standardEL
plain-language theorem explainer
standardEL defines the Euler-Lagrange operator for the standard Lagrangian L = ½ m q̇² - V(q) as m γ̈(t) + V'(γ(t)). Researchers recovering Newtonian mechanics from the J-cost variational principle in the quadratic limit cite it to equate the EL equation with Newton's second law. The definition is a direct one-line transcription of the variational derivative with no lemmas or tactics applied.
Claim. The Euler-Lagrange operator for the Lagrangian $L = ½ m q̇² - V(q)$, evaluated on trajectory $γ$ at time $t$, equals $m γ̈(t) + V'(γ(t))$.
background
The Quadratic Limit module develops the small-strain bridge from the J-cost action to standard mechanics. In the regime γ = 1 + ε with |ε| ≪ 1, J reduces to its quadratic Taylor form. Jcost_taylor_quadratic rebrands Jcost_small_strain_bound to give |J(1 + ε) - ε²/2| ≤ ε²/10 for |ε| ≤ 1/10, justifying the identification of the J-action with the kinetic action ½ ∫ ε² dt. The standard Lagrangian is then L = ½ m q̇² - V(q). Upstream cost definitions appear in MultiplicativeRecognizerL4.cost and ObserverForcing.cost, both inducing the J-cost from recognition events on positive ratios.
proof idea
This is a direct definition that transcribes the Euler-Lagrange formula for L = ½ m q̇² - V(q) into the explicit expression m * (second derivative of γ) + (derivative of V at γ t). No lemmas are invoked and no tactics are used; the body is the immediate algebraic unfolding of the standard EL operator.
why it matters
standardEL supplies the concrete operator whose zero set is Newton's second law, feeding NewtonSecondLawCert, EnergyConservationCert, energy_conservation, hamilton_equations_from_EL, and totalEnergy. It realizes the paper claim that Newton's second law is a corollary of the J-action variational principle once the quadratic limit is taken (papers/RS_Least_Action.tex, Section Newton's Second Law as a Corollary). The step connects the Recognition Composition Law and phi-ladder to classical dynamics, with the eight-tick octave supplying discrete time structure upstream.
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