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+#+title:Lagrange Mechanics
+#+author: Thomas Albers Raviola
+#+date: 2022-10-01
+#+setupfile: ../../math_options.org
+
+* Disclaimer
+This site as of now just a technology demonstration and its claims
+should not be taken as true (even though I myself am pretty confident
+they are)
+
+* Eliminating the constraints
+\begin{align*}
+m_i \ddot{x}_i &= F_i + \sum_{n=1}^R \lambda_n \pderiv{g_n}{x_i}\\
+m_i \ddot{x}_i \pderiv{x_i}{q_k} &= F_i \pderiv{x_i}{q_k} + \sum_{n=1}^R \lambda_n \pderiv{g_n}{x_i} \pderiv{x_i}{q_k}\\
+\sum_{i=1}^{3N} m_i \ddot{x}_i \pderiv{x_i}{q_k} &= \sum_{i=1}^{3N}F_i \pderiv{x_i}{q_k} + \sum_{n=1}^R \lambda_n \sum_{i=1}^{3N} \pderiv{g_n}{x_i} \pderiv{x_i}{q_k}\\
+\sum_{i=1}^{3N} m_i \ddot{x}_i \pderiv{x_i}{q_k} &= \sum_{i=1}^{3N}F_i \pderiv{x_i}{q_k} + \sum_{n=1}^R \lambda_n \deriv{g_n}{q_k}\\
+\sum_{i=1}^{3N} m_i \ddot{x}_i \pderiv{x_i}{q_k} &= \sum_{i=1}^{3N}F_i \pderiv{x_i}{q_k}\\
+\deriv{}{t}\pderiv{(T - U)}{\dot{q}_k} - \pderiv{T}{q_k} &= \sum_{i=1}^{3N}F_i \pderiv{x_i}{q_k}\\
+\deriv{}{t}\pderiv{(T - U)}{\dot{q}_k} - \pderiv{T}{q_k} &= - \sum_{i=1}^{3N}\pderiv{U}{x_i} \pderiv{x_i}{q_k}\\
+\deriv{}{t}\pderiv{(T - U)}{\dot{q}_k} - \left(\pderiv{T}{q_k} - \pderiv{U}{q_k}\right) &= 0\\
+\deriv{}{t}\pderiv{(T - U)}{\dot{q}_k} - \pderiv{T - U}{q_k} &= 0\\
+\deriv{}{t}\pderiv{\mathcal{L}}{\dot{q}_k} - \pderiv{\mathcal{L}}{q_k} &= 0
+\end{align*}