Why Does The Hamiltonian Of RK4 And Leapfrog Method Differ So Much In 3-body Problem? Chaotic Orbit Peaks? Periodic Orbit?

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**Why does the Hamiltonian of RK4 and leapfrog method differ so much in 3-body problem? Chaotic orbit peaks? Periodic orbit?**

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Introduction

The 3-body problem is a fundamental problem in physics and astronomy that involves the motion of three celestial bodies interacting with each other through gravity. It is a classic example of a chaotic system, where small changes in the initial conditions can lead to drastically different outcomes. In this article, we will explore the differences in the Hamiltonian of the RK4 and leapfrog methods in simulating the 3-body problem, and discuss the implications of these differences on the resulting orbits.

What is the 3-body problem?

The 3-body problem is a mathematical problem that involves the motion of three celestial bodies, such as planets, stars, or galaxies, interacting with each other through gravity. The problem is to determine the motion of the three bodies over time, given their initial positions, velocities, and masses. The 3-body problem is a classic example of a chaotic system, where small changes in the initial conditions can lead to drastically different outcomes.

What is the Hamiltonian?

The Hamiltonian is a fundamental concept in physics that describes the total energy of a system. It is a function of the generalized coordinates and momenta of the system, and it is used to derive the equations of motion. In the context of the 3-body problem, the Hamiltonian is a function of the positions, velocities, and masses of the three bodies.

RK4 Method

The RK4 (Runge-Kutta 4th order) method is a numerical method used to solve ordinary differential equations (ODEs). It is a popular method for simulating the motion of celestial bodies, such as planets and stars. The RK4 method uses a 4th order Taylor series expansion to approximate the solution of the ODE.

Leapfrog Method

The leapfrog method is another numerical method used to solve ODEs. It is a simple and efficient method that uses a second order Taylor series expansion to approximate the solution of the ODE. The leapfrog method is often used in simulations of celestial mechanics, such as the motion of planets and stars.

Hamiltonian of RK4 and Leapfrog Methods

The Hamiltonian of the RK4 and leapfrog methods differ significantly in the 3-body problem. The RK4 method uses a 4th order Taylor series expansion, which results in a more accurate approximation of the Hamiltonian. In contrast, the leapfrog method uses a second order Taylor series expansion, which results in a less accurate approximation of the Hamiltonian.

Chaotic Orbit Peaks

Chaotic orbit peaks are a characteristic feature of chaotic systems, such as the 3-body problem. They occur when the system exhibits a sudden and dramatic change in behavior, often resulting in a large increase in energy. Chaotic orbit peaks are often associated with the presence of a saddle point in the Hamiltonian, which is a point where the system exhibits a sudden change in behavior.

Periodic Orbit

Periodic orbits are a type of orbit that repeats itself over time. They are often associated with the presence of a stable fixed point in the Hamiltonian, which is a point where the system exhibits a stable behavior. Periodic orbits are often used as a benchmark for testing the accuracy of numerical methods, such as the RK4 and leapfrog methods.

Q&A

Q: What is the main difference between the RK4 and leapfrog methods in simulating the 3-body problem? A: The main difference between the RK4 and leapfrog methods is the order of the Taylor series expansion used to approximate the solution of the ODE. The RK4 method uses a 4th order Taylor series expansion, while the leapfrog method uses a second order Taylor series expansion.

Q: Why does the Hamiltonian of the RK4 and leapfrog methods differ so much in the 3-body problem? A: The Hamiltonian of the RK4 and leapfrog methods differs significantly in the 3-body problem because of the difference in the order of the Taylor series expansion used to approximate the solution of the ODE. The RK4 method uses a more accurate approximation of the Hamiltonian, while the leapfrog method uses a less accurate approximation.

Q: What is the significance of chaotic orbit peaks in the 3-body problem? A: Chaotic orbit peaks are a characteristic feature of chaotic systems, such as the 3-body problem. They occur when the system exhibits a sudden and dramatic change in behavior, often resulting in a large increase in energy. Chaotic orbit peaks are often associated with the presence of a saddle point in the Hamiltonian.

Q: What is the significance of periodic orbits in the 3-body problem? A: Periodic orbits are a type of orbit that repeats itself over time. They are often associated with the presence of a stable fixed point in the Hamiltonian, which is a point where the system exhibits a stable behavior. Periodic orbits are often used as a benchmark for testing the accuracy of numerical methods, such as the RK4 and leapfrog methods.

Conclusion

In conclusion, the Hamiltonian of the RK4 and leapfrog methods differ significantly in the 3-body problem. The RK4 method uses a more accurate approximation of the Hamiltonian, while the leapfrog method uses a less accurate approximation. Chaotic orbit peaks and periodic orbits are characteristic features of chaotic systems, such as the 3-body problem. Understanding the differences between the RK4 and leapfrog methods is essential for simulating the motion of celestial bodies, such as planets and stars.

References

  • [1] Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations II: Stiff and differential-algebraic problems. Springer.
  • [2] Leimkuhler, B., & Reich, S. (2004). Simulating Hamiltonian dynamics. Cambridge University Press.
  • [3] Sussman, G. J., & Wisdom, J. (1992). Chaotic evolution of the solar system. Science, 257(5073), 56-62.

Code

import numpy as np
from scipy.integrate import odeint

def hamiltonian(q, p):
m1, m2, m3 = 1.0, 1.0, 1.0
r1, r2, r3 = q[0], q[1], q[2]
v1, v2, v3 = p[0], p[1], p[2]
U = -m1 * m2 / np.linalg.norm(r1 - r2) - m1 * m3 / np.linalg.norm(r1 - r3) - m2 * m3 / np.linalg.norm(r2 - r3)
H = 0.5 * (m1 * v12 + m2 * v22 + m3 * v3**2) + U
return H



def equations_of_motion(q, p, t):
m1, m2, m3 = 1.0, 1.0, 1.0
r1, r2, r3 = q[0], q[1], q[2]
v1, v2, v3 = p[0], p[1], p[2]
a1 = -m2 * (r1 - r2) / np.linalg.norm(r1 - r2)**3 - m3 * (r1 - r3) / np.linalg.norm(r1 - r3)**3
a2 = -m1 * (r2 - r1) / np.linalg.norm(r2 - r1)**3 - m3 * (r2 - r3) / np.linalg.norm(r2 - r3)**3
a3 = -m1 * (r3 - r1) / np.linalg.norm(r3 - r1)**3 - m2 * (r3 - r2) / np.linalg.norm(r3 - r2)**3
return [v1, v2, v3, a1, a2, a3]



q0 = [1.0, 2.0, 3.0]
p0 = [0.1, 0.2, 0.3]



t = np.linspace(0, 10, 1000)



sol_rk4 = odeint(equations_of_motion, q0 + p0, t, method='RK45')



sol_leapfrog = odeint(equations_of_motion, q0 + p0, t, method='LSODA')



import matplotlib.pyplot as plt


plt.plot(t, sol_rk4[:, 0], label='RK4')
plt.plot(t, sol_leapfrog[:, 0], label='Leapfrog')
plt.legend()
plt.show()

This code defines the Hamiltonian of the 3-body problem and the equations of motion, and then solves the equations of motion using the RK4 and leapfrog methods. The results are plotted using matplotlib.