Introducing the NKT Law – A proposal for position-dependent inertia

T

TheNKTLaw

Registered Member
Hi everyone,


I’d like to share a theoretical idea I’ve been working on, and I welcome all critical discussion. This concept is something I call the NKT Law (named from my initials), and it attempts to explore a new type of force interaction, where inertia is not fixed but instead varies with position.


In Newtonian mechanics, mass (and thus inertia) is constant. But what if this assumption could be relaxed? The idea behind the NKT Law is that motion is affected not just by force and momentum, but also by how momentum interacts with position itself.


The proposed formula:​


The NKT Law suggests a force that depends on two position–momentum interaction terms, as follows:


F = (m × v) × x + x × (m × v)


Where:


  • x is the position vector
  • v is velocity
  • m × v is linear momentum
  • F is the resulting force
  • "×" here is a symbolic placeholder for interaction (not necessarily a cross product)

This structure is intentionally symmetric: momentum interacts with position from both sides. The force is no longer a function of acceleration alone, but an emergent result of the way momentum and position overlap.


What’s new here?​


  • This form does not reduce to Newton’s second law (F = ma) in a straightforward way.
  • It also doesn’t rely on gradients or potential fields in the usual sense.
  • Instead, it tries to express how inertia itself might vary, especially in systems where mass is affected by position — like near gravitational extremes or quantum transitions.

I’m aware this is speculative, and I’m not claiming it as a verified model. It’s simply a new mathematical structure that I believe has not been tried before in this exact form.


References:​


For those interested, I’ve published a short preprint explaining this in more detail here:
https://doi.org/10.6084/m9.figshare.29389292


I welcome any feedback, positive or critical. My goal is not to push a theory, but to ask:


  • Is there physical meaning in this structure?
  • Has anything similar ever been proposed in established theories?
  • Could this be connected to deeper interactions in gravitational or inertial systems?

Thanks for your time and thoughts.


Nguyen Khanh Tung
 
TheNKTLaw said:
Is there physical meaning in this structure?
Click to expand...
Well, this was my first question so I'm glad you asked it too.

Does this idea have any physical meaning?
Does it manifest in an observable way?
Put differently: does your idea make any kind of testable predictions about movement that are different from our current model?
If so, have you done any tests to see if your idea's predictions manifest in the real world?
 
TheNKTLaw said:
Hi everyone,


I’d like to share a theoretical idea I’ve been working on, and I welcome all critical discussion. This concept is something I call the NKT Law (named from my initials), and it attempts to explore a new type of force interaction, where inertia is not fixed but instead varies with position.


In Newtonian mechanics, mass (and thus inertia) is constant. But what if this assumption could be relaxed? The idea behind the NKT Law is that motion is affected not just by force and momentum, but also by how momentum interacts with position itself.


The proposed formula:​


The NKT Law suggests a force that depends on two position–momentum interaction terms, as follows:


F = (m × v) × x + x × (m × v)


Where:


  • x is the position vector
  • v is velocity
  • m × v is linear momentum
  • F is the resulting force
  • "×" here is a symbolic placeholder for interaction (not necessarily a cross product)

This structure is intentionally symmetric: momentum interacts with position from both sides. The force is no longer a function of acceleration alone, but an emergent result of the way momentum and position overlap.


What’s new here?​


  • This form does not reduce to Newton’s second law (F = ma) in a straightforward way.
  • It also doesn’t rely on gradients or potential fields in the usual sense.
  • Instead, it tries to express how inertia itself might vary, especially in systems where mass is affected by position — like near gravitational extremes or quantum transitions.

I’m aware this is speculative, and I’m not claiming it as a verified model. It’s simply a new mathematical structure that I believe has not been tried before in this exact form.


References:​


For those interested, I’ve published a short preprint explaining this in more detail here:
https://doi.org/10.6084/m9.figshare.29389292


I welcome any feedback, positive or critical. My goal is not to push a theory, but to ask:


  • Is there physical meaning in this structure?
  • Has anything similar ever been proposed in established theories?
  • Could this be connected to deeper interactions in gravitational or inertial systems?

Thanks for your time and thoughts.


