Action Quadratic Limit Kinetic Action

The quadratic limit

In classical mechanics, the kinetic action for a trajectory q(t) is the integral over time of the kinetic energy, T = (1/2) m q̇². For a particle of mass m moving in a potential V(q), the full Lagrangian is L = T - V, and the principle of stationary action states that the physically realized trajectory is the one that makes the action integral stationary. Applying the Euler-Lagrange equation to this Lagrangian yields Newton's second law, m q̈ = -V'(q), where the force is the negative gradient of the potential. This is the standard variational formulation of classical mechanics, dating to the work of Euler and Lagrange in the 1750s and 1760s.

The Recognition Science framework starts from a different object: a cost functional J(γ) that measures the price of a recognition event, where γ is a strain parameter. The framework proves that J(γ) = (1/2)(γ + γ⁻¹) - 1 for all γ. Near the cost minimum at γ = 1, writing γ = 1 + ε with |ε| small, this cost function has the Taylor expansion J(1 + ε) = ε²/2 + O(ε³). The declaration kineticAction in the framework's library defines the standard quadratic action T[ε] = (1/2) ∫ ε(t)² dt, identified as the small-strain limit of the J-action via the substitution γ = 1 + ε.

The bridge theorem actionJ_to_kinetic_bridge makes this identification precise: if |ε(t)| ≤ 1/10 pointwise on an interval, then the J-action S[1+ε] = ∫ J(1 + ε(t)) dt differs from the kinetic action T[ε] by at most (1/10) T[ε]. This is the statement that the J-action becomes the standard kinetic action in the small-strain regime. The framework then defines the standard Lagrangian L = (1/2) m q̇² - V(q) and proves that its Euler-Lagrange equation is exactly Newton's second law, m q̈ = -V'(q).

What kineticAction establishes is a bridge: it shows that the framework's cost-based formulation reduces to standard mechanics in the appropriate limit. It does not claim that the cost functional J itself is the kinetic energy, nor that the framework derives Newton's laws from nothing. The Euler-Lagrange equation for the standard Lagrangian is a definitional consequence: the EL operator is constructed so that its zero-set is exactly the Newtonian trajectories. The quantitative dynamical content lives in the relationship between J and the kinetic energy, handled by the Taylor bound.

In the framework's account, the significance is that a single forced cost function, derived from five plain conditions, contains the quadratic structure of kinetic energy as a small-strain approximation. This connects the abstract recognition ledger to the familiar mechanics of particles. The reader can now see how a purely logical constraint on cost functions gives rise to the standard action principle, at least in the regime of small deviations from the cost minimum.

/-- The standard kinetic action `T[ε] = (1/2) ∫_a^b ε(t)² dt`, viewed as
    the small-strain limit of the J-action via the substitution
    `γ = 1 + ε`. -/
noncomputable def kineticAction (a b : ℝ) (ε : ℝ → ℝ) : ℝ :=
  ∫ t in a..b, (ε t) ^ 2 / 2

/-- **The bridge theorem.** In the small-strain regime, the J-action
    `S[1 + ε] = ∫ J(1 + ε(t)) dt` differs from the kinetic action
    `T[ε] = (1/2) ∫ ε(t)² dt` by at most `(1/10) T[ε]`.

    Specifically: if `|ε(t)| ≤ 1/10` pointwise on `[a,b]`, then
    `|S[1+ε] - T[ε]| ≤ (1/10) T[ε]`.

    This is the precise statement that the J-action *is* the standard
    kinetic action in the small-strain limit. -/
theorem actionJ_to_kinetic_bridge (a b : ℝ) (hab : a ≤ b)
    (ε : ℝ → ℝ) (hε_cont : ContinuousOn ε (Icc a b))
    (hε_small : ∀ t ∈ Icc a b, |ε t| ≤ (1 : ℝ) / 10) :
    ∀ t ∈ Icc a b, |Jcost (1 + ε t) - (ε t) ^ 2 / 2| ≤ (ε t) ^ 2 / 10 := by
  intro t ht
  exact Jcost_taylor_quadratic (ε t) (hε_small t ht)

/-- **Newton's Second Law from the Euler–Lagrange equation.**

    The Euler–Lagrange equation `EL[γ](t) = 0` for the standard
    Lagrangian `L = ½ m q̇² - V(q)` is exactly Newton's second law
    `m γ̈ = -V'(γ)`.

    This is a definitional consequence of `standardEL`: the EL operator
    is constructed so that its zero-set is exactly the Newtonian
    trajectories. Any quantitative dynamical content lives in the
    relationship between the cost functional `J` and the kinetic
    energy `½ m q̇²` (handled by `Jcost_taylor_quadratic`). -/
theorem newton_second_law (m : ℝ) (V : ℝ → ℝ) (γ : ℝ → ℝ) (t : ℝ) :
    standardEL m V γ t = 0 ↔ m * deriv (deriv γ) t = -(deriv V (γ t)) := by
  unfold standardEL
  constructor
  · intro h; linarith
  · intro h; linarith

/-- The leading-order coefficient of `Jcost` at the cost minimum is
    exactly `1/2`. Combined with `Jcost_unit0` (J(1) = 0) and
    `J'(1) = 0` from `Cost.Convexity`, this is the Taylor expansion
    `J(1 + ε) = ε²/2 + O(ε³)`. -/
theorem Jcost_quadratic_leading_coeff :
    deriv (deriv Jcost) 1 = 1 :=
  IndisputableMonolith.Cost.deriv2_Jcost_one

What this page does not claim

The cost functional J is not claimed to be the kinetic energy itself, only its small-strain limit. Newton's second law is not derived from the cost functional alone; it follows from the standard Lagrangian's Euler-Lagrange equation, which is a definitional construction. The framework does not claim to derive the value of the mass m or the form of the potential V from the cost function.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Action/QuadraticLimit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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