Consider a particle of mass m = 1 moving in a three-dimensional space under the influence of a spherical potential well characterized by a Lagrangian:
\(L = \frac{p^2}{ 2m} - V(r),\)
where \(V(r) = ar\) for r < R and V(r) = 0 for r ≥ R. Here, R and a are fixed positive constants, and r stands for the radial coordinate of the particle in the spherical polar coordinate system.
What is the Lagrangian equation of motion for the radial distance r?
1
\( m \times r¨ = - a + \frac{r˙²}{r}\)
2
\(m \times r¨ = - a - \frac{ r˙²} { r}\)
3
\(m \times r¨ = a - \frac{ r˙²} {r }\)
4
\(m \times r¨ = a + \frac{ r˙² }{ r}\)