A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants

Adopt the datum of 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: one momentum chart of a hyperregular potential presentation, its $C^2$ Hamiltonian $H$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ on $I_0\times V\times {\mathbb{R}}^m$, open sets $J^{\mathrm{\prime }}\subseteq I_0$, $U^{\mathrm{\prime }}\subseteq V$ and $A^{\mathrm{\prime }}\subseteq {\mathbb{R}}^m$, and a $C^3$ map $S:J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}\to \mathbb{R}$ that solves the Hamilton-Jacobi equation associated with $H$ on $J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}$. 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) Complete integral. Let $(t_0,q_0,{\alpha }_0)\in J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times A^{\mathrm{\prime }}$. The solution $S$ is a complete integral of the model's Hamilton-Jacobi equation at $(t_0,q_0,{\alpha }_0)$ when the mixed derivative

$S_{q\alpha }(t,q,\alpha ):=(\frac{{\mathrm{\partial }}^{2}S}{\mathrm{\partial }{q}^a\mathrm{\partial }{\alpha }_j}(t,q,\alpha ){)}_{a,j=1}^m\in {\mathbb{R}}^{m\times m}$ is invertible at the base point, $\det S_{q\alpha }(t_0,q_0,{\alpha }_0)\mathrm{\ne }0$ (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix, Invertible Euclidean linear maps, mixed partials being those of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder). This is exactly the mathematical notion of a complete integral of The Hamilton-Jacobi equation and a complete integral of it applied to the function $H$ on the chart domain $J^{\mathrm{\prime }}\times U^{\mathrm{\prime }}\times {\mathbb{R}}^m$.

(b) Parameters and their dimensions. The third variable $\alpha =({\alpha }_1,\dots ,{\alpha }_m)$ is read with the declared coordinate dimensions ${\gamma }_a$ of $q$ in the fixed chart, $[{\alpha }_j]={\gamma }_j$. With this reading the entries of $S_{q\alpha }$ have dimensions $\mu {\mathrm{\ell }}^2{\tau }^{-1}\mathrm{/}({\gamma }_a{\gamma }_j)$. In Cartesian length coordinates these are mass/time dimensions. Changing units rescales rows and columns by nonzero factors, so invertibility is unit-independent; the momentum map below carries the momentum dimension.

(c) Momentum map and Jacobi constants. For a complete integral $S$ the momentum map and the Jacobi constants of $S$ are the maps

$p(t,q,\alpha ):=S_q(t,q,\alpha ),\beta (t,q,\alpha ):=S_{\alpha }(t,q,\alpha ),$ so that $p$ has the chart-momentum dimensions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, $[p_a]=\mu {\mathrm{\ell }}^2{\tau }^{-1}{\gamma }_a^{-1}$, and $\beta$ has the same dimensions as $p$, $[{\beta }_j]=\mu {\mathrm{\ell }}^2{\tau }^{-1}{\gamma }_j^{-1}$: the product ${\beta }_j{\alpha }_j$ has the action dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}$ of $S$. The pair $(\beta ,\alpha )$ is the Jacobi data of the complete integral at the base point.

(d) Status. The item defines data attached to the presentation and to the chosen solution. It asserts neither that a complete integral exists for a given presentation, nor that its characteristic curves are motions of the model, nor that $\beta$ is constant along such curves; those are the claims of the page's characteristics theorem. It also asserts no global statement: $\det S_{q\alpha }\mathrm{\ne }0$ is required at the base point only, and the theorem below records its local consequences.

• The mixed derivative is the local-coordinate hypothesis. The condition $\det S_{q\alpha }(t_0,q_0,{\alpha }_0)\mathrm{\ne }0$ gives the local inverse function theorem conclusions used by the characteristics theorem: the momentum equation $p=S_q$ can be solved locally for $\alpha$, and the constants equation $\beta =S_{\alpha }$ can be solved locally for $q$. If the determinant vanishes, those local-coordinate conclusions are not supplied by that theorem; a characteristic projection may develop a caustic. The companion page verifies this degeneration for the harmonic model at its conjugate times, where the displayed $S_{x\alpha }$ is unbounded and the label-to-position derivative vanishes. Its Kepler example separately treats simple turning points of a radial action integral.

• Finite-dimensional reading. No topology of the parameter set beyond the open subset $A^{\mathrm{\prime }}\subseteq {\mathbb{R}}^m$ is used, and no choice principle is invoked by this definition: $m$ is a finite number fixed by the chart.