About the dissipative Newton equation

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a child on a swing. In the perfect, ideal world of classical physics (the kind taught in high school), if you give the swing a push, it would swing back and forth forever, never slowing down, unless you explicitly added a "friction" rule to stop it. In this ideal world, the swing's momentum (how hard it's moving) is directly tied to its speed. Double the speed, double the momentum. Simple, right?

But in the real world, things slow down. Air resistance, friction at the pivot, and other invisible forces eat up energy. This is called dissipation.

This paper, titled "About the Dissipative Newton Equation," asks a bold question: What if dissipation isn't just an annoying extra rule we add to physics, but the fundamental rule itself? What if "perfect" motion is just a special, frictionless limit of a much messier, more realistic reality?

Here is the breakdown of the paper's ideas using everyday analogies:

1. The Thermodynamic "Rulebook"

The author, P. Ván, argues that we should look at physics through the lens of Thermodynamics (the study of heat and energy flow), specifically the Second Law of Thermodynamics. This law says that in any real process, "disorder" (entropy) must increase.

Think of it like a bank account. In an ideal world, you can move money back and forth without losing a cent. In the real world, every transaction has a fee. The author suggests that the laws of motion (Newton's laws) should be derived from this "fee" (entropy production) rather than starting with the ideal laws and trying to tack on friction later.

2. The "Dual Variable" Trick

To make this work, the author uses a clever mathematical trick called "Dual Internal Variables."

The Analogy: Imagine you are trying to describe a car. Usually, you just say "It is moving at 60 mph."
The Twist: The author says, "No, let's describe it with two things: its speed and a hidden 'memory' of the forces acting on it."
By treating the position and momentum as these two linked variables, the math naturally produces both the smooth, ideal motion and the messy, slowing-down motion in one single equation.

3. The Big Surprise: Momentum is "Sticky"

In standard physics, Momentum = Mass × Speed. It's a straight line.
In this new theory, the author finds that Momentum is not just about speed. It also depends on the Force pushing or pulling the object.

The Metaphor: Imagine you are pushing a heavy box across a floor.
- Old View: The harder you push, the faster it goes. The "heaviness" (momentum) is just how fast it's moving.
- New View: The "heaviness" of the box actually changes slightly depending on how hard you are pushing it. If you push harder, the box feels slightly "heavier" or "lighter" in a way that isn't just about its speed.
- Why? Because the act of pushing creates a tiny bit of "heat" or internal friction that changes how the object carries its momentum.

4. The "Spring" Prediction

The most exciting part of the paper is a specific prediction that can be tested in a lab.
The author predicts that the damping (how fast a swinging object slows down) doesn't just depend on the air or the material. It depends on a specific combination of:

The Mass of the object.
The Stiffness of the spring (or the force) holding it.

The Analogy: Imagine two pendulums. One is heavy and attached to a stiff spring; the other is light and attached to a loose spring.
- Standard Physics: The damping is just a constant number based on the air.
- This Paper's Prediction: The heavy/stiff pendulum will slow down at a different rate than the light/loose one, not just because of the air, but because the mass and the spring stiffness interact in a new way. The "friction" changes based on how hard the spring is pulling.

5. The Experiment: The Twisting Balance

To prove this, the author designed a special experiment using a Torsion Balance (a device that twists like a hanging door).

The Setup: It's a balance beam with weights that can slide in and out. This allows the scientists to change the Moment of Inertia (how hard it is to spin the beam) without changing the total weight.
The Goal: They will spin the beam and measure exactly how fast it slows down. If the slowing-down rate changes in the specific way predicted by the math (depending on the mass and the spring constant), it proves that Newton's laws need this "thermodynamic update."

6. Why Does This Matter?

Currently, physicists often treat "ideal motion" and "friction" as two separate things.

Ideal: Perfect, no energy loss.
Real: Add a "friction term" to the equation.

This paper suggests that Ideal Motion is just a special case where the friction is zero. By starting with the "Real" (thermodynamic) view, we might discover new effects that standard physics misses. It's like realizing that "perfectly smooth ice" is just a special case of "slippery ground," and by studying the ground, you understand the ice better.

Summary

The paper argues that Newton's laws are incomplete because they ignore the fundamental role of entropy (disorder). By fixing this, the author predicts that momentum is "force-dependent" and that friction behaves differently than we thought, specifically depending on the mass and the spring stiffness of the system. They have built a machine to test this, hoping to find a tiny, new "glitch" in the laws of physics that confirms the universe is more thermodynamic than we thought.

1. Problem Statement

The paper addresses a fundamental theoretical gap in classical mechanics: the lack of a unified framework that naturally incorporates dissipation (entropy production) alongside inertia (conservative dynamics).

Current Limitations: Standard mechanics relies on Hamiltonian variational principles, which describe ideal, non-dissipative systems. Dissipation is typically added as an ad-hoc extension (e.g., friction terms) rather than derived from first principles.
The Conflict: Thermodynamic evolution (relaxational, driven by entropy maximization) and mechanical evolution (inertial, driven by momentum conservation) are often treated as incompatible. Existing attempts to unify them (e.g., GENERIC framework) often require doubling the theoretical structure or adding independent postulates.
Core Question: Can ideal Newtonian mechanics be viewed as a specific limit (zero dissipation) of a more general, thermodynamically consistent theory? If so, does this general theory predict novel, experimentally testable effects not present in standard mechanics?

