Ordinary Differential Equation-2: 2nd Order ODE (RK2 Method) - Computational Physics Formulation & Simulation

Program 1

2nd Order ODE (RK2 Method)

Python 3.11 Ready

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Damped Harmonic Oscillator ODE Solution

This program uses numerical methods to solve the ordinary differential equation (ODE) for a damped harmonic oscillator. The ODE is given by:

$\frac{d^2x}{dt^2} + 2 \zeta \omega_n \frac{dx}{dt} + \omega_n^2 x = 0$

Initial Conditions: $x(t=0)=1 \quad \frac{dx}{dt}_{t=0}=0$

Program 2

2nd Order ODE (RK2 Method)

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Parameters
m = 1.0    
b = 0.2    
k = 1.0    
F0 = 1.0   
omega = 1.0

def F(t):
    return F0 * np.sin(omega * t)

def rk2_step(x, v, t, dt):
    dxdt = v
    dvdt = (F(t) / m) - (b / m) * v - (k / m) * x
    
    k1_x = dxdt
    k1_v = dvdt
    
    x_mid = x + 0.5 * k1_x * dt
    v_mid = v + 0.5 * k1_v * dt
    
    dxdt_mid = v_mid
    dvdt_mid = (F(t + 0.5 * dt) / m) - (b / m) * v_mid - (k / m) * x_mid
    
    x_new = x + k1_x * dt
    v_new = v + dvdt_mid * dt
    
    return x_new, v_new

t_start = 0.0
t_end = 20.0
dt = 0.01
num_steps = int((t_end - t_start) / dt)

time = np.linspace(t_start, t_end, num_steps)
x = np.zeros(num_steps)
v = np.zeros(num_steps)

x[0] = 0.0  # Initial position
v[0] = 0.0  # Initial velocity

for i in range(1, num_steps):
    t = time[i - 1]
    x[i], v[i] = rk2_step(x[i - 1], v[i - 1], t, dt)

plt.plot(time, x, label='Position (x)')
plt.plot(time, v, label='Velocity (v)', color='orange')
plt.title('Forced Oscillation')
plt.xlabel('Time (s)')
plt.ylabel('Position (x) and Velocity (v)')
plt.legend()
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Solve a forced oscillation differential equation using the RK2 (Runge-Kutta 2nd order) method, we consider a standard form of the equation:

$$ m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \sin(\omega t) $$

Initial conditions:$$ x(0) = 0 \quad \frac{dx}{dt}\big|_{t=0} = 0 $$

Theoretical Principles & Computational Physics Guide

2nd Order ODE (RK2 Method): Differential Equations & Dynamical Simulation

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Dynamical Evolution: Solves differential equations governing kinematics, planetary orbits, electrical circuits, and wave propagation.

Step-by-Step Stepping: Advances continuous state vectors forward in discrete time intervals $\Delta t$ using slope estimates.

Conservation Laws: Evaluates preservation of physical invariants (total energy, angular momentum) across long integration horizons.

2. Mathematical Formulation & Algorithmic Pipeline

Vector initial value problem and numerical state advancement:

$$ \frac{d\mathbf{y}}{dt} = \mathbf{f}(t, \mathbf{y}), \quad \mathbf{y}(t_0) = \mathbf{y}_0 \implies \mathbf{y}_{n+1} = \mathbf{y}_n + \Delta t \cdot \mathbf{\Phi}(t_n, \mathbf{y}_n, \Delta t) $$

3. Numerical Mechanics & Implementation Logic

Coupled Reduction: Convert $n$-th order ODEs into systems of $n$ coupled first-order equations.
Intermediate Slopes: Compute weighted intermediate derivatives ($k_1, k_2, k_3, k_4$ in RK4).
State Update: Advance state variables simultaneously and verify energy/phase invariants.

4. Relevant Computational Practice Exercises

Exercise 1

Solve the differential equation for 2nd Order ODE (RK2 Method) over 1000 time steps and plot state coordinates vs time.

Exercise 2

Compare numerical solution stability for small vs large step sizes $h = \Delta t$.

Exercise 3

Plot phase space trajectory demonstrating physical stability or damping attractors.

Standard Literature & References:

John R. Taylor, "Classical Mechanics", University Science Books (2005).
Mark Newman, "Computational Physics", University of Michigan (2013).

Ch. 13.2 2nd Order ODE (RK2 Method)

2nd Order ODE (RK2 Method)

Mathematical Problem Formulation

Damped Harmonic Oscillator ODE Solution

2nd Order ODE (RK2 Method)

Mathematical Problem Formulation

1. Core Principles & Theoretical Framework

2. Mathematical Formulation & Algorithmic Pipeline

3. Numerical Mechanics & Implementation Logic

4. Relevant Computational Practice Exercises