Use the Euler-Lagrange equation

to find the equation of motion for \(\mathcal{L}=\frac{m\dot{\phi}^2}{2} - \frac{k_s a^2}{2}\left(\partial_x \phi\right)^2 - \frac{m^*}{2}\omega^2\phi^2\)

 * 1) Substitute into the general Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}} + d_t^2\frac{\partial \mathcal{L}}{\partial\ddot{\phi}} \cdots = 0\): \(\frac{\partial}{\partial\phi}\left(\frac{k_s a^2}{2}\left(\partial_x \phi\right)^2 + \frac{m^*}{2}\omega^2\phi^2\right) - d_t\frac{\partial}{\partial\dot{\phi}}\frac{m\dot{\phi}^2}{2} = 0\)
 * 2) Solve: \(-m\ddot{\phi}-m^*\omega^2\phi - k_s a^2 \partial_x^2 \phi = 0\)

to find the equation of motion for \(\mathcal{L}=\frac{m\dot{\phi}^2}{2} - \frac{k_s}{2}\left(\partial_x^2 \phi\right)^2\)

 * 1) Substitute into the general Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}} + d_t^2\frac{\partial \mathcal{L}}{\partial\ddot{\phi}} \cdots = 0\): \(\frac{\partial}{\partial\phi}\frac{k_s}{2}\left(\partial_x^2 \phi\right)^2 - d_t\frac{\partial}{\partial\dot{\phi}}\frac{m\dot{\phi}^2}{2} = 0\)
 * 2) Solve: \(-m\ddot{\phi}- k_s \partial_x^4 \phi = 0\)

to find the equation of motion for \(\mathcal{L}=\frac{m\dot{\phi}_i^2}{2} - \frac{m^*}{2}\omega^2\phi^2 - \frac{\eta}{4}\phi^4\)

 * 1) Substitute into the general Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}} + d_t^2\frac{\partial \mathcal{L}}{\partial\ddot{\phi}} \cdots = 0\): \(\frac{\partial}{\partial\phi}\left(\frac{m^*}{2}\omega^2\phi^2 - \frac{\eta}{4}\phi^4 \right) - d_t\frac{\partial}{\partial\dot{\phi}}\frac{m\dot{\phi}^2}{2} = 0\)
 * 2) Solve: \(-m\ddot{\phi}-m^*\omega^2\phi - \eta \phi^3 = 0\)

to find the equation of motion for \(\mathcal{L}=\sum_{i=1}^n\left[\frac{m\dot{\phi}_i^2}{2} - \frac{k_s}{2}\left(\partial_x^2 \phi_i\right)^2\right]\)

 * 1) Substitute into the general Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi_j} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}_j} + d_t^2\frac{\partial \mathcal{L}}{\partial\ddot{\phi}_j} \cdots = 0\): \(\frac{\partial}{\partial\phi_j}\left(\frac{k_s a^2}{2}\left(\partial_x \phi_i\right)^2 \right) - d_t\frac{\partial}{\partial\dot{\phi}_j}\frac{m\dot{\phi_i}^2}{2} = 0\)
 * 2) Solve: \(-m\ddot{\phi}_i- k_s a^2 \partial_x^2 \phi_i = 0\)

to find the equation of motion for \(\mathcal{L}=\frac{m|\dot{\phi}|^2}{2} - \frac{k_s a^2}{2}\left(\partial_x^2 |\phi_i|\right)^2\)

 * 1) Separate the complex \(|\phi|\) variable into two independent fields, \(\phi\) and \(\phi^*\): \(\mathcal{L}=\frac{m\dot{\phi}\dot{\phi}^*}{2} - \frac{k_s a^2}{2}\partial_x\dot{\phi}\partial_x\dot{\phi}^*\)
 * 2) Substitute into the general Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi_j} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}_j} + d_t^2\frac{\partial \mathcal{L}}{\partial\ddot{\phi}_j} \cdots = 0\): \(\frac{\partial}{\partial\phi_j}\frac{k_s a^2}{2}\partial_x\dot{\phi}\partial_x\dot{\phi}^* - d_t\frac{\partial}{\partial\dot{\phi}_j}\frac{m\dot{\phi}\dot{\phi}^*}{2} = 0\)
 * 3) Solve: \(-m\ddot{\phi}_i- k_s a^2 \partial_x^2 \phi_i = 0\)