Simple Pendulum: Energy and Phase Space Simulation

This repository contains MATLAB code to simulate the motion of a simple pendulum and analyze its behavior. The project demonstrates both the small-angle approximation and the non-linear pendulum equations of motion, focusing on visualizing kinetic energy (KE), potential energy (PE), and total energy (TE) for different initial angles.

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Features

• Energy Analysis:
  The code calculates and plots kinetic, potential, and total energy for small-angle and non-linear cases.

• Phase Space Trajectory:
  Visualization of the phase space (angular displacement vs. angular velocity) for various initial angles.

• Comparison of Small-Angle Approximation vs. Non-Linear Pendulum:
  The differences in energy and phase behavior are illustrated clearly.

• Support for Multiple Initial Angles:
  Simulations are run for a range of initial angles to observe variations in pendulum behavior.

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Files Included

• energy.m:
  MATLAB code to simulate and plot the kinetic, potential, and total energy of a simple pendulum.

• phase_space_trajectory.m:
  MATLAB code to simulate and plot the phase space trajectory (displacement vs. velocity) of the simple pendulum.

• solution_of_simple_pendulum.m:
  MATLAB code to solve the equation of motion for the simple pendulum and visualize its motion over time.

• README.md:
  Provides an overview of the project, its features, and instructions for running the simulations.

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Getting Started

1. Requirements:

   • MATLAB installed on your system or access to the online MATLAB compiler.
   • Basic understanding of MATLAB’s ode45 solver and pendulum dynamics (helpful, but not necessary).

2. Running the Code:

   • Open energy.m in MATLAB.
   • Open phase_space_trajectory.m in MATLAB.
   • Open solution_of_simple_pendulum.m in MATLAB.
   • Run the scripts to generate plots for kinetic energy, potential energy, total energy, and phase space.
   • You can modify the initial angles or simulation time as needed.

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Results

• Energy Graphs (energy.m):
  Simulates and plots the time evolution of kinetic, potential, and total energy for different initial angles.

• Phase Space Diagrams (phase_space_trajectory.m):
  Visualizes the phase space trajectory (displacement vs. velocity) of the pendulum, demonstrating behavior for both small-angle approximation and full non-linear cases.

• Solution of Motion (solution_of_simple_pendulum.m):
  Solves the pendulum’s equation of motion and plots its angular displacement and velocity over time.

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Future Enhancements

• Add damping effects to simulate real-world friction.
• Include support for driven pendulum simulations.
• Expand the comparison to chaotic pendulum behavior.