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Module 3.1 Coupled Nonlinear Oscillations & Parameter Tuning Studio

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Simulation 1

Double Pendulum Simulation Studio

Python 3.11 with Matplotlib Slider Ready

Interactive Physics Ready

Click "Run Code" to launch simulation and interactive parameter sliders.


                                

                            


                            
                                                        

                                

                                                                            

                                            
 Mathematical Problem Formulation

                                            
Double Pendulum Simulation Studio: Coupled Nonlinear Oscillations, Phase Space, Energy Conservation & Trajectory Trail

                                        

                                    
                                                                            

                                            
 Theoretical Background & Explanation

                                            

    

        

             Classical Mechanics: Double Pendulum Dynamics & Chaos
        

        

            The double pendulum is the quintessential archetype of deterministic chaos in classical mechanics. While completely governed by deterministic Newton-Euler-Lagrange equations, its motion at moderate and high energies exhibits extreme sensitivity to initial conditions, complex fractal trajectories, and phase space mixing.
        

    


    
    

        Double Pendulum Coordinate Geometry, Angles, and Forces Schematic
        

            Figure 3.1.1: Physical schematic of the double pendulum displaying generalized coordinates $(\theta_1, \theta_2)$, Cartesian positions, gravitational loads $m_i \vec{g}$, and tangential velocities.
        

    


    

        1. Coordinate Kinematics & Generalized Coordinates
    

    

        Consider two point masses $m_1$ and $m_2$ suspended by rigid, massless rods of lengths $L_1$ and $L_2$ from a fixed frictionless pivot $\mathcal{O}(0, 0)$. Defining generalized coordinates $\theta_1$ and $\theta_2$ as the counter-clockwise deflection angles relative to the downward vertical:
    

    

        $$x_1 = L_1 \sin\theta_1, \qquad y_1 = -L_1 \cos\theta_1$$
        $$x_2 = L_1 \sin\theta_1 + L_2 \sin\theta_2, \qquad y_2 = -L_1 \cos\theta_1 - L_2 \cos\theta_2$$
    

    

        Differentiating with respect to time $t$ yields the velocity components:
    

    

        $$\dot{x}_1 = L_1 \dot{\theta}_1 \cos\theta_1, \qquad \dot{y}_1 = L_1 \dot{\theta}_1 \sin\theta_1 \implies v_1^2 = L_1^2 \dot{\theta}_1^2$$
        $$\dot{x}_2 = L_1 \dot{\theta}_1 \cos\theta_1 + L_2 \dot{\theta}_2 \cos\theta_2, \qquad \dot{y}_2 = L_1 \dot{\theta}_1 \sin\theta_1 + L_2 \dot{\theta}_2 \sin\theta_2$$
        $$v_2^2 = \dot{x}_2^2 + \dot{y}_2^2 = L_1^2 \dot{\theta}_1^2 + L_2^2 \dot{\theta}_2^2 + 2 L_1 L_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2)$$
    


    

        2. Kinetic & Potential Energy Formulation
    

    

        The total kinetic energy $T$ of the two-particle system is:
    

    

        $$T = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2}(m_1 + m_2) L_1^2 \dot{\theta}_1^2 + \frac{1}{2} m_2 L_2^2 \dot{\theta}_2^2 + m_2 L_1 L_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2)$$
    

    

        Taking gravitational potential energy reference $V = 0$ at the downward equilibrium position ($\theta_1 = \theta_2 = 0$):
    

    

        $$V(\theta_1, \theta_2) = (m_1 + m_2) g L_1 (1 - \cos\theta_1) + m_2 g L_2 (1 - \cos\theta_2)$$
    

    

        The Lagrangian function $\mathcal{L} = T - V$ fully defines the system's dynamics:
    

    

        $$\mathcal{L} = \frac{1}{2}(m_1 + m_2) L_1^2 \dot{\theta}_1^2 + \frac{1}{2} m_2 L_2^2 \dot{\theta}_2^2 + m_2 L_1 L_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2) - (m_1 + m_2) g L_1 (1 - \cos\theta_1) - m_2 g L_2 (1 - \cos\theta_2)$$
    


    

        3. Euler-Lagrange Equations of Motion
    

    

