TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical $(β, α)$ labels

Statement

Let $n \geq 1$ , let $J \subseteq R$ be a nonempty open interval, let $U \subseteq R^{n}$ and $A \subseteq R^{n}$ be nonempty open sets, let $H : J \times U \times R^{n} \to R$ be of class $C^{2}$ , and let $S : J \times U \times A \to R$ be a $C^{3}$ complete integral of the Hamilton-Jacobi equation associated with $H$ at $(t_{0}, q_{0}, α_{0}) \in J \times U \times A$ (The Hamilton-Jacobi equation and a complete integral of it); write $β_{0} : = S_{α} (t_{0}, q_{0}, α_{0})$ . Then:

(characteristics) for every $α$ sufficiently close to $α_{0}$ there is an open interval $I_{α} ∋ t_{0}$ and a curve $q : I_{α} \to U$ with $q (t_{0}) = q_{0}$ solving $q^{'} (t) = H_{p} (t, q (t), S_{q} (t, q (t), α))$ , and with $p (t) : = S_{q} (t, q (t), α)$ the pair $(q, p)$ solves Hamilton's equations $q^{'} = H_{p}$ , $p^{'} = - H_{q}$ on $I_{α}$ ;
(Jacobi constants) $β (t) : = S_{α} (t, q (t), α)$ is constant on $I_{α}$ , equal to $S_{α} (t_{0}, q_{0}, α)$ ;
(local converse) for $(α, β)$ near $(α_{0}, β_{0})$ the equation $β = S_{α} (t, q, α)$ can be solved locally for $q = Q (t, α, β)$ , and with $p : = S_{q} (t, Q (t, α, β), α)$ the curve $t \mapsto (Q, p)$ solves Hamilton's equations;
(canonical labels) on the domain of the local diffeomorphism $(q, p) \mapsto (β, α)$ obtained by composing $(q, p) \mapsto (q, α)$ with $(q, α) \mapsto (β, α)$ , one has the two-form identity $d q \land d p = d β \land d α$ at every fixed $t$ ; that is, $(β, α)$ are local canonical coordinates, conjugate to $(q, p)$ , in which the $β$ are constants of the motion of clause 2.

The words canonical and the finite-regularity form operations here use Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, with the displayed $C^{2}$ maps; smoothness is not inferred.

Facts & Assumptions

Given: The open sets $J, U, A$ , the $C^{2}$ Hamiltonian $H$ , the $C^{3}$ complete integral $S$ , the base point $(t_{0}, q_{0}, α_{0})$ and $β_{0} = S_{α} (t_{0}, q_{0}, α_{0})$ .

[F1]

Definition of the data (The Hamilton-Jacobi equation and a complete integral of it): on $J \times U \times A$ , $S_{t} + H (t, q, S_{q}) = 0$ , $\det S_{q α} (t_{0}, q_{0}, α_{0}) \neq 0$ , and $S$ is $C^{3}$ , so all second derivatives used below are continuous.

[F2]

Local existence for the characteristic ODEs (Picard-Lindelöf local existence and uniqueness for first-order systems, First-order systems, initial value problems, and solutions on intervals): the map $(t, q) \mapsto H_{p} (t, q, S_{q} (t, q, α))$ is $C^{1}$ on a neighbourhood of $(t_{0}, q_{0})$ , so the initial value problem with $q (t_{0}) = q_{0}$ has a unique local solution on some interval $I_{α} ∋ t_{0}$ .

[F3]

Calculus (Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, The Hessian of a $C^{2}$ scalar field is symmetric, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case): for the $C^{3}$ function $S$ the mixed partials commute, so the second derivatives appearing in the chain-rule computations below are symmetric in their two differentiation variables; and the finite-coordinate formula gives $d (d S) = 0$ by that mixed-partial equality.

[F4]

Inverse function theorem (The Euclidean inverse function theorem, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, For every square matrix over a commutative ring, $\det (A^{T}) = \det (A)$ ): the mixed partials of the $C^{3}$ function $S$ commute, so $S_{α q} = (S_{q α})^{T}$ and $\det S_{α q} = \det S_{q α}$ by $\det (A^{T}) = \det (A)$ ; since $\det S_{q α} (t_{0}, q_{0}, α_{0}) \neq 0$ , the map $(t, q, α) \mapsto (t, S_{α} (t, q, α), α)$ is a local $C^{2}$ diffeomorphism at the base point, so $β = S_{α} (t, q, α)$ can be solved locally for $q$ ; likewise the map $(q, p) \mapsto (q, α)$ given by $p = S_{q} (t, q, α)$ is a local diffeomorphism because its derivative is block triangular with invertible block $S_{q α}$ (Invertible Euclidean linear maps, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