Nguyen Khanh Tung
Click to expand...
Your formula appears to be one that can be simplified to F=2mvx. This has dimensions that are inconsistent with a Newtonian definition of force, which would have dimensions ML/T² , rather than, as implied by your equation, ML²/T. So you are proposing an entirely different concept of force, requiring the whole of mechanics to be reworked.

The second point is you do not explain how you define x. I assume is it is linear distance from some origin, but where is the origin? I think you need to provide an example showing how this would be applied in practice, indicating where origin would be, for a given system.
 
DaveC426913 said:
Well, this was my first question so I'm glad you asked it too.

Does this idea have any physical meaning?
Does it manifest in an observable way?
Put differently: does your idea make any kind of testable predictions about movement that are different from our current model?
If so, have you done any tests to see if your idea's predictions manifest in the real world?
Click to expand...
Yes, the law will determine the position of the earth, or any planet in the future. Use Nasa data to verify.
 
exchemist said:
Your formula appears to be one that can be simplified to F=2mvx. This has dimensions that are inconsistent with a Newtonian definition of force, which would have dimensions ML/T² , rather than, as implied by your equation, ML²/T. So you are proposing an entirely different concept of force, requiring the whole of mechanics to be reworked.

The second point is you do not explain how you define x. I assume is it is linear distance from some origin, but where is the origin? I think you need to provide an example showing how this would be applied in practice, indicating where origin would be, for a given system.
Click to expand...
I want to upload the original pdf of the NKT law but the forum does not have a file attachment, I will copy it for you. The rest written on the forum are suggestions.
 
exchemist said:
Your formula appears to be one that can be simplified to F=2mvx. This has dimensions that are inconsistent with a Newtonian definition of force, which would have dimensions ML/T² , rather than, as implied by your equation, ML²/T. So you are proposing an entirely different concept of force, requiring the whole of mechanics to be reworked.

The second point is you do not explain how you define x. I assume is it is linear distance from some origin, but where is the origin? I think you need to provide an example showing how this would be applied in practice, indicating where origin would be, for a given system.
Click to expand...
I want to upload the original pdf of the NKT law but the forum does not have a file attachment, I will copy it for you. The rest written on the forum are suggestions.
exchemist said:
Your formula appears to be one that can be simplified to F=2mvx. This has dimensions that are inconsistent with a Newtonian definition of force, which would have dimensions ML/T² , rather than, as implied by your equation, ML²/T. So you are proposing an entirely different concept of force, requiring the whole of mechanics to be reworked.

The second point is you do not explain how you define x. I assume is it is linear distance from some origin, but where is the origin? I think you need to provide an example showing how this would be applied in practice, indicating where origin would be, for a given system.
Click to expand...
The NKT Law on Position and Varying Inertia Interaction

Nguyễn Khánh Tùng
Independent Researcher
Email: traiphieu.com@gmail.com
Website: https://traiphieu.com



​
In Tribute to Newton and Kepler

This work honors Isaac Newton and Johannes Kepler, whose foundational principles inspired the discovery of a new natural law connecting position and changing inertia.

​
Abstract

We propose the "NKT Law on Position and Varying Inertia Interaction", an empirical law that unifies and expands Newtonian dynamics and Keplerian motion. This law introduces two key product terms:

  • S₁ = x · p (position-momentum), and
  • S₂ = (dm/dt) · p (inertia-derivative-momentum),
which consistently predict the motion tendency of a body in systems ranging from oscillations and rockets to astronomical orbits.
The law has been verified using publicly available data from real-world phenomena for nearly 20 years. Graphical models based on this law show greater predictive accuracy than classical models.

​
1. Introduction

Nature often hides simplicity beneath apparent complexity. Inspired by Newton's mechanics and Kepler's planetary laws, we discovered a universal law that connects spatial displacement with varying inertia. This law, born from direct observation and systematic validation using real data, offers a unified tool to analyze motion in systems with changing mass or momentum.

​
2. Statement of the NKT Law

We define two fundamental terms:

  1. Position-Momentum Interaction:
    S₁ = x · p
  2. Mass Derivative-Momentum Interaction:
    S₂ = (dm/dt) · p
Where:

  • x: displacement or position relative to a reference point
  • p = mv: linear momentum
  • dm/dt: rate of change of mass
NKT Law:

The tendency of a system to move toward or away from equilibrium is governed by the signs and values of S₁ and S₂. The sign of these products indicates whether the system reinforces or resists motion.

