Question

A particle of mass is in a concentrated potential field V (𝑟). Determine the equation of motion using the Lagrangian! Is there a constant of motion? Explain!

Answer 1

Given,

Mass of the particle = m

Potential field =V(r)

$L=T-V$

$L=m(\dot{r})^2- V(r)$

Now, $\frac{d}{dt}(\frac{\partial L }{\partial \dot{r}})=\frac{\partial L}{\partial r}$

$m\ddot{r}=\frac{\partial V({r})}{\partial r}$