Using the Lagrangian and the principle of least action to derive the Euler-Lagrange equations

Introduction

Introduction

L = T-V
T = \sum_i^n(1/2)m_iv_i^2
V=G\sum_i^{n-1}\sum_{i+1}^n\frac{m_im_j}{|\bold{r}_i-\bold{r}_j|}

euler lagrange

\dfrac{d}{dt}(\dfrac{\delta L}{\delta \bold{r}^{dot}_i})-\dfrac{\delta L}{\delta \bold{r}_i}=0
\dfrac{\delta L}{\delta \bold{r}^{dot}_i}=m_i\bold{r}^{dot}_i=\bold{p}_i
\dfrac{\delta L}{\delta \bold{r}_i}=-G\sum_{j=1,j\ne i}^nm_im_j\dfrac{\bold{r}_i-\bold{r}_j}{|\bold{r}_i-\bold{r}_j|^3}

gives us

m_i\bold{r}^{dot dot}_i=-G\sum_{j=1,j\ne i}^nm_im_j\dfrac{\bold{r}_i-\bold{r}_j}{|r_i-r_j|^3}

which is

~ ma=F

also gives

\bold{r}^{dot dot}_i=-G\sum_{j=1,j\ne i}^nm_j\dfrac{\bold{r}_i-\bold{r}_j}{|r_i-r_j|^3}