Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_1=H$, pairwise in involution and independent on the regular locus

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.