Action Quadratic Limit Newton First Law

The quadratic limit

Newton's first law states that an object moves at constant velocity when no force acts on it. In the Recognition Science framework, this law appears as a theorem called newton_first_law, which is proved in a machine-checked library of formal theorems. The theorem says: if the potential energy is constant, so the force is zero, and the Euler-Lagrange equation holds, then the acceleration is zero. That is, the second derivative of position with respect to time vanishes, which means the velocity does not change.

The result is a corollary of a deeper bridge. The framework's cost functional, which measures the cost of recognition events, has a quadratic Taylor expansion near unity. For a small strain γ = 1 + ε, the cost J(γ) = ½(γ + γ⁻¹) - 1 reduces to ½ε². This makes the cost functional's action, the integral of J over time, match the standard kinetic action T[ε] = ½∫ε(t)² dt to within one tenth of T when |ε| ≤ 1/10. The Euler-Lagrange equation of the standard Lagrangian L = ½m q̇² - V(q) is then exactly Newton's second law, m q̈ = -V'(q).

The declaration newton_first_law itself is a narrow piece of that chain. It takes as premises a nonzero mass, a constant potential, and the Euler-Lagrange equation, and concludes that acceleration is zero. The proof is a direct algebraic manipulation of the second-law theorem, not a new physical principle. Its content is entirely definitional: the Euler-Lagrange operator is constructed so that its zero set is the set of Newtonian trajectories.

What the declaration does not do is derive Newton's first law from the cost functional alone. The bridge from the J-action to the kinetic action is a separate theorem, actionJ_to_kinetic_bridge, and it only holds in the small-strain regime. The declaration assumes the Euler-Lagrange equation already, rather than deriving it from the cost functional. It also says nothing about the physical interpretation of the cost functional or about how recognition events relate to forces; that bridge remains open. The theorem is a formal statement about a definition, not an empirical claim about the world.

/-- **Inertia (Newton's First Law).** When the potential is constant
    (`V' ≡ 0`), the EL equation reduces to `m γ̈ = 0`, i.e., constant
    velocity motion. -/
theorem newton_first_law (m : ℝ) (hm : m ≠ 0) (γ : ℝ → ℝ) (t : ℝ)
    (h_no_force : deriv (fun _ : ℝ => (0 : ℝ)) (γ t) = 0)
    (h_EL : standardEL m (fun _ => 0) γ t = 0) :
    deriv (deriv γ) t = 0 := by
  rw [newton_second_law] at h_EL
  rw [h_no_force, neg_zero] at h_EL
  exact (mul_left_cancel₀ hm (by rw [h_EL, mul_zero]))

/-- **Quadratic Taylor expansion of `Jcost` near 1.** This is just a
    rebrand of the existing `Cost.Jcost_small_strain_bound`:
    `|J(1 + ε) - ε²/2| ≤ ε²/10` whenever `|ε| ≤ 1/10`. -/
theorem Jcost_taylor_quadratic (ε : ℝ) (hε : |ε| ≤ (1 : ℝ) / 10) :
    |Jcost (1 + ε) - ε ^ 2 / 2| ≤ ε ^ 2 / 10 :=
  Jcost_small_strain_bound ε hε

/-- **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

What this page does not claim

The declaration does not derive Newton's first law from the cost functional alone. The declaration does not establish a physical bridge between recognition events and forces. The theorem does not apply outside the small-strain regime where |ε| ≤ 1/10.

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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