If in the first place the computer ran for some time and then suddenly shut off, and afterwards turns on and then off again; one problem could be overheating.
The fan and heat sink could be clogged up with dust which will need to be cleaned out. Also, the thermal paste may need replacing.
However, it could be a range of other things which oldturkey03 is trying to diagnose. For example, it could be an issue with RAM.