ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

The planar pendulum in Hamiltonian form: kinetic metric $m l^{2}$ , substituted Hamiltonian $p^{2} / (2 m l^{2}) - m g l \cos θ$ and the reduced equation

Physical scope

A worked check of the Hamiltonian reformulation on the planar pendulum of the framework: one label of mass m constrained without friction to a vertical circle of radius l under a uniform downward field, described by a two-chart angular atlas covering the entire circle, including rotations; the kinetic metric is ml^2, the substituted Hamiltonian is p^2/(2ml^2) - mgl cos(theta), Hamilton's equations reduce to theta-double-dot = -(g/l) sin(theta), and the system is equivalent to the ideal constrained motion with its reaction recovered by the elimination theorem. All quantities are model values of the declared idealization; no measurement is reported.

Example

Declare one particle label of mass $m > 0$ in the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, with time window $R$ , ambient configuration domain $Q = R^{3}$ , potential $U (x, y, z) = m g y$ , constant $g > 0$ , and no residual force. Thus the force is $(0, - m g, 0)$ . Fix $l > 0$ and impose the scleronomic regular constraint $G (x, y, z) = (x^{2} + y^{2} - l^{2}, z) = 0,$ with the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Its locus is the full circle $N = {x^{2} + y^{2} = l^{2}, z = 0}$ . Here $m$ is a mass in kilograms, $l$ a length in metres, and $g$ an acceleration magnitude in metres per second squared; their dimensions are $μ$ , $ℓ$ , and $ℓ τ^{- 2}$ . The two components of $G$ have dimensions $ℓ^{2}$ and $ℓ$ , respectively. Write $G_{k i n}$ for the kinetic metric, to distinguish it from $G$ .

The full angular variable is an equivalence class $[θ] \in R / (2 π Z)$ , with $θ \sim θ + 2 π k$ , $k \in Z$ . The smooth periodic map $X (θ) = (l \sin θ, - l \cos θ, 0)$ descends to the circle of constrained configurations; this identification is the rescaled and rotated circle parametrization of $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle. It is not a one-to-one map from $R$ and is not used as one global generalized-coordinate parametrization.

Instead use the two angular charts $Ω_{0} = (- π, π)$ and $Ω_{π} = (0, 2 π)$ , with restrictions $X_{0}, X_{π}$ of $X$ . The ambient open sets $Q_{0} = R^{3} ∖ {(0, y, z) : y \geq 0}, Q_{π} = R^{3} ∖ {(0, y, z) : y \leq 0}$ satisfy $X_{a} (Ω_{a}) = N \cap Q_{a}$ for $a = 0, π$ . Restrict the ambient force model and constraint datum to $Q_{a}$ when applying Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (d). Each restricted map is a one-to-one $C^{3}$ parametrization with injective derivative, covering its constrained arc exactly once. The two arcs cover $N$ . On each connected component of an overlap the angular coordinates differ by $2 π k$ , with derivative $1$ ; no chart uses the same constrained configuration twice.

In either angular chart, the following hold, with dimensionless radian angle $θ$ :

On each phase-coordinate chart $Ω_{a} \times R$ with coordinates $(θ, p)$ , define an alternating bilinear form on coordinate increments $u = (u_{θ}, u_{p})$ and $v = (v_{θ}, v_{p})$ in $R^{2}$ by $ω_{a} (u, v) : = u_{θ} v_{p} - v_{θ} u_{p} .$ The notation $d θ \land d p$ below denotes this explicitly defined form field.

$G_{k i n} = m l^{2}$ , $\tilde{U} = - m g l \cos θ$ , and $\tilde{L} (θ, \dot{θ}) = \frac{1}{2} m l^{2} {\dot{θ}}^{2} + m g l \cos θ .$
The generalized momentum is $p = m l^{2} \dot{θ}$ , of dimension action $μ ℓ^{2} τ^{- 1}$ . The fibre map is hyperregular with inverse $\dot{θ} = p / (m l^{2})$ , and $\tilde{H} (θ, p) = \frac{p^{2}}{2 m l^{2}} - m g l \cos θ$ is the mechanical energy of the constrained motion.
Hamilton's equations are $\dot{θ} = p / (m l^{2})$ , $\dot{p} = - m g l \sin θ$ , equivalently $θ^{''} = - (g / l) \sin θ$ . Locally these are equivalent to the ideal constrained motion, with its reaction recovered by Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation.
On overlaps $θ_{b} = θ_{a} + 2 π k$ and $p_{b} = p_{a}$ . The Lagrangian, Hamiltonian, equations, and canonical form $d θ \land d p$ therefore agree, so the local Hamilton systems give one global system on the phase cylinder $(R \times R) / ((θ, p) \sim (θ + 2 π k, p)) .$ This retains the entire pendulum, including rotations crossing either chart cut. Equivalence with the ideal constrained equations is local at every time and hence holds along any constrained motion, without pretending that one real angular chart is globally injective.

All quantities are declared model data; no measurement is reported.

Facts & Assumptions

Given: The positive data $m, l, g$ , the uniform downward potential, the constraint $G$ , its ideal-reaction law, and the two chart domains and parametrizations above.

[F1]

Generalized coordinates and ideal constraint reduction (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction, Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits, Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation): a one-to-one $C^{3}$ parametrization onto the constrained locus in its declared ambient open set, with injective derivative, gives the kinetic metric $D X^{T} M D X$ and substituted Lagrangian; its Lagrange equations are equivalent to the ideal constrained motion, with reaction recovered in the form $D G^{T} λ$ .

