Double Pendulum App

This project extends the Double Pendulum Repo, a Lagrangian formulation of the equations of motion of a double pendulum system.

Available at: www.double-pendulum.net

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The above figure shows simple pendulum suspended from another simple pendulum by a frictionless hinge.

• Both pendulums move in the same plane.
• In this system, the rods $OP_1$ and $P_1P_2$ are rigid, massless and inextensible.
• The system has two degrees of freedom and is uniquely determined by the values of $\theta_1$ and $\theta_2$

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We solve the Euler-Lagrange equations for $\textbf{q} = [\theta_1, \theta_2]$ such that,

$$ \frac{\text{d}}{\text{d}t}\left(\frac{\partial L}{\partial \dot{\textbf{q}}}\right)-\frac{\partial L}{\partial \textbf{q}}=0 $$

The result is a system of $|\textbf{q}|$ coupled, second-order differential equations

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The equations are uncoupled by letting $\omega_i = \frac{\text{d}}{\text{d} t}\theta_i$

So $\omega_i$ for $i=1,2$ represents the angular velocity with $\frac{\text{d}}{\text{d} t}\omega_i \equiv \frac{\text{d}^2}{\text{d} t^2}\theta_i$

Derivation

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The Application

This Dash-based web application extends our previous work on the exploration and derivation of the system's dynamics

Key Features

• Derivation: The equations of motion are derived symbolically with SymPy and abstracted as a series of dependent functions. A simple conditional logic structure controls which model is derived.

• Model Selection: Offers a choice between 'Simple' and 'Compound' pendulum models.

• DoublePendulum Class:

  • Instantiating a DoublePendulum object; clicking the 'Run Simulation' button, derives the symbolic equations.
  • The equations of motion are cached to reduce runtime for further simulations of the same model
  • The equations are numerically integrated using SciPy's solve_ivp function. Integrator arguments are available in the class structure but this functionality is yet to be added to the UI.

• Interactive User Interface: Built using Dash and Plotly, the interface allows users to input initial conditions (angles, angular velocities) and physical parameters (lengths, masses & acceleration due to gravity).

• Visualisation: Motion is rendered using Plotly and Matplotlib, including time graphs, phase diagrams, and animations.

• Error Handling: Features robust validation for user inputs, ensuring precise simulations.

• Educational Content: Utilises MathJax for $\LaTeX$ rendering.

The application is available at double-pendulum.net

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Chaos Section

Update 15/01/2024.

Introduces significant architectural changes to the Double_Pendulum application, with a focus on exploring chaotic dynamics and enhancing the modularity and maintainability of the codebase.

• Chaotic Dynamics Exploration Page: Allows for a more in-depth investigation of chaos theory within the context of the double pendulum system.

• Architectural Enhancements:

  • Page-Specific Callbacks: To accommodate the new multi-page setup;
    • callbacks have been restructured to be layout or page-specific. This ensures a more organized codebase where each page's logic is contained within its respective module, improving readability and ease of maintenance.
  • Layout Abstraction: The application's layout has been abstracted away from the main app.py file to support better scalability and code organization. This includes:
    • A main page that serves as the entry point and primary interface for the standard double pendulum simulation. (existing main branch)
    • A chaos page designed to explore chaotic behaviors and patterns that emerge under various conditions.

• Error Handling: Error handling has been improved with the separation of validation logic.

  • The validate_inputs() function has been refined to accept a list of initial conditions, allowing for more flexible input checks.

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Application @ double-pendulum.net

For a full list of dependencies, refer to the requirements.txt file.

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