Consider a body of mass m in a central repulsive potential V(r)

a) Show that the motion is confined to a plane

b) Write down the Lagrangian of the system in polar coordinates

c) Obtain the equation of motion with the minimum amount of variables and determine the constants of motion

d) Determine in which conditions the motion is one dimensional

e) Write down an expression for the total energy of the system

f) For a potential that decays with the square of the distance, write an equivalente one dimensional problem. Determine the effective potential and draw the corresponding graph. Discuss the form of the trajectory using the graph.