diff options
author | Thomas Albers <thomas@thomaslabs.org> | 2023-03-08 23:43:00 +0100 |
---|---|---|
committer | Thomas Albers <thomas@thomaslabs.org> | 2023-03-08 23:43:00 +0100 |
commit | 61b5ce20f25c5785e41574998a12c6d06eb05a5e (patch) | |
tree | 20e2225b4f30b15d8dee30351041d1f33d42b34a /src/math/lagrange.org | |
parent | 561bac75579391c14e47eaccfabdf9eda98855da (diff) |
Diffstat (limited to 'src/math/lagrange.org')
-rw-r--r-- | src/math/lagrange.org | 23 |
1 files changed, 23 insertions, 0 deletions
diff --git a/src/math/lagrange.org b/src/math/lagrange.org new file mode 100644 index 0000000..565c7e7 --- /dev/null +++ b/src/math/lagrange.org @@ -0,0 +1,23 @@ +#+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*} |