        Applying Hamilton's principle of stationary action $\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{\theta}_i}\right) - \frac{\partial \mathcal{L}}{\partial \theta_i} = -\gamma \dot{\theta}_i$:
    

    

        $$(m_1 + m_2) L_1 \ddot{\theta}_1 + m_2 L_2 \ddot{\theta}_2 \cos(\theta_1 - \theta_2) = -m_2 L_2 \dot{\theta}_2^2 \sin(\theta_1 - \theta_2) - (m_1 + m_2) g \sin\theta_1 - \gamma \dot{\theta}_1$$
        $$m_2 L_1 \ddot{\theta}_1 \cos(\theta_1 - \theta_2) + m_2 L_2 \ddot{\theta}_2 = m_2 L_1 \dot{\theta}_1^2 \sin(\theta_1 - \theta_2) - m_2 g \sin\theta_2 - \gamma \dot{\theta}_2$$
    

    

        Letting $\Delta = \theta_1 - \theta_2$, we rewrite the system in compact matrix form $\mathbf{M}(\vec{\theta}) \ddot{\vec{\theta}} = \vec{R}(\vec{\theta}, \dot{\vec{\theta}})$:
    

    

        $$\begin{pmatrix} (m_1 + m_2) L_1 & m_2 L_2 \cos\Delta \\ m_2 L_1 \cos\Delta & m_2 L_2 \end{pmatrix} \begin{pmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{pmatrix} = \begin{pmatrix} R_1 \\ R_2 \end{pmatrix}$$
    

    

        The determinant of the mass matrix is strictly positive for all physical parameters:
        $$\det \mathbf{M} = m_2 L_1 L_2 \left[ (m_1 + m_2) - m_2 \cos^2\Delta \right] = m_2 L_1 L_2 \left[ m_1 + m_2 \sin^2\Delta \right] > 0$$
        By matrix inversion (Cramer's rule), the angular accelerations $\ddot{\theta}_1, \ddot{\theta}_2$ are computed unconditionally without singularities:
    

    

        $$\ddot{\theta}_1 = \frac{R_1 m_2 L_2 - R_2 m_2 L_2 \cos\Delta}{\det \mathbf{M}}, \qquad \ddot{\theta}_2 = \frac{R_2 (m_1 + m_2) L_1 - R_1 m_2 L_1 \cos\Delta}{\det \mathbf{M}}$$
    


    

        4. Numerical Runge-Kutta 4 (RK4) Integration & Conservation of Energy
    

    

        The 2nd-order coupled ODEs are transformed into four 1st-order state equations for state vector $\vec{Y} = (\theta_1, \theta_2, \omega_1, \omega_2)^T$. The Runge-Kutta 4th Order (RK4) integration algorithm updates the state across time step $\Delta t$:
    

    

        $$\vec{Y}_{n+1} = \vec{Y}_n + \frac{\Delta t}{6} \left( \vec{k}_1 + 2\vec{k}_2 + 2\vec{k}_3 + \vec{k}_4 \right)$$
    

    

        In the absence of damping ($\gamma = 0$), the total mechanical energy $E = T + V$ is a rigorous constant of motion ($dE/dt = 0$). The live simulation displays the energy breakdown and monitors energy conservation precision ($|\Delta E / E_0| < 0.05\%$).
    


    

        5. Student-Friendly Parameter Tuning Guide
    

    

        Use the live sliders below the workbench to explore exciting physics regimes:
        
• Small Angles ($|\theta_{1,2}| < 20^\circ$): Observe smooth, predictable, periodic/quasi-periodic normal mode oscillations.
        
• High Angles ($|\theta_{1,2}| > 90^\circ$): Witness the onset of deterministic chaos! The lower bob flips unpredictably over the top.
        
• Mass Dominance ($m_1 \gg m_2$ vs $m_1 \ll m_2$): When $m_1 \gg m_2$, the upper pendulum behaves almost as a simple pendulum driving the chaotic whip of $m_2$.
        
• Zero-Gravity ($g = 0$): Observe pure conservation of angular momentum and inertial tumbling!
        
• Damping ($\gamma > 0$): Watch the chaotic strange attractor collapse into the stable downward equilibrium sink $(0, 0)$.