First put in initial conditions for phi ( initial phi, initial phidot) and r (r_0, rdot_0)

PHIC={0,.3};RC={1,0};

Now solve the equations of motion (which have been massaged a bit, as we did in lecture)

SOL={ϕ[t],r[t]}/.NDSolve[{r''[t]-r[t] PHIC[[2]]^2 (RC[[1]]/r[t])^4==-1/r[t]^2,ϕ'[t]==PHIC[[2]]RC[[1]]^2/r[t]^2,r[0]==RC[[1]],r'[0]==RC[[2]],ϕ[0]==PHIC[[1]]},{ϕ[t],r[t]},{t,0,50},Method->"StiffnessSwitching"][[1]]

Now we animate the position of the mass as a function of time to visualize

Animate[ListPolarPlot[{SOL/.{t->T}},PlotRange->{{-1,1},{-1,1}}],{T,0,50},AnimationRate->.5]