Lagrange's Equations
Deriving Lagrange’s Equations using Alembert’s Principle #
Position vectors of the particles are: $\vec{r_i}=\vec{r_i}(q_1,q_2,q_3,…q_n,t)$ where t is the time and q_k are the generalized coordinates.
Differentiating $\vec{r_i}$ with respect to t: $\frac{d\vec{r_i}}{dt}=\frac{\partial \vec{r_i}}{\partial q_1}\frac{\partial q_1}{\partial t}+\frac{\partial \vec{r_i}}{\partial q_2}\frac{\partial q_2}{\partial t}+…+\frac{\partial \vec{r_i}}{\partial q_k}\frac{\partial q_k}{\partial t}+…+\frac{\partial \vec{r_i}}{\partial q_n}\frac{\partial q_n}{\partial t}+\frac{\partial \vec{r_i}}{\partial t}$ $$\vec{v_i}=\sum \limits_{k=1}^n \frac{\partial \vec{r_i}}{\partial q_k} q_k’\\ +\\ \frac{\partial \vec{r_i}}{\partial t}$$ $$ \delta \vec{r_i}=\sum_{k=1}^n \frac{\partial \vec{r_i}}{\partial q_k} \delta q_k $$
- i: (N particles)
- k: (n generalized coordinates)
From 3. D’Alembert’s Principle we know that $\sum_{i=1}^N (\vec{F_i^a}-\vec{p_i’})\cdot \delta \vec{r_i}=0$
LHS #
$$ \begin{align} \sum_i^N \vec{F_i} \cdot \delta \vec{r_i}=\sum_i^N \vec{F_i} \cdot \sum_k^n \frac{\partial \vec{r_i}}{\partial q_k} \delta q_k\\ =\sum_k^n \sum_i^N \left(\vec{F_i}\frac{\partial \vec{r_i}}{\partial q_k}\right) \delta q_k=\sum_k^nG_k\delta q_k\\ where\\ G_k=\sum_i^N \left(\vec{F_i}\cdot\frac{\partial \vec{r_i}}{\partial q_k}\right) \end{align} $$
If V is potential, $$ \begin{align} \vec{F_i}=-\vec{\nabla _i}V=\left< \frac{\partial V}{\partial x_i},\frac{\partial V}{\partial y_i},\frac{\partial V}{\partial z_i}\right>\\ So,\\ G_k=\sum_i^N -\left(\frac{\partial V}{\partial x_i}\frac{\partial x_i}{\partial q_k}+\frac{\partial V}{\partial y_i}\frac{\partial y_i}{\partial q_k}+\frac{\partial V}{\partial z_i}\frac{\partial z_i}{\partial q_k}\right)\\ =-\frac{\partial V}{\partial q_k} \end{align} $$
RHS #
$$ \begin{align} \sum _{i=1}^N \vec{p_i}’ \cdot \delta \vec{r_i}=\sum_i \left(m\vec{r_i}’’ \cdot \sum_k^n \frac{\partial \vec{r_i}}{\partial q_k} \delta q_k\right)\\ =\sum_k^n \sum_i\left(m_i\vec{r_i}’’ \cdot \frac{\partial \vec{r_i}}{\partial q_k} \right)\delta q_k\\ =\sum_k^n \sum_i\left(\frac{d}{dt}(m_i\vec{r_i}’\cdot \frac{\partial \vec{r_i}}{\partial q_k})- m_i \vec{r_i}’\frac{d}{dt}(\frac{\partial \vec{r_i}}{\partial q_k}) \right)\delta q_k\\ =\sum_k^n \sum_i\left(\frac{d}{dt}(m_i\vec{v_i}\cdot \frac{\partial \vec{v_i}}{\partial q_k’})- m_i \vec{v_i}(\frac{\partial \vec{v_i}}{\partial q_k}) \right)\delta q_k\\ =\sum_k^n \left(\frac{d}{dt}(\frac{\partial }{\partial q_k’}(\sum_i\frac{1}{2}m_i\\ \vec{v_i}\cdot\vec{v_i}))- (\frac{\partial}{\partial q_k}(\sum_i\frac{1}{2}m_i\\ \vec{v_i}\cdot\vec{v_i})) \right)\delta q_k\\ =\sum_k^n \left(\frac{d}{dt}(\frac{\partial }{\partial q_k’}(T))- (\frac{\partial}{\partial q_k}(T)) \right)\delta q_k \end{align} $$
LHS-RHS=0:
$$ \frac{d}{dt}(\frac{\partial T}{\partial q_k’})-\frac{\partial T}{\partial q_k}=G_k=-\frac{\partial V}{\partial q_k} $$
DERIVED!!
Generalized Force #
Associated with generalized coordinates qₖ. Gₖ may have any dimensions but dimensions of Gₖδqₖ are those of work (joules). $$ G_k=\sum_{i=1}^N \vec{F_i}\cdot \frac{\partial \vec{r_i}}{\partial q_k} $$
Lagrange’s Equations #
General Form $$ \frac{d}{dt}\left(\frac{\partial T}{\partial q_k’}\right)-\frac{\partial T}{\partial q_k}=G_k $$
Another form L=T-V: kinetic energy-potential energy $$ \frac{d}{dt}\left(\frac{\partial L}{\partial q_k’}\right)-\frac{\partial L}{\partial q_k}=0 $$