Newton’s vanishing areas and the proposed action scale

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The research anchor is Galileo’s inertial horizontal line compared with the falling parabola. Their enclosed area is \(v_0F\varepsilon^3/(6m)\), so \((3F/v_0)A=\varepsilon\delta E\). The matched-endpoint chord lens is half that area and gives the action difference \(\varepsilon\delta E/12\). Newton’s Lemmas X–XI supply the geometric anchors; Kepler’s swept sectors are a separate quantity. Calculation recorded 2026-09-05; target clarified by the user on 2026-09-14.

The N01 comparison connects this anchor to an explicit closed quantum phase experiment. The programme seeks a physical obstruction to indefinitely shrinking the difference as a distinguishable classical alternative. The deterministic energy gain and a quantum energy uncertainty must be kept distinct when assessing that obstruction.

1. What was read

We downloaded the Motte/Chittenden 1846 text and read Definitions I–VIII and Scholium, the Laws with corollaries and Scholium, Book I Section I, and Section II Propositions I–II. A Motte 1729 excerpt edited by Wilkins supplies readable formulae for Section I. The reading coverage is these later English witnesses and the passages listed below.

The closest passages are quite specific:

Newton writes in the closing Scholium that he means “evanescent divisible quantities” and defines the construction through limiting ratios. His example of an “ultimate velocity” means the limiting velocity at an event. These passages give the historical vocabulary for the continuity argument.

The source HTML contains modern editorial image descriptions. Some equations are images; the Wilkins PDF was used where HTML extraction omitted them. Selected historical figures were visually inspected: projectile, central polygon.

2. Constant-force geometry

Take a particle of mass \(m>0\), a constant force \(F>0\) in the positive \(y\) direction, \(V(y)=-Fy\), and initial data

\[ x(0)=y(0)=0,\qquad \dot x(0)=v_0>0,\qquad \dot y(0)=0. \]

For an interval \(\varepsilon>0\) the exact classical solution is

\[ x(t)=v_0t,\qquad y(t)=\frac{F}{2m}t^2. \]

The particle deflects along the force and loses potential energy. Reversing the vertical convention reverses the signs while preserving magnitudes. Define the energy transfer by

\[ \delta E:=K(\varepsilon)-K(0) =-\bigl[V(\varepsilon)-V(0)\bigr] =F\Delta y=\frac{F^2\varepsilon^2}{2m}. \]

Total mechanical energy is constant. Here \(\delta E\) denotes deterministic work; quantum energy spread will be written \(\sigma_H\) below. The proportionality constant in \(\Delta y=K_{\rm conv}\delta E\) is \(K_{\rm conv}=1/F\), which has dimensions of inverse force.

Let \(A=(0,0)\), \(B=(v_0\varepsilon,0)\) be the inertial endpoint and \(D=(v_0\varepsilon,F\varepsilon^2/(2m))\) the actual endpoint. The triangle \(ABD\) has

\[ \mathcal A_\triangle=\frac12\Delta x\Delta y =\frac{v_0F}{4m}\varepsilon^3 =\frac{v_0}{2F}\,\varepsilon\delta E. \]

This confirms the proposed proportionality, with an explicit factor and energy definition. Multiplying this configuration-space area by \(F/v_0\) gives action units. Section 3 obtains the action coefficient by direct integration.

For \(y(x)=Fx^2/(2mv_0^2)\) the other two areas are

\[ \mathcal A_{\rm tangent,curve}=\int_0^{v_0\varepsilon}y(x)\,dx =\frac23\mathcal A_\triangle,\qquad \mathcal A_{\rm chord,curve}=\frac13\mathcal A_\triangle =\frac{v_0F\varepsilon^3}{12m}. \]

These are the ratios in Lemma XI, Corollary 5, exact for this parabola. The tangent triangle is cubic in \(\varepsilon\) because the initial vertical velocity is zero. About a later point, deflection from the local tangent is again quadratic, while absolute vertical displacement also includes the existing vertical velocity.

3. Matched-endpoint action difference

The same-endpoint straight chord is an admissible off-shell comparison path. On \(0\leq t\leq\varepsilon\), set

\[ y_{\rm cl}(t)=\frac{F}{2m}t^2,\qquad y_{\rm ch}(t)=\frac{F\varepsilon}{2m}t,\qquad x(t)=v_0t. \]

Compare both with the same Lagrangian

\[ S[x,y]=\int_0^\varepsilon \left[\frac m2(\dot x^2+\dot y^2)+Fy\right]dt. \]

For \(y=y_{\rm cl}+\eta\), with \(\eta(0)=\eta(\varepsilon)=0\), integration by parts cancels the linear variation because \(m\ddot y_{\rm cl}=F\). Consequently

\[ S[y_{\rm cl}+\eta]-S[y_{\rm cl}] =\frac m2\int_0^\varepsilon\dot\eta^2\,dt. \]

Here \(\eta=F t(\varepsilon-t)/(2m)\) for the chord. Therefore

\[ \boxed{\Delta S_{\rm ch,cl} =\frac{F^2\varepsilon^3}{24m} =\frac{F}{2v_0}\mathcal A_{\rm chord,curve} =\frac{\varepsilon\delta E}{12}.} \]

The action integral fixes the coefficient relating the lens to the action difference. Adding the same point-function term \(dG(q,t)/dt\) to the Lagrangian preserves the difference because endpoints and times agree.

