TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Complementarity, momentum balance, and energy loss at a scalar impact

Statement

For any resolved output of First-impact step for a one-dimensional unilateral contact, let $q_{*}$ be the reported gap (zero at an impact, the free-flight endpoint gap otherwise), let $J$ be the reported impulse, and let $v_{*}$ be the reported endpoint velocity. Then $q_{*} \geq 0$ , $J \geq 0$ , and $q_{*} J = 0$ . On a free-flight branch, $J = 0$ and $m v_{*} = m v + m a h$ . On an impact branch at time $s$ , with incoming velocity $v^{-} = v + a s$ , the post-impact velocity satisfies $m v^{+} = m v^{-} + J$ and $\frac{1}{2} m (v^{+})^{2} - \frac{1}{2} m (v^{-})^{2} = - \frac{1}{2} m (1 - e^{2}) (v^{-})^{2} \leq 0.$ When $a = 0$ , the full resolved step has nonincreasing kinetic energy. For $a \neq 0$ , the displayed energy statement concerns only the instantaneous impact jump; work during free flight is not covered by it.

Facts & Assumptions

Given: The inputs and resolved output of the algorithm in the Statement.

[F1]

First-impact step for a one-dimensional unilateral contact: an impact output has $q_{*} = 0$ , $v^{-} < 0$ , $J = - m (1 + e) v^{-}$ , and $v^{+} = - e v^{-}$ ; a free-flight output has $J = 0$ , $q_{*} = q + h v + \frac{1}{2} a h^{2} \geq 0$ , and $v_{*} = v + a h$ .

Proof

1.1F1givenalgebra

On an impact branch, [F1] gives $q_{*} = 0$ , $J = - m (1 + e) v^{-} > 0$ , and $v^{+} = - e v^{-} \geq 0$ , since $m > 0$ , $e \in [0, 1]$ , and $v^{-} < 0$ . Hence $q_{*} J = 0$ . On a free-flight branch, [F1] gives $q_{*} \geq 0$ and $J = 0$ , so complementarity also holds.

1.2F1givenalgebra

On a free-flight branch, $m v_{*} = m (v + a h) = m v + m a h$ . On an impact branch, [F1] gives $m v^{-} = m (v + a s)$ and $J = - m (1 + e) v^{-}$ . Therefore $m v^{-} + J = m v^{-} - m (1 + e) v^{-} = - m e v^{-} = m v^{+}$ , using $v^{+} = - e v^{-}$ .

2.1F1step 1.1givenalgebra∎

On an impact branch, [F1] gives $(v^{+})^{2} = e^{2} (v^{-})^{2}$ , so the kinetic-energy difference is $\frac{1}{2} m (e^{2} - 1) (v^{-})^{2} = - \frac{1}{2} m (1 - e^{2}) (v^{-})^{2} \leq 0$ . If $a = 0$ and the branch is free flight, $v_{*} = v$ and kinetic energy is unchanged; impact branches have the nonincrease just established. For nonzero $a$ , no claim is made about total free-flight work.

Depends on

First-impact step for a one-dimensional unilateral contact

Used by

Inelastic and elastic endpoints of a scalar impact step Example

Dependency tree · one level

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Sources

David E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints (standard reference, not scraped)
Dimitri P. Bertsekas, 6.253 Convex Analysis and Optimization, Complete Lecture Notes (standard reference, not scraped)