Physics:Newton and Binet-Obidi Equation in the Theory of Entropicity(ToE)

Introduction

In the Theory of Entropicity(ToE), as first formulated and developed by John Onimisi Obidi, motion is not driven by fundamental forces but emerges from gradients in a scalar entropy field \( S(x^\mu) \). This article presents the entropic analogs of Newtonian trajectory equations, including orbital motion, relativistic paths, and irreversible corrections due to the Entropic Time Limit (ETL).

Nonrelativistic Entropic Motion

Entropic Lagrangian

A particle of mass \( m \) moving in an entropic field \( S(x,t) \) has the Lagrangian:

[math]\displaystyle{ L = \tfrac{1}{2} m \dot{\boldsymbol{x}}^{\,2} - m\,S(\boldsymbol{x},t) }[/math]

Equation of Motion

Euler–Lagrange yields:

[math]\displaystyle{ m\,\ddot{\boldsymbol{x}}(t) = -\,m\,\nabla S(\boldsymbol{x},t) }[/math]

Hamiltonian Form

[math]\displaystyle{ H = \tfrac{1}{2m}\,\boldsymbol{p}^{2} + m\,S(\boldsymbol{x},t) }[/math]

[math]\displaystyle{ \dot{\boldsymbol{x}} = \frac{\partial H}{\partial \boldsymbol{p}},\quad \dot{\boldsymbol{p}} = -\frac{\partial H}{\partial \boldsymbol{x}} }[/math]

Constant Entropic Gradient

If \( \nabla S = -\boldsymbol{a}_E = \text{const} \), then:

[math]\displaystyle{ \boldsymbol{x}(t) = \boldsymbol{x}_0 + \boldsymbol{v}_0 t + \tfrac{1}{2} \boldsymbol{a}_E t^2 }[/math]

[math]\displaystyle{ \boldsymbol{v}(t) = \boldsymbol{v}_0 + \boldsymbol{a}_E t }[/math]

Central Entropic Field and Orbital Motion

Let \( S = S(r) \), with \( r = |\boldsymbol{x}| \). The entropic force is:

[math]\displaystyle{ \boldsymbol{F}_{\text{ent}} = -m\,S'(r)\,\hat{\boldsymbol{r}} }[/math]

Angular Momentum

[math]\displaystyle{ h = r^2 \dot{\phi} = \text{constant} }[/math]

Binet–Obidi Equation

In terms of \( u(\phi) = 1/r \), the famous Binet-Obidi Equation in the Theory of Entropicity(ToE) is given as:

[math]\displaystyle{ \frac{d^2 u}{d\phi^2} + u = \frac{S'(r)}{h^2 u^2}\Big|_{r=1/u} }[/math]

Example: Inverse-Radius Field

If \( S(r) = -\frac{G M}{r} \), then:

[math]\displaystyle{ \frac{d^2 u}{d\phi^2} + u = \frac{G M}{h^2} }[/math]

Solution:

[math]\displaystyle{ r(\phi) = \frac{p}{1 + e \cos(\phi - \phi_0)},\quad p = \frac{h^2}{G M} }[/math]

Relativistic Entropic Trajectories

Worldline Action

[math]\displaystyle{ S_{\text{p}} = -m \int d\tau - m \int S(x(\tau))\,d\tau }[/math]

Equation of Motion

[math]\displaystyle{ m\,\frac{D u^\mu}{D\tau} = -m\,\left(\delta^\mu_\nu + u^\mu u_\nu\right)\nabla^\nu S }[/math]

Irreversibility and ETL Corrections

ToE introduces a minimal time scale \( \tau_E \) for entropy exchange:

[math]\displaystyle{ \tau_E = \sqrt{\frac{\lambda}{k_B^2 \langle (\nabla S)^2 \rangle}} }[/math]

Rayleigh Dissipation Model

Modified equation:

[math]\displaystyle{ m\,\ddot{\boldsymbol{x}} = -m\,\nabla S - \frac{m}{\tau_E}\,\dot{\boldsymbol{x}} }[/math]

Closed Form for Constant Gradient

[math]\displaystyle{ v(t) = (v_0 - a_E \tau_E) e^{-t/\tau_E} + a_E \tau_E }[/math]

[math]\displaystyle{ x(t) = x_0 + (v_0 - a_E \tau_E)\tau_E(1 - e^{-t/\tau_E}) + a_E \tau_E t }[/math]

Non-Markovian Memory Kernel

Generalized equation:

[math]\displaystyle{ m\,\ddot{\boldsymbol{x}}(t) = -m\,\nabla S(\boldsymbol{x},t) - \int_0^t \Gamma(t - t')\,\dot{\boldsymbol{x}}(t')\,dt' }[/math]

Example kernel:

[math]\displaystyle{ \Gamma(\Delta t) = \frac{m}{\tau_E^2} e^{-\Delta t/\tau_E} \Theta(\Delta t) }[/math]

Summary Table

Concept	Expression	Role
Entropic force	[math]\displaystyle{ \boldsymbol{F}_{\text{ent}} = -m\,\nabla S }[/math]	Drives motion
Equation of motion	[math]\displaystyle{ m\,\ddot{\boldsymbol{x}} = -m\,\nabla S }[/math]	Nonrelativistic dynamics
Orbital equation	[math]\displaystyle{ \frac{d^2 u}{d\phi^2} + u = \cdots }[/math]	Central field trajectories
Relativistic motion	[math]\displaystyle{ \frac{D u^\mu}{D\tau} = -(\perp^\mu{}_\nu)\nabla^\nu S }[/math]	Worldline evolution
ETL correction	[math]\displaystyle{ \ddot{\boldsymbol{x}} + \frac{1}{\tau_E} \dot{\boldsymbol{x}} }[/math]	Irreversibility
Memory kernel	[math]\displaystyle{ \int \Gamma(t - t') \dot{\boldsymbol{x}}(t') dt' }[/math]	Nonlocal entropic delay

See Also

• Physics:Theory of Entropicity
• Physics:Entropic gravity
• Physics:Information geometry
• Physics:General relativity
• Physics:Rayleigh dissipation
• Physics:Binet equation