The neighbouring paths form a continuous family. Choosing \(\eta=\alpha t(\varepsilon-t)\) gives

\[ \Delta S(\alpha)=\frac{m\alpha^2\varepsilon^3}{6}\longrightarrow0 \quad\text{as }\alpha\to0. \]

Positive action differences in this family therefore have infimum zero. A physical distinguishability relation or a restriction of allowed states changes the admissible comparison; defining that change is the next modelling question.

4. Polygonal approximation and its error

Proposition I proves equality of finite swept triangles already at the polygonal stage, using central impulses. Its swept triangles sum to the swept area over a fixed time. By contrast, chord–curve lenses measure the error of the polygonal approximation; their total has a different limit.

For the constant-force parabola, dividing a fixed duration \(T\) into \(N\) equal intervals gives \(N\) chord lenses, each of area \(v_0F(T/N)^3/(12m)\). Their sum is

\[ \mathcal A_{\rm error,total}=\frac{v_0FT^3}{12mN^2}\to0. \]

This gives the exact \(N^{-2}\) convergence of the polygonal error. The constant parallel force belongs to the projectile Scholium and Lemmas X–XI; Proposition I assumes a force directed towards a fixed centre.

For constant force the displayed solution exists globally. Singular central forces require collision and continuation analysis. The approximation above takes a mesh limit; a semiclassical calculation instead varies the parameter \(\hbar\).

5. Phase resolution, uncertainty and reference frames

The Feynman chapter supplies the quantum interpretation through phases \(e^{iS/\hbar}\). Equal phase occurs when an action difference is \(nh\), since \(h=2\pi\hbar\). This is periodicity of the relative phase. Classical stationarity concerns neighbourhoods of paths and their phase cancellation.

For our specific chord comparison,

\[ \Delta\varphi=\frac{\Delta S_{\rm ch,cl}}{\hbar} =\frac{F^2\varepsilon^3}{24m\hbar}. \]

At fixed \(\hbar>0\), \(\varepsilon\to0\) makes these two phases closer. Requiring order-one relative phase would define a comparison scale

\[ \varepsilon_{\rm phase}\sim\left(\frac{24m\hbar}{F^2}\right)^{1/3}. \]

This comparison scale marks an order-one phase difference for the selected pair of paths. The phase rule introduces a specified \(\hbar\) as quantum input. The technical paper turns relative phase into an exact resolution threshold for a fixed measurement protocol and copy count.

For a time-independent observable \(B\), the quantum commutator variance inequality and Heisenberg evolution imply

\[ \sigma_H\sigma_B\geq\tfrac12|\langle[H,B]\rangle|, \qquad \tau_B:=\frac{\sigma_B}{|d\langle B\rangle/dt|}, \qquad \sigma_H\tau_B\geq\frac\hbar2, \]

when domains and variances are appropriate and the denominator is nonzero. Here \(\sigma_H\) is a state energy spread and \(\tau_B\) an observable-change timescale. The earlier \(\delta E\) and \(\varepsilon\) describe deterministic work and a chosen trajectory interval. A wavepacket uncertainty area concerns the distribution of conjugate observables.

The spatial area also changes under a horizontal Galilean boost. In the frame moving at \(v_0\), horizontal launch velocity and the plotted spatial area vanish. The action difference above stays finite and unchanged: both paths share the same horizontal motion, whose action contribution cancels. Thus the area conversion uses a frame with \(v_0\neq0\), while the matched-endpoint action difference survives the boost.

For \(v_0<c\), the constant-force solution stays below \(c\) whenever \(0<\varepsilon<(m/F)\sqrt{c^2-v_0^2}\). Arbitrarily small intervals remain possible. This checks the speed of a finite Newtonian segment. Section 6 of the technical paper supplies a corresponding small-variation calculation for the free relativistic Lagrangian.

These results lead to an operational question: how does the action difference determine distinguishability under specified preparations and measurements? The complementary spectral question studies the effects of state space, boundary conditions and periodicity.

6. Historical interpretation and reproduction

The modern interpretation starts from Newton’s use of arbitrarily fine geometric limits. The proposed implicit \(h\to0\) reading is recorded as a historical conjecture. Its chronology, relationship to the Classical Scholia and place in the existing literature are tasks H02/H03.

The written derivation connects the geometric construction to the action integral. The saved diagram and earlier check output remain historical artifacts. Their legacy generator is inactive because it also executes symbolic verification.