[F2]

Mechanical momentum and Hamiltonian (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, 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, The mechanical Hamiltonian is the model's energy in the inertial chart, its balance is the energy balance, and the substituted-chart Hamiltonian is the energy of time-independent coordinates): a positive kinetic metric gives momentum $p = G_{k i n} v$ , global inverse in each momentum fibre, Hamiltonian $\frac{1}{2} p \cdot G_{k i n}^{- 1} p + \tilde{U}$ , and local equivalence of the Lagrange and Hamilton equations; in a static chart its value is the mechanical energy.

[F3]

Circle parametrization and calculus ( $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle, The derivatives of sine and cosine are cosine and minus sine, Sine and cosine defined by their real power series, Parity and the Pythagorean identity for sine and cosine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the rescaled periodic angular parametrization is one-to-one modulo $2 π$ and onto the circle, and sine/cosine have the stated derivatives and identity $\sin^{2} + \cos^{2} = 1$ .

[F4]

Model, units and matrix conventions (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, Linear map between vector spaces over the same field, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms, Submersions and immersions between Euclidean open sets): $M = m I_{3}$ , the force is $(0, - m g, 0)$ , and all derivatives, products and units use the declared chart conventions.

[F5]

The alternating bilinear form on each phase-coordinate chart is the field $ω_{a}$ defined in the Example prose above, denoted there by $d θ \land d p$ .

Verification

technique · direct; establish an actual angular atlas, apply the mechanical formulas and constraint reduction in each chart, and verify agreement on the overlaps

1.1F1F3F4algebra

The constraint differential has rows $(2 x, 2 y, 0)$ and $(0, 0, 1)$ , independent on $N$ because $x^{2} + y^{2} = l^{2} > 0$ , so the datum is regular and scleronomic. By [F3], two values of $X$ agree exactly when their angles differ by $2 π k$ ; no two distinct points in either open interval $Ω_{a}$ do so. Its derivative is $(l \cos θ, l \sin θ, 0)$ with norm $l > 0$ . On $Ω_{0}$ the only missing circle point is $(0, l, 0)$ , and on $Ω_{π}$ it is $(0, - l, 0)$ , exactly the respective intersections with the removed ambient half-planes. Thus $X_{a}$ maps bijectively onto $N \cap Q_{a}$ with injective derivative. Its inverse is smooth locally because at each point either the $x$ or $y$ derivative is nonzero, giving a smooth scalar inverse by The Euclidean inverse function theorem; these local inverses agree by injectivity and give the smooth inverse on the arc. Consequently both are legitimate parametrizations for [F1], and their arcs cover the full circle.

2.1F1F2F3F4step 1.1algebra

In either chart $G_{k i n} = D X_{a}^{T} (m I_{3}) D X_{a} = m l^{2}$ and $\tilde{U} = U (X_{a}) = - m g l \cos θ$ . Therefore $\tilde{L} = \frac{1}{2} m l^{2} {\dot{θ}}^{2} + m g l \cos θ$ and its velocity gradient is $p = m l^{2} \dot{θ}$ . Since $m l^{2} > 0$ , this fibre map is globally invertible with inverse $p / (m l^{2})$ , and [F2] gives $\tilde{H} = p^{2} / (2 m l^{2}) - m g l \cos θ$ , equal to the mechanical energy.

3.1F1F2F3step 2.1algebra

Differentiating the Hamiltonian gives ${\tilde{H}}_{p} = p / (m l^{2})$ and ${\tilde{H}}_{θ} = m g l \sin θ$ . Thus Hamilton's equations reduce to $\dot{θ} = p / (m l^{2})$ and $\dot{p} = - m g l \sin θ$ , whence $m l^{2} θ^{''} = - m g l \sin θ$ . By [F1] and [F2], within each chart these equations are equivalent to the ideal constrained equations, and the reaction is recovered as $D G^{T} λ$ with its unique two-component multiplier.

4.1F1F2F3F5step 1.1step 3.1algebra∎

On an overlap the two angle representatives differ by a continuous integer multiple of $2 π$ , hence by one constant multiple on each connected component, by Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ along each overlap interval. Their time derivatives agree, so $p_{b} = m l^{2} {\dot{θ}}_{b} = m l^{2} {\dot{θ}}_{a} = p_{a}$ ; the transition map has identity derivative in $(θ, p)$ , and the definition [F5] therefore gives $ω_{b} = ω_{a}$ , that is, $d θ_{b} \land d p_{b} = d θ_{a} \land d p_{a}$ . Periodicity of sine and cosine makes the formulas for $\tilde{L}$ , $\tilde{H}$ and the equations identical on overlaps. This defines the stated phase cylinder by gluing the charts (equivalently taking the displayed equivalence classes), and the local solutions and reactions agree wherever both charts apply. Every event of a constrained curve lies in one of the two arcs; continuity puts the curve in that arc for a neighbourhood of its parameter value, where step 3.1 applies. Thus the equivalence holds at every time and the full pendulum claim, including crossing the cuts, is retained.

Remarks

The angular generalized momentum $m l^{2} \dot{θ}$ has action units; the tangential kinetic momentum $m l \dot{θ}$ has linear-momentum units. The atlas replaces the invalid globally injective real-angle parametrization without restricting the pendulum to one arc or excluding rotations. Its static charts preserve the energy interpretation; a moving chart with a velocity-linear term has the separate interpretation given by the companion example.

The planar pendulum in Hamiltonian form: kinetic metric $m l^{2}$ , substituted Hamiltonian $p^{2} / (2 m l^{2}) - m g l \cos θ$ and the reduced equation

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The planar pendulum in Hamiltonian form: kinetic metric ml2ml2, substituted Hamiltonian p2/(2ml2)−mglcos⁡θp2/(2ml2)−mglcosθ and the reduced equation

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The planar pendulum in Hamiltonian form: kinetic metric $m l^{2}$ , substituted Hamiltonian $p^{2} / (2 m l^{2}) - m g l \cos θ$ and the reduced equation