blob: 565c7e78e56e9e353eba9be4239403a54e7ce03b (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
|
#+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*}
|
