Hamilton-Jacobi equation §

Is a general first order PDE of the form

$$G(Du,u_t,u,x,t)=u_t+H(Du,x)=0$$

call $y=(\mathbf{x},t)$ a $n+1$ vector, and $\mathbf{p}=(u_{x_1},\dots ,u_{x_n},u_t)$ the equation and $z=u(x,t)$ (this notation is used in the method of characteristic) becomes

$$G(p,z,y)=0$$

so

$$D_{p}G=(D_{p}H,1)$$ $$D_{y}G=(D_{x}H,0)$$ $$G_z=0$$

so that the characteristic equations are

$$\{\begin{array}{l}\dot{\mathbf{p}}(s)=-(D_{x}H,0)\\ \dot{z}(s)=\mathbf{p}(s)\cdot (D_{p}H,1)\\ \dot{\mathbf{y}}(s)=(D_{p}H,1)\end{array}$$

these equation for the time component means that we can idenitify the parameter $s$ with the time $t$. For the other coordinates

$$\{\begin{array}{l}\dot{\mathbf{p}}(s)=-D_{x}H\\ \dot{\mathbf{x}}(s)=D_p\end{array}$$

which are the Hamilton equations. We can ignore $z(s)$ since it doesn’t appear in $H$.