The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_t+H(t,q,S_q)=0$ in one momentum chart

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U$ and residual forces $d$ of the standing regularity (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). Assume the presentation is hyperregular on its charts, with the Hamiltonians $H$ and $\tilde{H}$ and inverse Legendre branches of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Hyperregular potential presentations and their inverse Legendre branches in the model's charts, Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian. The momentum charts, their forms and their canonical coordinates are those of The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart and Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $\omega ={\sum }_{a=1}^{m}d{\theta }^a\wedge d{\theta }^{m+a}$ with $m$ the chart's half-dimension, and the model's own momentum charts; dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) The chart datum. Fix one momentum chart of the presentation: either the Cartesian chart with the coordinates $(t,x,p)$ on $I_0\times Q\times {\mathbb{R}}^{3N}$ and the Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions clause (a), or the chart of a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r$ coordinates, the coordinates $(t,q,p)$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ and the Hamiltonian $\tilde{H}$ of clause (b). Write $m\in \{3N,r\}$ for the number of momentum pairs, let $V\subseteq {\mathbb{R}}^m$ denote the configuration-coordinate domain of the chart ($Q$ in the Cartesian case, $\mathrm{\Omega }$ in the other), and write $H$ for the corresponding Hamiltonian,

$H:I_0\times V\times {\mathbb{R}}^m⟶\mathbb{R},(t,q,p)⟼H(t,q,p),$

a function of class $C^2$ and dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (energy) by Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian. Let $J^{\mathrm{\prime }}\subseteq I_0$, $U^{\mathrm{\prime }}\subseteq V$ and $A^{\mathrm{\prime }}\subseteq {\mathbb{R}}^m$ 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, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

(b) Solutions of the equation. A map

$S:J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}⟶\mathbb{R},(t,q,\alpha )⟼S(t,q,\alpha ),$

of class $C^3$ solves the Hamilton-Jacobi equation associated with $H$ on $J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}$ when

$S_t(t,q,\alpha )+H(t,q,S_q(t,q,\alpha ))=0((t,q,\alpha )\in J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}),$ where $S_t$ is the partial derivative in $t$ and $S_q=(S_{q^1},\dots ,S_{q^m})$ is the gradient in $q$ (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); the momentum slot of $H$ is substituted by $p=S_q(t,q,\alpha )$. This is exactly the mathematical Hamilton-Jacobi equation of The Hamilton-Jacobi equation and a complete integral of it associated with the function $H$ on the open set $J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times {\mathbb{R}}^m$, read in the chart coordinates $(t,q,p)$; the two definitions determine the same class of functions.

(c) Dimensions and chart consistency. The equation forces $[S]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (action): $S_t$ has dimension $[S]{\tau }^{-1}$ and $H$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, while each $S_{q^a}$ has dimension $[S]\mathrm{/}{\gamma }_a$ with ${\gamma }_a$ the declared dimension of the coordinate $q^a$ in this chart; the substitution is dimensionally consistent in every chart because the chart momenta of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta have dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}{\gamma }_a^{-1}$. In the Cartesian chart all ${\gamma }_a=\mathrm{\ell }$ and $[S]\mathrm{/}\mathrm{\ell }=\mu \mathrm{\ell }{\tau }^{-1}$, the momentum dimension.

(d) Status. The equation is attached to the declared datum: replacing the chart or the potential representative changes $H$ by the rules recorded in clause (d) of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions and hence changes the equation, and a solution family $S$ is a solution for that datum only. On each fixed hyperregular chart the inverse Legendre map is unique; it is not an independent branch choice. The item defines a class of functions; whether a given datum admits solutions, and how solutions produce curves of the model, is not asserted here. In particular no existence theorem, no global solution, no equation of motion for the model, no conservation law and no measured value is claimed.

• Why the parameters are left free. The third variable $\alpha$ ranges over an open subset of the chart-coordinate space and is not constrained by this definition. A complete integral additionally requires the nondegeneracy of the mixed derivative $S_{q\alpha }$; that requirement and the physical reading of the parameters are the subject of A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants.

• Time-dependent Hamiltonians are included. The equation as written allows ${\mathrm{\partial }}_{t}U\equiv ̸0$ and $d\equiv ̸0$ in the branch Hamiltonian only through the declared potential presentation: the residual forces of $d$ are not part of $H$ unless they are presented by a potential (Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian; the exact criterion is clause 4 of the companion page's Hamiltonian-field item pthm-cm13-hamiltonian-field-integrates-the-model-equations).