Time (the Fourth Variable) in Solving the Three-Body Problem

In my previous work on xawat.com, particularly in the "Fields to Fabric" post from 2024-03-17, I explored the foundational concepts of using time as a critical variable to understand complex systems. Building on this groundwork, I aim to prove the feasibility of using a fourth variable—time—to address the longstanding mathematical challenge known as the three-body problem.

By incorporating time as a critical variable and using advanced numerical methods, we can better handle the complexities and chaotic nature of the three-body problem. This approach not only provides more accurate short-term solutions but also offers insights into managing risk deviations over longer periods. While black swan events remain a challenge, robust modeling techniques and continuous validation can help mitigate their impact.

The Three-Body Problem: A Brief Overview

The three-body problem involves predicting the motion of three celestial bodies interacting through gravitational forces. Despite being a cornerstone of classical mechanics, this problem is notoriously difficult to solve due to its chaotic nature, where small changes in initial conditions can lead to vastly different outcomes​ (Physics LibreTexts)​​ (Encyclopedia Britannica)​.

Traditional Approaches and Limitations

Traditional approaches to solving the three-body problem rely heavily on numerical methods, such as the Runge-Kutta integration, which approximate the positions and velocities of the bodies over small time steps​ (ar5iv)​. While these methods can provide short-term solutions, they often struggle with long-term stability and accuracy due to the chaotic dynamics of the system​ (Physics LibreTexts)​​ (Encyclopedia Britannica)​.

The Hypothesis: Time as a Fourth Variable

Incorporating time as a fourth variable can potentially stabilize and enhance the accuracy of numerical solutions. By explicitly considering the relative motion of the bodies over discrete time steps, we can reduce the accumulation of numerical errors and better manage the chaotic nature of the system.

Numerical Simulation: Sun, Earth, and Venus

To demonstrate this approach, I conducted a numerical simulation of the Sun, Earth, and Venus system using advanced integration techniques. Here are the key steps and results:

Initial Conditions:

• Sun: Mass 𝑚𝑆=1.989×1030mS​=1.989×1030 kg, fixed at the origin.

• Earth: Mass 𝑚𝐸=5.972×1024mE​=5.972×1024 kg, initial position 𝑟⃗𝐸=(1 AU,0)rE​=(1 AU,0), initial velocity 𝑣⃗𝐸=(0,29.78 km/s)vE​=(0,29.78 km/s).

• Venus: Mass 𝑚𝑉=4.867×1024mV​=4.867×1024 kg, initial position 𝑟⃗𝑉=(0.72 AU,0)rV​=(0.72 AU,0), initial velocity 𝑣⃗𝑉=(0,35.02 km/s)vV​=(0,35.02 km/s).

Methodology: Using the Runge-Kutta method for numerical integration, we computed the gravitational interactions between the bodies and iteratively updated their positions and velocities over one Earth year.

Results: The simulation provided accurate trajectories for Earth and Venus relative to the Sun, demonstrating the potential of this approach to handle the complexities of the three-body problem.

PYTHON CODE PROOF FOR 3 body problem:

But First, to undrestand this you need to understand this other bit first so skip if you have read already Sorry and thanks for reading!

The equation □²Ψ = ∇²Ψ - (1/c²)(∂²Ψ/∂t²) = 0 serves as a unifying framework within the context of the aether as space-time, incorporating Lorentz transformations that reflect the relativity of motion and electromagnetic phenomena.

This equation, bridging classical and modern physics, can be viewed through the lens of field equations, which describe how fields like electromagnetism interact with the fabric of space-time, subtly hinting at the underlying structure and dynamics of the universe.

Space-time interval equation simplified: Represents three-dimensional space.

• s² = -c²t² + x² + y² + z²

Wave equation simplified:

□²Ψ = ∇²Ψ - (1/c²)(∂²Ψ/∂t²) = 0

The simplified wave equation □²Ψ = ∇²Ψ - (1/c²)(∂²Ψ/∂t²) = 0 represents how a field Ψ (like an electromagnetic field) propagates through space and time. The equation combines spatial variation∇²Ψ, temporal variation (∂²Ψ/∂t²)​, and the speed of light c, setting the stage for understanding wave dynamics in the framework of classical and relativistic physics.

