$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ ###[Eikonal and Hamilton-Jacobi equations](id:sect-5.2) > 1. [Eikonal and Hamilton-Jacobi equations](#sect-5.2.1) > 2. [Solving non-linear first order PDEs](#sect-5.2.2) #### [Eikonal and Hamilton-Jacobi equations](id:sect-5.2.1) Plugging into [(5.1.4)](./L5.1.html#mjx-eqn-eq-5.1.4) solution $u=e^{ih^{-1}S(x)}A(x)$, using [(5.1.6)](./L5.1.html#mjx-eqn-eq-5.1.6) 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](id:example-5.2.1).** In particular for wave equation [(5.1.1)](./L5.1.html#mjx-eqn-eq-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](id:example-5.2.2).** In particular for Schrödinger equation [(5.1.3)](./L5.1.html#mjx-eqn-eq-5.1.3) 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](http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.3.html#mjx-eqn-eq-10.3.8). In classical mechanics function $S$ is called *action.* #### [Solving non-linear first order PDEs](id:sect-5.2.2) Solution of such PDEs (with initial data $S|\_{t=0}=S\_0(x)$) is descrbed in [Subsection 2.2.2 from PDE Textbook](http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.2.html#sect-2.2.2). See [equations (2.2.11)--(2.2.13) from PDE Textbook](http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.2.html#mjx-eqn-eq-2.2.11): \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](id:example-5.2.3).** In particular for *eikonal equation* after we rewrite it as \begin{equation} S\_t + c(x)|\nabla S| \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} **[Example 4](id:example-5.2.4).** 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](id:def-5.2.1).** (\ref{eq-5.2.12})--(\ref{eq-5.2.14}) define a *Hamiltonian flow* $\Psi\_t$, $\Psi\_0=I$. **[Theorem 1](id:thm-5.2.1).** 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 $x$. 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. 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 $S(x,t)=S(\pi\_x^{-1}(x),t)$. 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](id:def-5.2.2).** a. This surface $\Lambda\_t$ we call a *Lagrangian manifold*. b. Points where $\pi\_x$ is a local diffeomorphism we call *regular points* and all other points we call *singular points*. **[Remark 1](id:rem-5.2.1).** a. Under reasonable assumptions $\Lambda\_t$ is defined globally, for all $t\ge 0$ [^1]. b. 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. c. Still globally defined $\Lambda\_t$ will be of prime significance for construction of asymptotics. [^1]: We are interested only in $t\ge 0$ ________ [$\Leftarrow$](./L5.1.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./L5.3.html)