Double Pendulum Simulation

This repository contains a simulation of a double pendulum system using Java and solved these equations by Euler-Lagrange differential equation.

Overview

A double pendulum consists of two pendulums attached end to end. The motion of a double pendulum is chaotic and highly sensitive to initial conditions, making it a fascinating subject for study in physics and mathematics.

Modeling

System Description

The double pendulum consists of two point masses connected by massless rods:

• Mass 1 (m₁): Connected to a fixed pivot point
• Mass 2 (m₂): Connected to the end of the first pendulum
• Length 1 (L₁): Length of the first rod
• Length 2 (L₂): Length of the second rod
• Gravity (g): Gravitational acceleration

Coordinate System

The system uses angular coordinates:

• θ₁: Angle of the first pendulum from vertical (downward)
• θ₂: Angle of the second pendulum from vertical (downward)
• ω₁ = θ̇₁: Angular velocity of the first pendulum
• ω₂ = θ̇₂: Angular velocity of the second pendulum

Position of the Masses

The Cartesian coordinates of each mass are derived from the angular positions:

Mass 1:

x₁ = L₁ sin(θ₁)
y₁ = -L₁ cos(θ₁)

Mass 2:

x₂ = L₁ sin(θ₁) + L₂ sin(θ₂)
y₂ = -L₁ cos(θ₁) - L₂ cos(θ₂)

Velocities

Taking the time derivative of the positions, we get the velocities:

Mass 1:

ẋ₁ = L₁ ω₁ cos(θ₁)
ẏ₁ = L₁ ω₁ sin(θ₁)

Mass 2:

ẋ₂ = L₁ ω₁ cos(θ₁) + L₂ ω₂ cos(θ₂)
ẏ₂ = L₁ ω₁ sin(θ₁) + L₂ ω₂ sin(θ₂)

The speed squared for each mass:

v₁² = ẋ₁² + ẏ₁² = L₁² ω₁²
v₂² = ẋ₂² + ẏ₂² = L₁² ω₁² + L₂² ω₂² + 2L₁L₂ ω₁ω₂ cos(θ₁ - θ₂)

Lagrangian Formulation

The Lagrangian (L) is defined as the difference between kinetic energy (T) and potential energy (V):

L = T - V

Kinetic Energy:

T = ½ m₁ v₁² + ½ m₂ v₂²
T = ½ m₁ L₁² ω₁² + ½ m₂ [L₁² ω₁² + L₂² ω₂² + 2L₁L₂ ω₁ω₂ cos(θ₁ - θ₂)]

Potential Energy:

V = m₁ g y₁ + m₂ g y₂
V = -m₁ g L₁ cos(θ₁) - m₂ g [L₁ cos(θ₁) + L₂ cos(θ₂)]

Complete Lagrangian:

L = ½(m₁ + m₂)L₁² ω₁² + ½ m₂ L₂² ω₂² + m₂ L₁ L₂ ω₁ω₂ cos(θ₁ - θ₂)
    + (m₁ + m₂) g L₁ cos(θ₁) + m₂ g L₂ cos(θ₂)

Euler-Lagrange Equations

The equations of motion are derived from the Euler-Lagrange equation:

d/dt(∂L/∂θ̇ᵢ) - ∂L/∂θᵢ = 0

For our system with two degrees of freedom (θ₁ and θ₂), we get two coupled differential equations.

After applying the Euler-Lagrange equations and simplifying, we obtain the angular accelerations:

Angular Acceleration of θ₁:

         -g(2m₁ + m₂)sin(θ₁) - m₂g sin(θ₁ - 2θ₂) - 2sin(θ₁ - θ₂)m₂[ω₂²L₂ + ω₁²L₁cos(θ₁ - θ₂)]
θ̈₁ = ───────────────────────────────────────────────────────────────────────────────────────────
                            L₁[2m₁ + m₂ - m₂cos(2θ₁ - 2θ₂)]

Angular Acceleration of θ₂:

         2sin(θ₁ - θ₂)[ω₁²L₁(m₁ + m₂) + g(m₁ + m₂)cos(θ₁) + ω₂²L₂m₂cos(θ₁ - θ₂)]
θ̈₂ = ─────────────────────────────────────────────────────────────────────────────────
                         L₂[2m₁ + m₂ - m₂cos(2θ₁ - 2θ₂)]

Numerical Integration

The simulation uses Euler's method for numerical integration:

1. Calculate accelerations using the Euler-Lagrange equations at time t:

   α₁ = θ̈₁(θ₁, θ₂, ω₁, ω₂)
   α₂ = θ̈₂(θ₁, θ₂, ω₁, ω₂)
   

2. Update angular velocities:

   ω₁(t + Δt) = ω₁(t) + α₁ · Δt
   ω₂(t + Δt) = ω₂(t) + α₂ · Δt
   

3. Update angles:

   θ₁(t + Δt) = θ₁(t) + ω₁(t + Δt) · Δt
   θ₂(t + Δt) = θ₂(t) + ω₂(t + Δt) · Δt
   

Where:

• Δt is the time step (smaller values = more accurate but slower)
• α₁, α₂ are the angular accelerations
• ω₁, ω₂ are the angular velocities (θ̇₁, θ̇₂)
• θ₁, θ₂ are the angles

Understanding the Derivatives

In the Lagrangian formulation:

• θ: Position (angle) - describes where the pendulum is
• θ̇ (omega, ω): First derivative - describes how fast the angle is changing (angular velocity)
• θ̈ (alpha, α): Second derivative - describes how fast the velocity is changing (angular acceleration)

The relationship:

θ̇ = dθ/dt  (velocity is the rate of change of position)
θ̈ = dω/dt = d²θ/dt²  (acceleration is the rate of change of velocity)

Energy Conservation

In an ideal system with no friction, the total energy should remain constant:

E = T + V = constant

The simulation tracks energy to verify numerical accuracy. Small energy drift over time indicates numerical errors from the integration method.

Algorithm Implementation

The core algorithm in the Lagrange.java class follows these steps:

1. calculateAccelerations(): Computes θ̈₁ and θ̈₂ from the current state
2. integrate(): Updates the system state using Euler's method
3. calculateEnergy(): Computes total energy for verification
4. calculateLagrangian(): Computes L = T - V for analysis

The simulation loop:

for each time step:
    1. Calculate accelerations from current angles and velocities
    2. Update velocities using accelerations
    3. Update angles using new velocities
    4. Draw the pendulum at new position
    5. Record trajectory

Features

• Real-time visualization of the double pendulum motion
• Trajectory tracking showing the path of the second mass
• Energy and Lagrangian value display
• Pause/resume functionality
• Adjustable simulation parameters
• Custom dark-themed UI

Usage

Run the simulation with custom parameters:

java -jar lagrange-double-pendulum.jar --theta1=90 --theta2=45 --m1=10 --m2=10 --L1=150 --L2=150 --g=9.81

Or use a configuration file:

java -jar lagrange-double-pendulum.jar --config=myconfig.txt

Controls

• ESC: Pause/Resume simulation
• R: Reset to initial conditions