The Hamilton-Jacobi equation and a complete integral of it

Let $n\ge 1$, let $J\subseteq \mathbb{R}$ be a nonempty open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $U\subseteq {\mathbb{R}}^n$ be a nonempty open set, and let

$H:J\times U\times {\mathbb{R}}^n⟶\mathbb{R}$

be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms); write $p=(p_1,\dots ,p_n)$ for the third variable, so that $(t,q,p)$ range over phase-type data and $H$ is a Hamiltonian in the sense of The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$. Let $J^{\mathrm{\prime }}\subseteq J$, $U^{\mathrm{\prime }}\subseteq U$ and $A^{\mathrm{\prime }}\subseteq {\mathbb{R}}^n$ be nonempty open sets (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and let $S:J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}\to \mathbb{R}$ be of class $C^3$; write $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ for the third variable.

$S$ solves the Hamilton-Jacobi equation associated with $H$ on $J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}$ if

$S_t(t,q,\alpha )+H(t,q,S_q(t,q,\alpha ))=0((t,q,\alpha )\in J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}),$

where $S_t$ is the partial derivative in $t$, $S_q=(S_{q^1},\dots ,S_{q^n})$ is the gradient in $q$ and $H$ is evaluated at the momentum $p=S_q(t,q,\alpha )$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). $S$ is a complete integral of the Hamilton-Jacobi equation at $(t_0,q_0,{\alpha }_0)$ if in addition

$\det S_{q\alpha }(t_0,q_0,{\alpha }_0)\mathrm{\ne }0,$

where $S_{q\alpha }$ is the $n\times n$ matrix with entries ${\mathrm{\partial }}^{2}S\mathrm{/}\mathrm{\partial }{q}^i\mathrm{\partial }{\alpha }_j$, called the mixed derivative.

If $S$ is a complete integral, the associated momentum map is $p(t,q,\alpha ):=S_q(t,q,\alpha )$ and the associated constants $\beta (t,q,\alpha ):=S_{\alpha }(t,q,\alpha )$ are the Jacobi constants of the complete integral. This is the Hamiltonian case of the general first-order equation $u_t+H(t,q,D_{q}u)=0$: the $n$ parameters $\alpha$ enter essentially through the nonvanishing mixed derivative, and $S_t$ is determined algebraically by $S_q$ on the solutions.