newtonSecondLawCert
Newton's second law is recovered as the Euler-Lagrange equation of the small-strain J-action. A physicist chaining variational principles to classical mechanics cites this witness when embedding Newtonian dynamics inside the Recognition Science J-cost framework. The definition is a direct record that packages three theorems already established in QuadraticLimit.
claimThe certificate asserts that for mass $m$, potential $V$, trajectory $γ$, and time $t$, the Euler-Lagrange equation of $L=½mẋ²-V(x)$ vanishes if and only if $mγ̈(t)=-V'(γ(t))$; that $|J(1+ε)-ε²/2|≤ε²/10$ for $|ε|≤1/10$; and that the second derivative of $J$ at unity equals 1.
background
The J-cost function supplies the recognition cost whose quadratic expansion near the ground state reproduces the standard kinetic term. NewtonSecondLawCert is the structure that bundles the EL-Newton equivalence, the quadratic Taylor bound on Jcost, and the Hessian calibration J''(1)=1. The module sits in the Action domain and draws directly on the d'Alembert calibration of J together with the small-strain results proved in QuadraticLimit.
proof idea
The definition is a one-line wrapper that records the three theorems from QuadraticLimit: the EL-Newton equivalence is taken from newton_second_law, the Taylor bound from Jcost_taylor_quadratic, and the Hessian identity from Jcost_quadratic_leading_coeff.
why it matters in Recognition Science
The witness is consumed by the downstream theorem newton_second_law_one_statement that proves the certificate is inhabited. It closes the small-strain limit step that derives Newton's second law from the J-action variational principle, as stated in the companion paper RS_Least_Action.tex §Newton's Second Law as a Corollary. The construction relies on the phi-forcing chain and the eight-tick octave but remains inside the continuous approximation.
scope and limits
- Does not extend the EL-Newton equivalence beyond the small-strain regime |ε|≤1/10.
- Does not incorporate relativistic or quantum corrections to the action.
- Does not derive the explicit form of the potential V.
- Does not treat multi-particle or field-theoretic generalizations.
Lean usage
theorem newton_second_law_one_statement : Nonempty NewtonSecondLawCert := ⟨newtonSecondLawCert⟩formal statement (Lean)
55def newtonSecondLawCert : NewtonSecondLawCert where
56 el_iff_newton := QuadraticLimit.newton_second_law
proof body
Definition body.
57 quadratic_taylor_bound := Jcost_taylor_quadratic
58 hessian_one := Jcost_quadratic_leading_coeff
59
60/-- One-statement summary: Newton's second law is the EL equation of
61the small-strain J-action. -/