POC Bounded Classical – Law L02: Newton’s Second Law – SSM

Extends F = m * a into a two-lane form (m,a) so you can see whether a 20 N force is calm, borderline, or stressed.

Domain. Mechanics / Robotics / Motion Control

Classical law. F = m * a

What this shows.

In this law POC, we take Newton’s Second Law and show how:

  • The Classical calculation gives the correct scalar force F, and
  • The Shunyaya Symbolic Mathematics (SSM) version keeps that force magnitude intact but adds a bounded alignment lane a in (-1,+1), indicating whether this particular F = m * a situation is calm, borderline, or stressed.

When the motion is smooth and mass is well-known, Shunyaya Symbolic Mathematics (SSM) collapses to the classical result and you effectively get the same answer.

When acceleration is jerky or mass is uncertain, the alignment lane a and its band surface posture that the classical scalar F alone does not show.

POC display policy (simple).

For this POC, we use:

  • a_semantics = "drift-positive"

so larger printed a means more drift / more risk. This is a display choice only; you can flip the semantics (e.g., make +a mean more stability) without changing the math or phi((m,a)) = m.

1) Setup (inputs)

Imagine a small robotic cart on a test rig. The cart is commanded to move forward gently. You know its mass to a reasonable approximation, and you record two acceleration samples a short time apart:

  • The first sample is during a slightly jerky start.
  • The second sample is when motion is more settled.

We model:

  • Mass of the cart (approximate):
    • m_m = 20.0 kg, m_a = +0.05
    • The cart mass is fairly well known (small alignment, near-calm).
  • Acceleration measurements (instantaneous):
    • a1_m = 0.90 m/s^2, a1_a = +0.65 — jerkier instant (controller still settling)
    • a2_m = 1.10 m/s^2, a2_a = +0.10 — smoother instant (steady push)

In SSM:

  • Each quantity is represented as (m, a) with a in (-1,+1).
  • Collapse is phi((m,a)) = m — we drop the alignment lane and keep the magnitude.

Our target.

  • Use F = m * a
  • Compare Classical force vs SSM force (m_F, a_F).

2) Classical calculation

Ignoring alignment, we do the usual Newton’s law with an average acceleration:

# classical illustration (no external packages required)

m = 20.0        # kg (approx mass of the cart)
a1 = 0.90       # m/s^2 (jerkier instant)
a2 = 1.10       # m/s^2 (calmer instant)

a_avg = 0.5 * (a1 + a2)
F     = m * a_avg

print(a_avg)  # 1.0
print(F)      # 20.0

Classical result.

  • a_avg = 1.0 m/s^2
  • F = 20.0 N

A standard lab report would simply state: Force ≈ 20 N, with no indication of how jerky or smooth the motion really was.

3) SSM calculation (same magnitude + bounded alignment lane)

In Shunyaya Symbolic Mathematics (SSM), we:

  1. Treat each acceleration sample as (m, a) with a in (-1,+1).
  2. Compute an alignment for acceleration using the weighted sum pooling rule:
    • For each sample, clamp and map into rapidity space:
      • a_c := clamp(a, -1+eps, +1-eps)
      • u := atanh(a_c)
    • Accumulate with weights w := |m|^gamma (default gamma = 1):
      • U += w * u
      • W += w
    • Collapse back to the alignment lane:
      • a_a_out := tanh( U / max(W, eps) )
  3. Combine the acceleration alignment with the mass alignment using product chaining for the law F = m * a:
    • a_F := tanh(atanh(a_m_c) + atanh(a_a_out_c))
  4. Keep the force magnitude as the classical result m_F = m_m * a_avg:
    • phi((m_F, a_F)) = m_F = m_m * a_avg

So:

  • The magnitude of force is still 20.0 N.
  • The bounded alignment lane a_F tells us how smooth or jerky the realized F = m * a is, given our two acceleration samples and the slightly uncertain mass.

Intuitively:

  • One acceleration sample has high drift (a1_a ≈ +0.65 under drift-positive semantics).
  • The other is closer to calm (a2_a ≈ +0.10).
  • Their pooled acceleration alignment a_a_out lands in a moderate risk region.
  • Combining with a mostly calm mass (m_a ≈ +0.05) yields a force alignment a_F that is noticeably non-zero, but not “worst case”.

