Mathematical Curves Shaping the Universe: From Classical Geometry to Astrophysics, Quantum Mechanics and High-Energy Physics

Mathematics provides powerful tools for understanding the universe. Some classical curves, like the Witch of Agnesi, the Cissoid of Diocles, and the Folium of Descartes, transcend their historical origins to find applications in quantum physics, astrophysics, and beyond. These curves help solve complex problems such as quantum resonances, relativistic trajectories, matter distributions in galactic collisions, and particle dynamics in quantum fields.

Below, we explore these connections with problem statements, analytical resolutions, and computational examples.

1. Witch of Agnesi: Resonances and Lorentzian Distributions

History and Properties

The Witch of Agnesi, defined by:

is renowned for its symmetry and smooth decay. These properties make it ideal for modeling Lorentzian distributions, which are essential in describing resonances in particle colliders and astrophysical systems.

Applications

1. Quantum Resonances: Describes resonance profiles in subatomic particles.
2. Perturbed Gravitational Lenses: Models angular perturbations in asymmetric gravitational lenses.

Problem 1.1: Analyzing Resonances in the LHC

Statement:

In a particle collider like the LHC, resonances with a central energy E0 and a width Γ can be modeled using the Witch of Agnesi:

where a is related to Γ as a=Γ/2.

Tasks:

1. Adjust a for Γ=2.5?GeV.
2. Find the energies E where P(E) drops to 50% of its maximum value.

Here is the fully revised and properly structured document, written in English with clear problem statements, solutions, and computational implementations.

Analytical Solution:

The relationship between a and Γ is:

a=Γ/2

To find the energies where P(E)=0.5?Pmax solve:

Thus, the energies are:

E=E0±2a

import numpy as np
import matplotlib.pyplot as plt

# Parameters
E0 = 91.2  # Central energy (GeV)
Gamma = 2.5  # Resonance width (GeV)
a = Gamma / 2  # Half-width parameter
E = np.linspace(85, 97, 1000)  # Energy range (GeV)

# Probability function
P = 8 * a**3 / (4 * a**2 + (E - E0)**2)

# Energies where P drops to 50%
P_max = np.max(P)
E_half_lower = E0 - 2 * a
E_half_upper = E0 + 2 * a

# Plot
plt.plot(E, P, label="P(E)")
plt.axvline(E_half_lower, color='r', linestyle='--', label=f'50% at {E_half_lower:.2f} GeV')
plt.axvline(E_half_upper, color='r', linestyle='--', label=f'50% at {E_half_upper:.2f} GeV')
plt.title("Z Boson Resonance")
plt.xlabel("Energy (GeV)")
plt.ylabel("Probability")
plt.legend()
plt.grid()
plt.show()        

Problem 1.2: Intensity in Binary Star Lenses

Statement:

In binary star systems, gravitational lenses can have intensity profiles approximated by the Witch of Agnesi.

Tasks:

1. Determine the maximum angular deviation induced by the lens.
2. Analyze how this deviation affects the detection of exoplanets.

Analytical Solution:

The angular deviation θ is proportional to the derivative of the Witch of Agnesi:

θ=?dP/dE

The maximum intensity occurs at E0, where dP/dE=0.

import numpy as np
import matplotlib.pyplot as plt

# Parameters
E0 = 1.5  # Central energy in arbitrary units
a = 0.5  # Width parameter
E = np.linspace(-3, 3, 1000)

# Intensity function
P = 8 * a**3 / (4 * a**2 + (E - E0)**2)

# Derivative of intensity
dP_dE = -16 * a**3 * (E - E0) / (4 * a**2 + (E - E0)**2)**2

# Plot
plt.plot(E, P, label="P(E)")
plt.plot(E, dP_dE, label="dP/dE", linestyle="--")
plt.axvline(E0, color='r', linestyle='--', label=f'Max at {E0}')
plt.title("Intensity in Gravitational Lenses")
plt.xlabel("Energy (arbitrary units)")
plt.ylabel("Intensity / Derivative")
plt.legend()
plt.grid()
plt.show()        

2. Cissoid of Diocles: Trajectories and Geodesics

History and Properties

The Cissoid of Diocles is defined as:

This curve is particularly useful in modeling complex trajectories under strong interactions or intense gravitational fields.

Applications

1. Quark Trajectories: Approximates trajectories in confinement potentials in quantum chromodynamics.
2. Relativistic Geodesics: Describes particle trajectories around compact objects like neutron stars or black holes.

