How the NKT Law Aligns with Classical Mechanics – ██FR█████ █INTELL███████████

This content originally appeared on DEV Community and was authored by Nguyen Khanh Tung

“The NKT Law does not reject Newton, Euler, or Kepler—it complements them by modeling motion in mass-varying systems with surprising clarity.”

Classical mechanics, built on the foundations laid by Newton, Euler, and Kepler, has successfully described motion in a vast range of systems for over three centuries. From falling apples to planetary orbits, these laws have stood the test of time.

But what happens when we consider systems where mass is not constant?

This is where the NKT Law comes in—not to overthrow classical mechanics, but to extend its language to better describe modern, dynamic systems like rockets, evaporating droplets, or aggregating particles.

Classical Frameworks in Brief

Let’s recall three central frameworks:

Newton’s Second Law: F = ma (or F = dp/dt)
Euler’s Equations of Motion: Used to analyze rigid bodies with rotating parts
Kepler’s Laws: Describe the motion of planets based on fixed-mass gravitational systems

All of these assume that mass is constant unless stated otherwise. In many real-world cases, however, mass varies — and that’s where extensions are needed.

What is the NKT Law?

The NKT Law proposes two interaction terms:

S₁ = x · p → Position-Momentum Interaction
S₂ = (dm/dt) · p → Inertia-Change-Momentum Interaction

Where:

x: displacement from reference
p = mv: linear momentum
dm/dt: rate of change of mass

These terms help describe the tendency of a system to move toward or away from equilibrium, even when mass is changing.

Alignment with Newton

The NKT Law assumes Newtonian principles still hold — p = mv, and motion arises from interactions of force and mass.

In fact:

S₁ = x · p behaves like a dynamical indicator of momentum spread relative to position (similar to energy gradient direction).
S₂ = (dm/dt) · p incorporates Newton’s own formulation of momentum (F = dp/dt) but in non-constant mass systems.

Thus, the NKT Law can be seen as a layer on top of Newton, not a contradiction.

Alignment with Kepler

Kepler described orbits as elliptical, but didn’t explain why speed changes near perihelion/aphelion.

With the NKT Law:

S₁ flips sign near turning points, indicating convergence or divergence in orbital motion.
Combined with momentum data, this matches the elliptical speed variance predicted by Kepler.

It offers a momentum-centric interpretation of Kepler’s area law.

Alignment with Euler

Euler’s equations are ideal for rotational systems and complex rigid bodies.

The NKT Law doesn’t deal with angular momentum directly, but S₁ = x · p can be interpreted geometrically when p is not aligned with x, yielding angular-like tendencies — without assuming torques.

It offers an accessible entry point to pre-angular dynamics before introducing full Euler formalism.

Why This Matters

Many real systems don’t fit neatly into Newton’s or Kepler’s assumptions:

Rockets expel mass (dm/dt ≠ 0)
Planets don’t move with constant speed
Biological systems grow or decay

The NKT Law doesn’t reject classical mechanics — it respects and extends it by handling variable-mass systems with clarity and minimal math.

Learn More

Thanks for reading. I’d love to hear your thoughts — especially how you see this model aligning (or diverging) from classical mechanics in your field.

This content originally appeared on DEV Community and was authored by Nguyen Khanh Tung