4) Tiny script (copy-paste)

Below is a small script implementing this Newton’s law POC in ASCII-only Python:

# scenario_L02_newton_fma.py  (ASCII-only, top-level prints)

import math

def clamp(a, e=1e-6):
    return max(-1 + e, min(1 - e, float(a)))

def ssm_align_weighted(pairs, gamma=1.0, eps=1e-12):
    """
    pairs: iterable of (a_raw, m)
    weight w := |m|^gamma
    """
    U = 0.0
    W = 0.0
    for a_raw, m in pairs:
        a = clamp(a_raw)
        # atanh(a) = 0.5 * ln((1+a)/(1-a))
        u = 0.5 * math.log((1.0 + a) / (1.0 - a))
        w = abs(float(m)) ** gamma
        U += w * u
        W += w
    return math.tanh(U / max(W, eps))

def ssm_align_product(a1_raw, a2_raw, eps=1e-6):
    """
    Product chaining for alignment lane:
    a_out := tanh(atanh(a1_c) + atanh(a2_c))
    """
    a1 = clamp(a1_raw, eps)
    a2 = clamp(a2_raw, eps)
    u1 = 0.5 * math.log((1.0 + a1) / (1.0 - a1))
    u2 = 0.5 * math.log((1.0 + a2) / (1.0 - a2))
    return math.tanh(u1 + u2)

# 1) law-specific inputs: Newton's second law F = m * a

# mass (m, a) of the cart
m_m, m_a = 20.0, +0.05   # kg, mostly calm

# acceleration measurements (m, a) at two instants
a1_m, a1_a = 0.90, +0.65  # m/s^2, jerkier instant
a2_m, a2_a = 1.10, +0.10  # m/s^2, calmer instant

# 2) classical magnitude: average acceleration, then F = m * a
a_avg = 0.5 * (a1_m + a2_m)
F_m   = m_m * a_avg

# 3) SSM alignments
# pooled acceleration alignment
a_accel = ssm_align_weighted(
    [(a1_a, a1_m), (a2_a, a2_m)],
    gamma=1.0,
    eps=1e-12,
)

# force alignment from mass and acceleration
a_F = ssm_align_product(m_a, a_accel, eps=1e-6)

print("Classical:", f"{F_m:.4f}")            # 20.0000
print("SSM:", f"m={F_m:.4f}, a={a_F:+.4f}")  # a_F ~ +0.48 (drift-positive)


You can later add band assignment in the overall runner, for example:

|a| < 0.20         -> A+ (calm)
0.20 <= |a| < 0.50 -> A0 (borderline)
|a| >= 0.50        -> A- (stressed)

5) What to expect

Running the script gives:

  • Classical: F ≈ 20.0000 N
  • SSM: m = 20.0000 N, a ≈ +0.4253 (under drift-positive semantics)

Under a simple policy:

  • |a| < 0.20 → A+ (calm)
  • 0.20 ≤ |a| < 0.50 → A0 (borderline)

So in this example:

  • a ≈ +0.4253 sits in the upper part of A0 (borderline), below the stressed A- region.
  • The underlying force magnitude F = 20 N is perfectly classical, but SSM makes visible that the underlying acceleration samples are not uniformly calm.

If both acceleration samples were smooth and mass alignment small, we would get:

  • a_accel ≈ 0, a_F ≈ 0, and SSM ≡ Classical in posture as well.

6) Why this helps in the real world

  • Robotics engineers can distinguish between “20 N applied smoothly” vs “20 N coming from jerky control”, using the alignment lane a as a quick indicator of motion quality.
  • Testing teams can flag borderline or stressed F = m * a situations even when the scalar force looks fine, prompting better tuning of controllers and actuators.
  • Dashboards and logs can color or band force values by alignment, making it easier to spot which trials or operating points were genuinely smooth and which were numerically correct but posturally noisy.

7) License and scope

  • License. CC BY-NC 4.0 (non-commercial, attribution required).
  • Scope. Observation-only; not for critical use.

This POC is intended for thinking, experimentation, and education around bounded classical laws. It is not a safety case, design guarantee, or regulatory tool.

Navigation

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Next: POC Bounded Classical – Law L03: Hooke’s Law – SSM


Disclaimer (summary):
All Shunyaya Symbolic Mathematics (SSM) POCs for popular laws are observation-only examples and must not be used for design, certification, or safety-critical decisions.