Problem 2.1: Motion in Curved Fields Near Black Holes

Statement:

In Schwarzschild spacetime, a particle follows non-trivial trajectories influenced by the black hole's gravity. The effective potential is given by:

Tasks:

1. Calculate the effective potential for different radial distances r.
2. Determine how angular momentum L affects the trajectory.

Analytical Solution:

The terms in Veff represent:

• ?GM/r: Gravitational attraction.
• L^2}/{2r^2} - {GML^2}/{r^3}: Centrifugal and relativistic corrections due to angular momentum L.

import numpy as np
import matplotlib.pyplot as plt

# Parameters
G = 6.674e-11  # Gravitational constant (m^3/kg/s^2)
M = 1.989e30  # Solar mass (kg)
L = 4e10  # Angular momentum (m^2/s)
r = np.linspace(1e7, 1e10, 1000)  # Radial distance (m)

# Effective potential
V_eff = -G * M / r + L**2 / (2 * r**2) - G * M * L**2 / r**3

# Plot
plt.plot(r, V_eff)
plt.title("Effective Potential Around a Black Hole")
plt.xlabel("Radial Distance (m)")
plt.ylabel("V_eff (J)")
plt.grid()
plt.show()        

3. Advanced Astrophysics Problem: Galactic Collisions and Dark Matter Dynamics

Problem 3.1: Galactic Collisions and Angular Momentum Evolution

Statement:

During a galactic collision, the surface density can be modeled by a Folium of Descartes:

x^3+y^3=3axy

Tasks:

1. Compute the density evolution over time.
2. Determine the change in angular momentum of the system.

Analytical Solution:

The density evolution is described by:

Critical points are obtained by solving:

import numpy as np
import matplotlib.pyplot as plt

# Parameters
a = 10
Sigma_0 = 1
x = np.linspace(-20, 20, 100)
y = np.linspace(-20, 20, 100)
X, Y = np.meshgrid(x, y)

# Surface density
Sigma = Sigma_0 * np.exp(-(X**3 + Y**3) / (3 * a * X * Y))

# Plot
plt.contourf(X, Y, Sigma, levels=50, cmap="viridis")
plt.title("Surface Density in Galactic Collision")
plt.xlabel("x (kpc)")
plt.ylabel("y (kpc)")
plt.colorbar(label="Density")
plt.show()        

Problem 3.2: Dark Matter Filament Dynamics

Statement:

Dark matter forms filaments connecting galaxies. These filaments can be locally approximated by Folium profiles, which help model their gravitational impact and density evolution.

Tasks:

1. Model the gravitational impact of a dark matter filament on nearby galaxies.
2. Calculate the density evolution within a galaxy cluster.

Analytical Solution:

The gravitational impact of a dark matter filament can be modeled using the potential:

where Σ(x,y) is the surface density from the Folium of Descartes.

The density evolution in a galaxy cluster can be described by the time-dependent profile:

where λ is the decay rate.

import numpy as np
import matplotlib.pyplot as plt

# Parameters
a = 10
Sigma_0 = 1
lambda_decay = 0.01  # Decay rate
time = 10  # Time in arbitrary units
x = np.linspace(-20, 20, 100)
y = np.linspace(-20, 20, 100)
X, Y = np.meshgrid(x, y)

# Surface density at time t
Sigma = Sigma_0 * np.exp(-(X**3 + Y**3) / (3 * a * X * Y)) * np.exp(-lambda_decay * time)

# Gravitational potential
Phi = -Sigma / np.sqrt(X**2 + Y**2)
Phi[np.isinf(Phi)] = 0  # Handle singularity at the origin

# Plot density evolution
plt.contourf(X, Y, Sigma, levels=50, cmap="viridis")
plt.title(f"Density Evolution at t = {time}")
plt.xlabel("x (kpc)")
plt.ylabel("y (kpc)")
plt.colorbar(label="Density")
plt.show()

# Plot gravitational potential
plt.contourf(X, Y, Phi, levels=50, cmap="plasma")
plt.title("Gravitational Potential of Dark Matter Filament")
plt.xlabel("x (kpc)")
plt.ylabel("y (kpc)")
plt.colorbar(label="Potential")
plt.show()        

Conclusion

These problems demonstrate how classical curves like the Witch of Agnesi, Cissoid of Diocles, and Folium of Descartes can be applied to modern challenges in high-energy physics and astrophysics. They provide a bridge between historical mathematical tools and cutting-edge research, from resonances in colliders to dark matter dynamics and galactic interactions.

If you have additional ideas or applications for these curves, feel free to contribute to this ongoing exploration of geometry and physics.