leftarrow$  $\Uparrow$  $\Rightarrow$