Hamilton–Jacobi and integrability

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).

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.

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 momentum charts and two-form datum 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, the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian, the bracket of The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart and the branch Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, writing $H$ for the Hamiltonian of the chosen chart ($\tilde{H}$ in generalized coordinates). The model reading is Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential: when $d\equiv 0$, Cartesian solutions are model motions; generalized-coordinate solutions satisfy the substituted Lagrange equations. An arbitrary lower-dimensional immersion alone does not identify those solutions with ambient unconstrained motions. 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) Phase functions and the domain. Let $W_0$ be a nonempty open subset of one momentum phase domain of the presentation. A phase function of the presentation on $W_0$ of class $C^k$ is a map $f:W_0\to \mathbb{R}$ of class $C^k$, of a declared dimension.

(b) First integrals. Let $f:W_0\to \mathbb{R}$ be of class $C^1$. Then $f$ is a first integral of the branch Hamiltonian $H$ on $W_0$ when for every interval $J\subseteq I_0$ with nonempty interior and every solution $z:J\to W_0$ of the Hamiltonian system $\dot{z}(t)=X_{H_t}(z(t))$ (First-order systems, initial value problems, and solutions on intervals) the composite $t\mapsto f(z(t))$ is constant on $J$. In the Cartesian chart with $d\equiv 0$, this is equivalently constancy along every model motion whose momentum-phase curve lies in $W_0$, by clause 1 of Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential. In generalized coordinates with $d\equiv 0$, the same definition concerns solutions of the substituted Lagrange equations, by its clause 2.

(c) The bracket criterion. Let $H$ be autonomous on $I_0\times W_0$ (${\mathrm{\partial }}_{t}U\equiv 0$ and $d\equiv 0$) and let $f:W_0\to \mathbb{R}$ be autonomous of class $C^2$. Then $f$ is a first integral of $H$ on $W_0$ if and only if

$\{f,H\}=0\text{everywhere on }{W}_0,$

the bracket being that of The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart. For sufficiency, clause 2 of Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals makes $f\circ z$ constant on every solution when the bracket vanishes. For necessity, fix $w\in W_0$ and a time $t_{\ast }$ in the interior of $I_0$. The chart field $X_H=(H_p,-H_q)$ is $C^1$, hence locally Lipschitz in state, since the chart Hamiltonian is $C^2$. Applying Picard-Lindelöf local existence and uniqueness for first-order systems on the open domain $\mathrm{int}(I_0)\times W_0$ gives a local Hamiltonian solution $z$ through $z(t_{\ast })=w$. If $f$ is a first integral, differentiating the constant composite at $t_{\ast }$ and using clause 1 of Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals gives $0=(f\circ z{)}^{\mathrm{\prime }}(t_{\ast })=\{f,H\}(w)$. This holds for every $w$.

(d) Involution. Phase functions $f_1,\dots ,f_k:W_0\to \mathbb{R}$ of class $C^2$ are in involution (or pairwise Poisson-commuting) when

$\{f_i,f_j\}=0\text{on }{W}_0\text{ for all }i,j\in \{1,\dots ,k\};$

a family of first integrals in involution is a family whose members are first integrals of $H$ and which is in involution. The notion is the presentation reading of the root-library notion of First integral and Poisson-commuting functions under the bracket of The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart, which is the bracket of Poisson bracket on a symplectic manifold for the two-form 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.

(e) Differential independence. Phase functions $f_1,\dots ,f_k$ of class $C^1$ are independent at $z\in W_0$ when the differentials $d{f}_1(z),\dots ,d{f}_k(z)$ are linearly independent as elements of the dual of the tangent space at $z$ (Linear independence: a finite list $v:n\to V$ is independent when ${\sum }_{i<n}{\lambda }_i{v}_i=0_V$ forces every ${\lambda }_i=0_F$, and a subset $S\subseteq V$ is independent when every injective finite list into $S$ is independent); the set of points at which this holds is the independence locus of the family, $\{z\in W_0:\mathrm{rank}(d{f}_1(z),\dots ,d{f}_k(z))=k\}$ (The rank of a derivative and constant-rank Euclidean maps, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). The family is independent on $W_0$ when it is independent at every $z\in W_0$.

(f) Status. The item defines model properties of phase functions and establishes the bracket equivalence using observable evolution and local existence through every phase point. It asserts no existence of a nonconstant first integral for a given force law, no completeness of any flow, no conservation law of the model and no measured value; in particular every closure or constancy statement about brackets is a theorem of this page or of Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals, not part of the definition.