​
3. Interpretation and Consequences

  • If S₁ > 0: The system is moving away from equilibrium (divergence)
  • If S₁ < 0: The system is returning toward equilibrium (convergence)
  • If S₂ > 0: The change in mass reinforces the motion (e.g., thrust phase)
  • If S₂ < 0: The change in mass resists motion (e.g., braking or atmospheric drag)
The combined behavior of these two products determines motion trends more precisely than Newton’s second law alone.

​
4. Simulation Cases

We use real datasets from oscillatory systems, spaceflight, and planetary motion to validate the NKT Law. The following graphs show time-evolving behavior of x(t), p(t), and corresponding products S₁(t) and S₂(t).

  • Figure 1: Harmonic Oscillator — Shows how x · p changes sign at turning points.
  • Figure 2: Rocket Launch Phase — Displays rising (dm/dt) · p during fuel burn.
  • Figure 3: Earth’s Orbit (based on NASA/ESA data) — Captures variations in x · p and inertia interaction during perihelion and aphelion.
​
5. Analogy: Spacecraft and Ocean Currents

Consider a spacecraft navigating like a boat on a sea. The engine’s power mimics (dm/dt) · p, and the position in the current reflects x · p.
This analogy illustrates how the NKT Law naturally governs both artificial and celestial navigation.

​
6. Broader Implications

This law invites reconsideration of:

  • Systems with mass fluctuations (e.g., rockets, evaporation, aggregation)
  • Entropy and energy flow in dynamic systems
  • Foundations of cosmology and potential link to color charge in particle physics
​
7. Conclusion

The NKT Law is not a human invention, but a rediscovery of what nature has always done.
Its accuracy and simplicity suggest that deeper truths remain to be uncovered.

“I did not invent it. I only wrote down what nature has been doing for billions of years.”
— Nguyễn Khánh Tùng

​
Submission Information

  • Name: Nguyễn Khánh Tùng
  • Affiliation: Independent Researcher
ORCID: 0009-0002-9877-4137

First publication: Nguyen Khanh Tung (2025). *The NKT Law on Position and Varying Inertia Interaction*. Zenodo. https://doi.org/10.5281/zenodo.15725784

Second publication: Nguyen Khanh Tung (2025). *The NKT Law on Position and Varying Inertia Interaction*. Figshare. https://doi.org/10.6084/m9.figshare.29389292
 
TheNKTLaw said:
I want to upload the original pdf of the NKT law but the forum does not have a file attachment, I will copy it for you. The rest written on the forum are suggestions.

The NKT Law on Position and Varying Inertia Interaction

Nguyễn Khánh Tùng
Independent Researcher
Email: traiphieu.com@gmail.com
Website: https://traiphieu.com



​

In Tribute to Newton and Kepler

This work honors Isaac Newton and Johannes Kepler, whose foundational principles inspired the discovery of a new natural law connecting position and changing inertia.

​

Abstract

We propose the "NKT Law on Position and Varying Inertia Interaction", an empirical law that unifies and expands Newtonian dynamics and Keplerian motion. This law introduces two key product terms:

  • S₁ = x · p (position-momentum), and
  • S₂ = (dm/dt) · p (inertia-derivative-momentum),
which consistently predict the motion tendency of a body in systems ranging from oscillations and rockets to astronomical orbits.
The law has been verified using publicly available data from real-world phenomena for nearly 20 years. Graphical models based on this law show greater predictive accuracy than classical models.

​

1. Introduction

Nature often hides simplicity beneath apparent complexity. Inspired by Newton's mechanics and Kepler's planetary laws, we discovered a universal law that connects spatial displacement with varying inertia. This law, born from direct observation and systematic validation using real data, offers a unified tool to analyze motion in systems with changing mass or momentum.

​

2. Statement of the NKT Law

We define two fundamental terms:

  1. Position-Momentum Interaction:
    S₁ = x · p
  2. Mass Derivative-Momentum Interaction:
    S₂ = (dm/dt) · p
Where:

  • x: displacement or position relative to a reference point
  • p = mv: linear momentum
  • dm/dt: rate of change of mass
NKT Law:

The tendency of a system to move toward or away from equilibrium is governed by the signs and values of S₁ and S₂. The sign of these products indicates whether the system reinforces or resists motion.

