Why it doesn't works ?

The reader does not really know what "does not work" is. The reader does not know what you are modeling, and what these functions suppose to do. You have also some syntax errors. May be it is due to cut/paste. You define function f and g but there is no trailing _ around the symbols there.

This below works, in the sense that Manipulate is showing something. But I do not know what is it one is supposed to see. May be you need to change the range of the sliders, or change the plot ranges to see what ever is it one is suppose to see? If I move the sliders to some specific locations, I see something:

Manipulate[
 ParametricPlot[{f[a, r, t, v, omega], g[a, r, t, v, omega]}, {t, 0, 10}],
 {r, 0, 1},
 {a, 1, 20},
 {v, 0, 20},
 {omega, 0, 2 Pi},
 Initialization :>
  {
   f[a_, r_, t_, v_, omega_] := v t (1 + (1/(1 - (r/a) Cos[omega*t])));
   g[a_, r_, t_, v_, omega_] := r Sin[omega t]/(1 - (r/a) Cos[omega t])
   }
 ]