First integrals of the model's Hamiltonian and families in involution: the bracket criterion and differential independence

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.