Let's solve **Problem 2.4: Spring-coupled masses** step by step using Lagrange's equations. We'll assume a typical setup: two masses \( m_1 \) and \( m_2 \) connected by a spring of stiffness \( k \). Let \( z_1 \) and \( z_2 \) be the displacements of the two masses. --- ## 1. **Define the System** - **State variables:** - \( x_1 = z_1 \) (position of mass 1) - \( x_2 = z_2 \) (position of mass 2) - \( x_3 = \dot{z}_1 \) (velocity of mass 1) - \( x_4 = \dot{z}_2 \) (velocity of mass 2) --- ## 2. **Kinetic and Potential Energy** - **Kinetic Energy (T):** \[ T = \frac{1}{2} m_1 \dot{z}_1^2 + \frac{1}{2} m_2 \dot{z}_2^2 \] - **Potential Energy (V):** \[ V = \frac{1}{2} k (z_2 - z_1)^2 \] --- ## 3. **Lagrangian** \[ L = T - V = \frac{1}{2} m_1 \dot{z}_1^2 + \frac{1}{2} m_2 \dot{z}_2^2 - \frac{1}{2} k (z_2 - z_1)^2 \] --- ## 4. **Lagrange's Equations** For each mass: \[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{z}_i} \right) - \frac{\partial L}{\partial z_i} = 0 \] ### For \( z_1 \): - \(\frac{\partial L}{\partial \dot{z}_1} = m_1 \dot{z}_1\) - \(\frac{d}{dt} (m_1 \dot{z}_1) = m_1 \ddot{z}_1\) - \(\frac{\partial L}{\partial z_1} = -k (z_2 - z_1) \cdot (-1) = k(z_2 - z_1)\) So: \[ m_1 \ddot{z}_1 + k(z_1 - z_2) = 0 \] --- ### For \( z_2 \): - \(\frac{\partial L}{\partial \dot{z}_2} = m_2 \dot{z}_2\) - \(\frac{d}{dt} (m_2 \dot{z}_2) = m_2 \ddot{z}_2\) - \(\frac{\partial L}{\partial z_2} = k(z_2 - z_1)\) So: \[ m_2 \ddot{z}_2 + k(z_2 - z_1) = 0 \] --- ## 5. **State-Space Representation** Let: - \( x_1 = z_1 \) - \( x_2 = z_2 \) - \( x_3 = \dot{z}_1 \) - \( x_4 = \dot{z}_2 \) Then: \[ \dot{x}_1 = x_3 \] \[ \dot{x}_2 = x_4 \] \[ \dot{x}_3 = \ddot{z}_1 = -\frac{k}{m_1}(x_1 - x_2) \] \[ \dot{x}_4 = \ddot{z}_2 = -\frac{k}{m_2}(x_2 - x_1) \] --- ## 6. **Final State Equations** \[ \boxed{ \begin{aligned} \dot{x}_1 &= x_3 \\ \dot{x}_2 &= x_4 \\ \dot{x}_3 &= -\frac{k}{m_1}(x_1 - x_2) \\ \dot{x}_4 &= -\frac{k}{m_2}(x_2 - x_1) \end{aligned} } \] --- **Explanation:** - \( x_1, x_2 \) are positions, \( x_3, x_4 \) are velocities. - Dynamics derived from Lagrange’s equations using the kinetic and potential energy of the coupled spring-mass system. If you have the actual figure or more specific details (such as external forces or damping), let me know for further refinement!