Proof

technique · direct; differentiate the Hamilton-Jacobi equation in $q$ and in $\alpha$, use the inverse function theorem for the converse, and take $d$ of the identity $dS=p\,dq+\beta\,d\alpha-H\,dt$

1.1F1F2F3algebra

Clause 1: by [F2] take the local solution $q$ of the characteristic equation and set $p : = S_{q} (t, q, α)$ . Then $p^{'} (t) = S_{t q} + S_{q q} q^{'}$ by the chain rule; differentiating the Hamilton-Jacobi equation in $q$ gives $S_{t q} + H_{q} + S_{q q} H_{p} = 0$ at $(t, q (t), α)$ ; substituting $q^{'} = H_{p}$ and using the symmetry of $S_{q q}$ from [F3] yields $p^{'} = S_{t q} + S_{q q} H_{p} = - H_{q}$ , while $q^{'} = H_{p}$ holds by construction. Hence $(q, p)$ solves Hamilton's equations on $I_{α}$ .

1.2F3F4algebra

Clause 4: from $d S = S_{t} d t + S_{q} d q + S_{α} d α = - H d t + p d q + β d α$ and [F3], applying $d$ gives $0 = d p \land d q + d β \land d α - d H \land d t$ on $J \times U \times A$ ; pulling back along the inclusion of a time slice ${t} \times U \times A$ kills the $d t$ terms and yields $d q \land d p = d β \land d α$ as forms on $U \times A$ at fixed $t$ . By [F4] both $(q, α) \mapsto (q, p)$ and $(q, α) \mapsto (β, α)$ are local diffeomorphisms (block triangular with invertible mixed block), so the composite $(q, p) \mapsto (β, α)$ is a local diffeomorphism whose pullback of $d β \land d α$ equals $d q \land d p$ : $(β, α)$ are local canonical coordinates.

2.1F1F3step 1.1algebra

Clause 2: by the chain rule, $\frac{d}{d t} S_{α_{j}} = S_{α_{j} t} + \sum_{i} S_{α_{j} q_{i}} q_{i}^{'} = S_{t α_{j}} + \sum_{i} H_{p_{i}} S_{q_{i} α_{j}}$ ; differentiating the Hamilton-Jacobi equation in $α_{j}$ gives exactly $S_{t α_{j}} + \sum_{i} H_{p_{i}} S_{q_{i} α_{j}} = 0$ , so $\frac{d}{d t} S_{α_{j}} = 0$ and $S_{α}$ is constant on $I_{α}$ .

2.2F1F3F4step 1.1algebra

Clause 3: by [F4], solve $β = S_{α} (t, q, α)$ locally as $q = Q (t, α, β)$ near $(t_{0}, α_{0}, β_{0})$ . Differentiating the identity $S_{α} (t, Q, α) = β$ in $t$ gives $S_{α t} + S_{α q} Q^{'} = 0$ ; differentiating the Hamilton-Jacobi equation in $α$ gives $S_{α t} + S_{α q} H_{p} = 0$ ; subtracting, $S_{α q} (Q^{'} - H_{p}) = 0$ , and $\det S_{α q} \neq 0$ gives $Q^{'} = H_{p}$ . With $p : = S_{q} (t, Q, α)$ , the computation of step 1.1 (which uses only $q^{'} = H_{p}$ and the $q$ -derivative of the Hamilton-Jacobi equation) gives $p^{'} = - H_{q}$ .

3.1step 1.1step 2.1step 2.2step 1.2∎

Clauses 1-4 are established; in particular a complete integral produces solutions of Hamilton's equations through its characteristics, its $α$ -derivatives are constants of those solutions, and its constants furnish local canonical coordinates.

Remarks

The mixed invertibility $\det S_{q α} \neq 0$ is the local no-caustic hypothesis: it is exactly what makes both the characteristic momentum map and the constants map local diffeomorphisms, and it excludes the caustics where the family of Jacobi constants fails to be a coordinate system.
Nothing here asserts a global solution or global completeness: the theorem is local in $(t, q, α)$ , and the characteristic flow is only integrated on the interval supplied by [F2].

A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical $(β, α)$ labels

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical (β,α)(β,α) labels

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

A complete integral produces Hamilton characteristics, the Jacobi constants, a local converse and canonical $(β, α)$ labels