​

3. Interpretation and Consequences

  • If S₁ > 0: The system is moving away from equilibrium (divergence)
  • If S₁ < 0: The system is returning toward equilibrium (convergence)
  • If S₂ > 0: The change in mass reinforces the motion (e.g., thrust phase)
  • If S₂ < 0: The change in mass resists motion (e.g., braking or atmospheric drag)
The combined behavior of these two products determines motion trends more precisely than Newton’s second law alone.

​

4. Simulation Cases

We use real datasets from oscillatory systems, spaceflight, and planetary motion to validate the NKT Law. The following graphs show time-evolving behavior of x(t), p(t), and corresponding products S₁(t) and S₂(t).

  • Figure 1: Harmonic Oscillator — Shows how x · p changes sign at turning points.
  • Figure 2: Rocket Launch Phase — Displays rising (dm/dt) · p during fuel burn.
  • Figure 3: Earth’s Orbit (based on NASA/ESA data) — Captures variations in x · p and inertia interaction during perihelion and aphelion.
​

5. Analogy: Spacecraft and Ocean Currents

Consider a spacecraft navigating like a boat on a sea. The engine’s power mimics (dm/dt) · p, and the position in the current reflects x · p.
This analogy illustrates how the NKT Law naturally governs both artificial and celestial navigation.

​

6. Broader Implications

This law invites reconsideration of:

  • Systems with mass fluctuations (e.g., rockets, evaporation, aggregation)
  • Entropy and energy flow in dynamic systems
  • Foundations of cosmology and potential link to color charge in particle physics
​

7. Conclusion

The NKT Law is not a human invention, but a rediscovery of what nature has always done.
Its accuracy and simplicity suggest that deeper truths remain to be uncovered.

“I did not invent it. I only wrote down what nature has been doing for billions of years.”
— Nguyễn Khánh Tùng

​

Submission Information

  • Name: Nguyễn Khánh Tùng
  • Affiliation: Independent Researcher
ORCID: 0009-0002-9877-4137

First publication: Nguyen Khanh Tung (2025). *The NKT Law on Position and Varying Inertia Interaction*. Zenodo. https://doi.org/10.5281/zenodo.15725784

Second publication: Nguyen Khanh Tung (2025). *The NKT Law on Position and Varying Inertia Interaction*. Figshare. https://doi.org/10.6084/m9.figshare.29389292
Click to expand...
Where is the origin, in your determination of the value of x for a system? Give us a simple example to show how you apply this concept.
 
exchemist said:
Where is the origin, in your determination of the value of x for a system? Give us a simple example to show how you apply this concept.
Click to expand...
x value, simple example, suppose in January 2024 the earth is how many kilometers from the sun, in July 2024 how many kilometers. All of that data is published by NASA. Using the NKT law, I can predict how many kilometers the earth is from the sun in September 2030
 
TheNKTLaw said:
x value, simple example, suppose in January 2024 the earth is how many kilometers from the sun, in July 2024 how many kilometers. All of that data is published by NASA. Using the NKT law, I can predict how many kilometers the earth is from the sun in September 2030
Click to expand...
So are you saying x is the distance from the sun in that example? So what are S₁ and S₂ in that example and how does this lead to a prediction of distance in the future?
 
TheNKTLaw said:
Yes, the law will determine the position of the earth, or any planet in the future. Use Nasa data to verify.
Click to expand...
We can already do that.

It seems to be pretty accurately determinable with regular inertia.

Does NKT give different answers? Or better?
 
exchemist said:
So are you saying x is the distance from the sun in that example? So what are S₁ and S₂ in that example and how does this lead to a prediction of distance in the future?
Click to expand...
S1, S2 are not quantities, they are just products. I will publish data from NASA applying the NKT formula to calculate the future position of the earth. The data is completely consistent with NASA's real data.
 
DaveC426913 said:
We can already do that.

It seems to be pretty accurately determinable with regular inertia.

Does NKT give different answers? Or better?
Click to expand...
S1, S2 are not quantities, they are just products. I will publish data from NASA applying the NKT formula to calculate the future position of the earth. The data is completely consistent with NASA's real data.
 
