This is a symptom of the start capaciytor being bad. The capacitor is responsible for getting the compressor going. Follow the wires to the capacitor , mark wires, remove, and take old with you to heating/cooling shop and get a new one.
I have a follow up question. The same capacitor "can" also serves the evaporator fan The capacitor can has 3 terminals. One capacitor is labled as 45 micro F and the other is 5 micro F. Have you ever seen where only one of the capacitors inside the can goes bad with out the other?
Yes, this very common for one or the other to go bad instead of both.
The only thing in common is the C terminal. They are 2 independent circuits inside.
Power fluctuation, power outage, hard start up...even just weak filaments can compromise the capacitor. Most capacitors (dual) that I change have only bad side.