2. Methodology

The author employs the Method of Dual Internal Variables within the framework of Extended Thermodynamics.

State Variables: The system is described by two internal variables: position ( $x$ ) and momentum ( $p$ ). These are treated as thermodynamic state variables.
Entropy Function: The entropy $S$ is defined as a function of total energy $E$ , position, and momentum: $S(E, x, p)$ . The internal energy $U$ is defined as the difference between total energy and the Hamiltonian ( $U = E - H$ ).
Second Law Constraint: The evolution equations for $\dot{x}$ and $\dot{p}$ are derived strictly by enforcing the Second Law of Thermodynamics (non-negative entropy production rate, $\dot{S} \geq 0$ ).
Linear Constitutive Relations: The evolution equations are formulated as linear relations between the fluxes ( $\dot{x}, \dot{p}$ $\overset{x}{˙}, \overset{p}{˙}$ ) and the thermodynamic forces (gradients of the Hamiltonian). This introduces a phenomenological matrix $L$ $L$ containing coefficients ( $l_1, l_2, l, k$ $l_{1}, l_{2}, l, k$ ).
- The symmetric part of $L$ governs dissipation.
- The antisymmetric part governs the ideal (Hamiltonian) evolution.
Derivation: The author derives the general equations of motion and then analyzes specific limits (e.g., setting specific coefficients to zero) to recover known mechanics and identify new terms.

3. Key Contributions

The paper makes several theoretical and predictive contributions:

A. Thermodynamic Derivation of Newton's Equation

The author demonstrates that the standard Newtonian equation ( $m\ddot{x} = F$ ) emerges naturally as the zero-dissipation limit of a general thermodynamic evolution equation. This unifies inertial and relaxational dynamics under a single principle (the Second Law) without requiring separate variational principles.

B. Novel Dissipative Momentum

A critical finding is that in the general dissipative framework, momentum is not strictly proportional to velocity.

The derived relation is: $p = m\dot{x} + \text{dissipative terms}$ .
Specifically, the momentum acquires a force-dependent component ( $l_1 F$ ). This term has no counterpart in standard Hamiltonian mechanics.
Equation (13) in the paper: $p(\dot{x}, x) = m\dot{x} + \frac{l_1 V'}{k-l}$ .

C. Modified Damping Coefficient

When applied to a harmonic oscillator, the theory predicts a damping coefficient that depends linearly on both the inertial mass ( $m$ ) and the spring constant ( $D$ ).

Standard mechanics predicts damping proportional to velocity ( $\gamma \dot{x}$ ) where $\gamma$ is a constant.
The thermodynamic prediction (Eq. 38): The effective damping term is $(m \hat{D} a + b)\dot{x}$ .
This implies that the damping behavior changes based on the stiffness of the system and the mass, a feature absent in classical idealized models.

D. Recovery of Known Equations

The framework successfully recovers several known equations as special cases:

Standard Damped Mechanics: Recovered when $l_1 = 0$ .
Eliezer-Ford-O'Connell (EFO) Equation: Recovered when $l_2 = 0$ . This equation describes radiation reaction and is known for eliminating "runaway" solutions in electrodynamics. The paper shows this arises naturally from thermodynamic constraints ( $l_2=0$ ) without invoking electrodynamics.

4. Results

General Equation of Motion: The derived equation of motion (Eq. 14) is:
$m\ddot{x} + (ml_1 V'' + l_2)\dot{x} + (l_1 l_2 - l_2 + k^2)V' = 0$
This reveals that the damping coefficient depends on the potential curvature ( $V''$ ) and mass, while the restoring force is enhanced by dissipative coefficients.
Stability: The thermodynamic constraints ensure that the equilibrium of the system is asymptotically stable, provided the coefficients satisfy specific inequalities (positive definiteness of the symmetric part of matrix $L$ ).
Experimental Design: The author proposes a specific experimental setup to test the prediction that damping depends on mass and spring constant.
- Device: A Bessel-type torsion balance with variable moment of inertia (masses movable along the arms).
- Goal: Measure the damping coefficient while varying the mass distribution and spring constant to verify the linear dependence predicted by Eq. (38).

5. Significance

Foundational Shift: The paper challenges the view that dissipation is merely an "extension" of mechanics. Instead, it posits that mechanics is the "marginal case" of a broader thermodynamic reality.
Predictive Power: Unlike many thermodynamic theories which are purely phenomenological, this framework generates a quantitative, experimentally testable prediction regarding the dependence of damping on mass and stiffness.
Unification: It provides a constructive method to derive evolution equations for systems with inertia using only the Second Law, avoiding the need for complex microscopic models or ad-hoc variational modifications.
Micro-Macro Bridge: The paper argues that while microscopic mechanisms (collisions, viscosity) explain standard damping, the novel force-dependent momentum term ( $l_1$ ) is a genuinely macroscopic thermodynamic prediction that may arise from memory effects, delays, or non-local couplings, regardless of the specific microscopic origin.

In summary, P. Ván presents a rigorous thermodynamic reconstruction of classical mechanics that predicts a force-dependent correction to momentum and a mass/spring-dependent damping effect, offering a pathway to experimentally validate the thermodynamic basis of Newton's laws.