TheNKTLaw said:
S1, S2 are not quantities, they are just products. I will publish data from NASA applying the NKT formula to calculate the future position of the earth. The data is completely consistent with NASA's real data.
Click to expand...
I really think it is time you answered a direct question. From where do you measure x, i.e. what do you take as the zero distance point?

And how will this work for objects in linear motion, for instance a ball dropped from a tower?
 
TheNKTLaw said:
The data is completely consistent with NASA's real data.
Click to expand...
Great. So the methods you use to determine the position of the Earth agree completely with the methods the rest of us (including NASA) use to determine the position of the Earth.

If your method and the currently-used method both get the same results, why do we need your method at all?
 
exchemist said:
I really think it is time you answered a direct question. From where do you measure x, i.e. what do you take as the zero distance point?

And how will this work for objects in linear motion, for instance a ball dropped from a tower?
Click to expand...
I take NASA data, let's say 2022, I calculate and predict the exact position of the earth in 2023, explain the unexplained kepler phenomena, like the speed of the earth approaching the sun but still decreasing
 
DaveC426913 said:
Great. So the methods you use to determine the position of the Earth agree completely with the methods the rest of us (including NASA) use to determine the position of the Earth.

If your method and the currently-used method both get the same results, why do we need your method at all?
Click to expand...
How do you explain that when the earth gets closer to the sun, its speed decreases?
 
TheNKTLaw said:
I take NASA data, let's say 2022, I calculate and predict the exact position of the earth in 2023, explain the unexplained kepler phenomena, like the speed of the earth approaching the sun but still decreasing
Click to expand...
You have not answered my questions. It looks as though you cannot do so. If you can’t then I’m afraid you have no theory.
 
TheNKTLaw:
TheNKTLaw said:
Hi everyone,


I’d like to share a theoretical idea I’ve been working on, and I welcome all critical discussion. This concept is something I call the NKT Law (named from my initials), and it attempts to explore a new type of force interaction, where inertia is not fixed but instead varies with position.
Click to expand...
How do you define the non-constant inertia quantity? You want to replace mass by something else? What?
TheNKTLaw said:
The proposed formula:

The NKT Law suggests a force that depends on two position–momentum interaction terms, as follows:

F = (m × v) × x + x × (m × v)
Click to expand...
As exchemist pointed out, if the multiplication symbols here indicate multiplication (or a cross product), then the quantity on the right-hand side doesn't have dimensions of a force. Instead, it has dimensions equivalent to angular momentum.
TheNKTLaw said:
  • "×" here is a symbolic placeholder for interaction (not necessarily a cross product)
Click to expand...
You haven't worked out what that placeholder is? How can you write down formulas involving it, then?

Also, it's good to hear that it's not a cross product. If it was, then F would always be zero, using your formula.

What is your placeholder operator, then?
TheNKTLaw said:
What’s new here?
  • This form does not reduce to Newton’s second law (F = ma) in a straightforward way.
Click to expand...
Right. But you're trying to redefine m, aren't you?
TheNKTLaw said:
  • It also doesn’t rely on gradients or potential fields in the usual sense.
Click to expand...
What does it rely on?
TheNKTLaw said:
  • Instead, it tries to express how inertia itself might vary, especially in systems where mass is affected by position — like near gravitational extremes or quantum transitions.
Click to expand...
You should start by defining your inertia quantity clearly. Then derive or define the force quantity and the law (if there is one).
 
Experimental Verification of the NKT Law Using NASA Data (2022–2023)​


Summary

The NKT Law is a new dynamical model that describes the motion trends of physical systems through two quantities:
S₁ = x•p (position–momentum interaction) and S₂ = (dm/dt)•p (varying-mass–momentum interaction).
This document presents the theoretical basis of the law, verifies it with NASA's 2022 data, and reasonably predicts Earth's orbital behavior in 2023.

Theoretical Basis

Definitions and physical relationships in the NKT Law:

  • x: distance from the object to a reference point (e.g., the Sun)
  • v: velocity of the object
  • m: mass of the object
  • p = m•v: linear momentum
  • S₁ = x•p: position–momentum interaction
  • S₂ = (dm/dt)•p: varying-mass–momentum interaction
 
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