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 momentum charts and two-form datum 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, the canonical coordinates of 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, the bracket of The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart, the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian and the Hamiltonian of the chosen chart of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions: $H$ in Cartesian coordinates or $\tilde{H}$ in generalized coordinates. Assume this chart Hamiltonian is autonomous and write it throughout this definition as $H:W\to \mathbb{R}$, on that same chart's phase domain $W$; by clauses 1–2 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ (clause 1 for the Cartesian domain and clause 2 for a generalized-coordinate domain), each momentum phase domain is a symplectic manifold for $\omega$, with $\omega$ the standard two-form in the chart's canonical coordinates (Symplectic form and symplectic manifold). 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 datum. Let $W_0$ be a nonempty open subset of one momentum phase domain of the presentation, of dimension $2n$, and let

$F=(F_1,\dots ,F_n),F_i:W_0⟶\mathbb{R},$

be phase functions of class $C^{\mathrm{\infty }}$ with $F_1=H{\mathrm{\mid }}_{W_0}$. The family $F$ is a completely integrable datum of the presentation on $W_0$ when

1. $F$ is in involution, $\{F_i,F_j\}=0$ on $W_0$ for all $i,j$ (First integrals of the model's Hamiltonian and families in involution: the bracket criterion and differential independence clause (d)); and
2. the differentials $d{F}_1,\dots ,d{F}_n$ are linearly independent on a dense open subset of $W_0$ (Linear independence: a finite list $v:n\to V$ is independent when ${\sum }_{i<n}{\lambda }_i{v}_i=0_V$ forces every ${\lambda }_i=0_F$, and a subset $S\subseteq V$ is independent when every injective finite list into $S$ is independent, The rank of a derivative and constant-rank Euclidean maps).

(b) Reading in the root library. Since $(W_0,\omega )$ is a $2n$-dimensional symplectic manifold and the bracket of The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart is the bracket of Poisson bracket on a symplectic manifold for $\omega$, a datum $F$ in the sense of clause (a), with $F_1=H$ and the autonomous Hamiltonian $H$ of the presentation, is exactly a completely integrable Hamiltonian system on $(W_0,\omega )$ in the sense of Completely integrable Hamiltonian system, with branches read through the presentation's Hamiltonians; the root notion of First integral and Poisson-commuting functions is read through the bracket criterion of First integrals of the model's Hamiltonian and families in involution: the bracket criterion and differential independence clause (c), whose sufficiency is Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals clause 2.

(c) Integral map, regular locus and fibres. Attached to a datum $F$ are:

1. the integral map $F:W_0\to {\mathbb{R}}^n$;
2. the regular locus $W_{\mathrm{r}\mathrm{e}\mathrm{g}}:=\{z\in W_0:\mathrm{rank}dF(z)=n\}$, which is open by continuity of the partial derivatives and dense in $W_0$ by clause (a) 2;
3. the regular values, namely the values $b\in F(W_0)$ such that $\mathrm{rank}dF(z)=n$ at every $z\in F^{-1}(b)$; for such a value the regular fibre $N_b:=F^{-1}(b)$ is the common level set of $F_1,\dots ,F_n$ through $b$;
4. the Hamiltonian fields $X_{F_1},\dots ,X_{F_n}$ of the family (The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian).

(d) Status. The item defines a datum class and its attached objects. It makes no assertion that a particular model is completely integrable, that a fibre is compact, connected or a torus, that action-angle coordinates exist, or that a measured value is known; the structural and motion statements are the torus theorem and the action-angle theorem of this page (items pthm-cm14-regular-tori and pthm-cm14-action-angle), conditional on the hypotheses stated there.

• Involution and independence are separate requirements. Poisson commuting alone does not give coordinates: Poisson-commuting phase functions with dependent differentials fail to form a datum of clause (a) and do not yield Liouville-Arnold coordinates; the companion page exhibits the failure in the model's own phase space.

• Regularity of the datum. The class $C^{\mathrm{\infty }}$ is part of the datum: the branch Hamiltonian of a potential presentation with a $C^2$ potential is only $C^2$ in general, so a completely integrable datum carrying $F_1=H$ requires a smoother presentation (for example $U$ and the chart data of class $C^{\mathrm{\infty }}$); merely smoothing other integrals cannot replace the required equality $F_1=H$. This is a hypothesis on the datum, not a consequence of the model's regularity.

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), the presentation being hyperregular on its 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 Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions). Fix one momentum chart of the presentation (Cartesian coordinates $(t,x,p)$, or the chart $(t,q,p)$ of a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces)), with $m$ pairs of coordinates and Hamiltonian $H$ of class $C^2$ on $J_0\times V\times {\mathbb{R}}^m$; let $S:J\times U\times A\to \mathbb{R}$ be a $C^3$ complete integral of the model's Hamilton-Jacobi equation at $(t_0,q_0,{\alpha }_0)\in J\times U\times A$ in the sense 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 and A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants ($J\subseteq J_0$, $U\subseteq V$, $A\subseteq {\mathbb{R}}^m$ open), and put

${\beta }_0:=S_{\alpha }(t_0,q_0,{\alpha }_0)\mathrm{.}$

Then:

