I was trying to find the acceleration of the harmonic accelerator using the Euler-Lagrange Equation, and I’m wondering if there is any error in this solution.
\begin{align} &\textit{Euler-Lagrange Eq. of Motion:}\\ & \frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) = 0 \\ \ \\ & \textbf{Harmonic Oscillator:}\\ & L = \frac 1 2 m \dot{x}^2 - \frac 1 2 kx^2, q=x \\ & \frac{\partial L}{\partial x} = -kx, \frac{\partial L}{\partial \dot{x}} = m\dot{x} \\ \therefore\; & -kx-\frac{d}{dt}(m\dot{x}) = 0, kx+m\ddot{x} = 0. \\ \therefore\; & \ddot{x} = -\frac k m x = -\omega^2x. \end{align}
I’m a beginner for this equation. I need help checking any issues for my sols and using this flexibly.
+) Is my proof for the equation correct(Since I’m studying alone, I barely have chance to check my proof…):
\begin{align} &\delta J = \delta \int_{t_1}^{t_2} L dt = \int_{t_1}^{t_2} \sum_{i}\left(\frac{\partial L}{\partial q_i}\delta q_i+\frac{\partial L}{\partial \dot{q_i}}\delta \dot{q_i}\right)dt = 0. \\ & \int_{t_1}^{t_2} \sum_i \left(\frac{\partial L}{\partial q_i}\delta q_i + \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_i}}\delta q_i\right) - \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q_i}}\right)\delta q_i\right)dt = 0. \\ &\sum_i \frac{\partial L}{\partial \dot{q_i}}\delta q_i \Bigg|_{t_1}^{t_2} + \int_{t_1}^{t_2} \sum_i \left(\frac{\partial L}{\partial q_i}-\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q_i}}\right)\right)\delta q_i dt = 0. \\ \therefore \; & \frac{\partial L}{\partial q_i} - \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q_i}}\right) = 0. \blacksquare \end{align}