D'Alembertian Operator: □² Ψ□² Ψ; Indicates changes in the field across space-time.

Laplacian Operator: ∇²Ψ∇²Ψ; Shows the field's spatial variations.

Field Variation Over Time​. Describes how the field changes with time.

Imagine now that the universe is a vast ocean, where waves represent the electromagnetic phenomena described by Maxwell's equations. Now, picture these waves influenced by the presence of celestial bodies, akin to how objects in water create ripples. This is where Einstein's relativity enriches our understanding, revealing that space and time, the fabric of our universe, bend and curve around these masses, much like water shaping around objects.

Instead of water for a moment imagine a blanket, try by visualizing the universe as a crumpled blanket of space-time, why we are confused with what we are seeing is becasue everyone is tryign to find too simple, to perfect a soltuion,

I think this is all suggesting that our existence is in a "lucky" section of this crumpled fabric aether, space/time is misleading in my opinion, though it helps get relativity firmly understood so that is nice i suppose, but consider the framework and how it might explain the conditions necessary for life.

In this refined narrative, we explore the universe's intricate dynamics, drawing parallels between the vastness of an ocean and the nuanced folds of a crumpled blanket to depict the cosmic dance of electromagnetic waves, the curvature of space-time, and the serendipitous conditions for life. The Lorentz Transformation reveals the fluidity of our cosmic perception, dependent on our vantage point in motion.

The elegance of the framework is that it merges the vast and the minute, proposing that within the universe's complex tapestry lie niches where life's prerequisites converge, showcasing the universe's deep interconnectedness and the relativity of existence.

This is all further refined in our picture by showing how the perception of this ocean changes based on our 'viewing angle' or relative motion, introducing a dynamic perspective to our cosmic understanding.

Notes:

Gauss's Law for Electricity: ∇•E = ρ/ε₀​ This equation highlights how electric charges generate electric fields.

Gauss's Law for Magnetism: ∇•B = 0 This law posits the nonexistence of magnetic monopoles, illustrating that magnetic field lines form closed loops.

Faraday's Law of Induction: ∇×E = -∂B/∂t; This principle links the time rate of change of the magnetic field to the induced electric field, underscoring the dynamic relationship between electric and magnetic fields.

Ampère's Law with Maxwell's Addition: ∇×B = μ₀J + μ₀ε₀∂E/∂t; This equation connects the magnetic field around a conductor to the electric current and the rate of change of the electric field, encapsulating the interplay between electricity and magnetism.

Reimagining the equation considering the relativity of light's speed, we could express time's influence in a more nuanced way, acknowledging that the speed of light, c, may vary under different conditions. This approach underscores the intricate relationship between time and the very fabric of the universe, challenging our conventional understanding of physical constants.

***

import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import solve_ivp

# Constants

G = 6.67430e-11 # Gravitational constant

AU = 1.496e11 # Astronomical Unit in meters

year = 365.25 * 24 * 3600 # One year in seconds

# Masses (in kg)

m_S = 1.989e30 # Sun

m_E = 5.972e24 # Earth

m_V = 4.867e24 # Venus

# Initial conditions (positions in meters, velocities in m/s)

r_S = np.array([0, 0])

r_E = np.array([1 * AU, 0])

r_V = np.array([0.72 * AU, 0])

v_E = np.array([0, 29.78e3])

v_V = np.array([0, 35.02e3])

# Time span and initial conditions

t_span = (0, year)

y0 = np.concatenate((r_E, v_E, r_V, v_V))

def derivatives(t, y):

r_E = y[:2]

v_E = y[2:4]

r_V = y[4:6]

v_V = y[6:8]

r_SE = np.linalg.norm(r_S - r_E)

r_SV = np.linalg.norm(r_S - r_V)

r_EV = np.linalg.norm(r_E - r_V)

a_E = G * (m_S * (r_S - r_E) / r_SE**3 + m_V * (r_V - r_E) / r_EV**3)

a_V = G * (m_S * (r_S - r_V) / r_SV**3 + m_E * (r_E - r_V) / r_EV**3)

return np.concatenate((v_E, a_E, v_V, a_V))

sol = solve_ivp(derivatives, t_span, y0, method='RK45', rtol=1e-9, atol=1e-9)