1. (characteristic curves) For every $\alpha$ sufficiently close to ${\alpha }_0$ there is an open interval $I_{\alpha }\ni t_0$ and a $C^1$ curve $q:I_{\alpha }\to U$ with $q(t_0)=q_0$ solving $q^{\mathrm{\prime }}(t)=H_p(t,q(t),S_q(t,q(t),\alpha ))$, and with $p(t):=S_q(t,q(t),\alpha )$ the pair $(q,p)$ satisfies Hamilton's equations $q^{\mathrm{\prime }}=H_p$, $p^{\mathrm{\prime }}=-H_q$ on $I_{\alpha }$. If the chart is the Cartesian chart and $d\equiv 0$ on the state domain, then $x:=q$ is a motion of the model on $I_{\alpha }$ and $p(t)=\mathsf{M}{x}^{\mathrm{\prime }}(t)$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta); if the chart is the chart of a generalized-coordinate map and $d\equiv 0$, then $q$ solves the declared Lagrange equations of the substituted Lagrangian and $p(t)=G(q(t))q^{\mathrm{\prime }}(t)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart).
2. (Jacobi constants) With $(q,p)$ as in clause 1, $\beta (t):=S_{\alpha }(t,q(t),\alpha )$ is constant on $I_{\alpha }$ and equal to $S_{\alpha }(t_0,q_0,\alpha )$.
3. (local converse) For $(\alpha ,\beta )$ sufficiently close to $({\alpha }_0,{\beta }_0)$ the equation $\beta =S_{\alpha }(t,q,\alpha )$ can be solved locally for $q=Q(t,\alpha ,\beta )$, and with $p:=S_q(t,Q(t,\alpha ,\beta ),\alpha )$ the curve $t\mapsto (Q,p)$ solves Hamilton's equations; in the Cartesian chart on the potential branch it is the momentum-phase curve of a motion of the model.
4. (canonical labels) On the domain of the local diffeomorphism $(q,p)\mapsto (\beta ,\alpha )$ obtained by composing $(q,p)\mapsto (q,\alpha )$ (solving $p=S_q$) with $(q,\alpha )\mapsto (\beta ,\alpha )$, one has $\omega =d\beta \wedge d\alpha$ at every fixed $t$, where $\omega$ is the two-form of the presentation in this chart (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); equivalently, $(\beta ,\alpha )$ are canonical coordinates of the presentation in the sense of clause (a) of 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.

• Locality is exactly the content of the nondegeneracy hypothesis. The characteristic curve in clause 1 exists for parameters $\alpha$ near ${\alpha }_0$ and on an interval $I_{\alpha }\ni t_0$ supplied by the local theory; clause 3 is local near $(t_0,{\alpha }_0,{\beta }_0)$. No solution is extended across a conjugate time or a caustic, and no global statement about the family of characteristics is made.

• The constants carry momentum dimensions. With the reading of clause (b) of A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants, the Jacobi constants $\beta =S_{\alpha }$ have the momentum-type dimensions of the chart's canonical momenta, and clause 4 says that $(\beta ,\alpha )$ is a canonical chart for the presentation's two-form; this is the precise content of the textbook statement that the integration constants of a complete integral provide $2m$ constants of the motion in canonical pairs.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). 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 momentum charts and two-form datum 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, the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian and the branch Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions; by clauses 1–2 of The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$ (clause 1 for Cartesian domains and clause 2 for generalized-coordinate domains), each momentum phase domain is a symplectic manifold for the presentation's two-form $\omega$ (Symplectic form and symplectic manifold). Let $W_0\subseteq W$ be a nonempty open phase domain of dimension $2n$ and let $F=(F_1=H{\mathrm{\mid }}_{W_0},\dots ,F_n)$ be a completely integrable datum of the presentation on $W_0$ (Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_1=H$, pairwise in involution and independent on the regular locus). Let $b_0$ be a regular value of the integral map whose regular fibre $N:=F^{-1}(b_0)\subseteq W_0$ is nonempty, compact and connected, and assume there are a saturated open neighbourhood $U\subseteq W_0$ of $N$ and a ball $B\subseteq {\mathbb{R}}^n$ of regular values with $F(U)=B$ such that $\pi :=F{\mathrm{\mid }}_U:U\to B$ is a proper submersion with connected fibres. Then:

1. (Lagrangian fibre) $N$ is a compact connected embedded $n$-dimensional Lagrangian submanifold of $(W_0,\omega )$, and for every $z\in N$ $T_{z}N=\mathrm{span}\{X_{F_1}(z),\dots ,X_{F_n}(z)\},$ the fields being those of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian.
2. (commuting flows and period lattice) The fields $X_{F_1},\dots ,X_{F_n}$ have pairwise commuting flows; these flows combine into a local ${\mathbb{R}}^n$ action on $W_{\mathrm{r}\mathrm{e}\mathrm{g}}$, and on the compact invariant fibre $N$ their restrictions are complete and generate a transitive action of ${\mathbb{R}}^n$ whose stabilizer at every point is one and the same full lattice $\mathrm{\Gamma }\subseteq {\mathbb{R}}^n$, with $N\cong {\mathbb{R}}^n\mathrm{/}\mathrm{\Gamma }$.
3. (torus) $N$ is diffeomorphic to the torus $T^n={\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$.

The statement is conditional on the adopted force branch, the declared presentation, the smooth completely integrable datum and the listed hypotheses on the fibre; it asserts no global product structure of $U$, no action-angle coordinates, no completeness of any flow outside the compact fibre and no empirical content.