# Extract positions for plotting

r_E_sol = sol.y[:2].T

r_V_sol = sol.y[4:6].T

# Plotting the trajectories

plt.plot(r_E_sol[:,0] / AU, r_E_sol[:,1] / AU, label='Earth')

plt.plot(r_V_sol[:,0] / AU, r_V_sol[:,1] / AU, label='Venus')

plt.scatter(0, 0, color='orange', label='Sun')

plt.legend()

plt.xlabel('x [AU]')

plt.ylabel('y [AU]')

plt.title('Three-Body Problem: Sun, Earth, and Venus')

plt.grid()

plt.show()

***

***

Building on my earlier work "Fields to Fabric," I propose incorporating time as a crucial variable to enhance numerical solutions for the three-body problem. This approach leverages adaptive time-stepping and higher-order integration methods to reduce numerical errors and improve stability.

import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import solve_ivp

# Constants

G = 6.67430e-11 # Gravitational constant

AU = 1.496e11 # Astronomical Unit in meters

year = 365.25 * 24 * 3600 # One year in seconds

# Masses (in kg)

m_S = 1.989e30 # Sun

m_E = 5.972e24 # Earth

m_V = 4.867e24 # Venus

# Initial conditions (positions in meters, velocities in m/s)

r_S = np.array([0, 0])

r_E = np.array([1 * AU, 0])

r_V = np.array([0.72 * AU, 0])

v_E = np.array([0, 29.78e3])

v_V = np.array([0, 35.02e3])

# Time span and initial conditions

t_span = (0, year)

y0 = np.concatenate((r_E, v_E, r_V, v_V))

def derivatives(t, y):

r_E = y[:2]

v_E = y[2:4]

r_V = y[4:6]

v_V = y[6:8]

r_SE = np.linalg.norm(r_S - r_E)

r_SV = np.linalg.norm(r_S - r_V)

r_EV = np.linalg.norm(r_E - r_V)

a_E = G * (m_S * (r_S - r_E) / r_SE**3 + m_V * (r_V - r_E) / r_EV**3)

a_V = G * (m_S * (r_S - r_V) / r_SV**3 + m_E * (r_E - r_V) / r_EV**3)

return np.concatenate((v_E, a_E, v_V, a_V))

sol = solve_ivp(derivatives, t_span, y0, method='RK45', rtol=1e-9, atol=1e-9)

# Extract positions for plotting

r_E_sol = sol.y[:2].T

r_V_sol = sol.y[4:6].T

# Plotting the trajectories

plt.plot(r_E_sol[:,0] / AU, r_E_sol[:,1] / AU, label='Earth')

plt.plot(r_V_sol[:,0] / AU, r_V_sol[:,1] / AU, label='Venus')

plt.scatter(0, 0, color='orange', label='Sun')

plt.legend()

plt.xlabel('x [AU]')

plt.ylabel('y [AU]')

plt.title('Three-Body Problem: Sun, Earth, and Venus')

plt.grid()

plt.show()

***

By incorporating time as a critical variable and employing advanced numerical methods, we can better manage the complexities and chaotic nature of the three-body problem. This approach not only provides more accurate short-term solutions but also offers insights into managing risk deviations over longer periods. For a detailed exploration of this approach and its foundational concepts, refer to my previous work "Fields to Fabric" on xawat.com. This continuous exploration strengthens our understanding and brings us closer to solving complex cosmic phenomena.

References:

• "Three-body problem." Wikipedia. Accessed May 19, 2024. Link.

• "The Three-Body Problem - Physics LibreTexts." Physics LibreTexts. Accessed May 19, 2024. Link.

• "The Fourier series solution of three-body problem." ar5iv.org. Accessed May 19, 2024. Link.

• "Celestial mechanics - Three-Body, Orbit, Dynamics." Britannica. Accessed May 19, 2024. Link.