My Dell 5000e unexpectidly turns itself off.
There are a few things this could be, most of them are motherboard related, but there could be a cooling issue or a power supply issue. Check the BIOS log for "thermal events". Things like CPU overheating can cause weird reactions in the machine, called System hysteresis. Basically, the computer goes crazy, opening things, crashing, rebooting, etc. Based on what youve said, "restarts normally", I dont think there is a CPU issue, nor a memory issue.
The first thing I would try is replacing the power supply and remove the battery (just as a test), and see if the machine starts behaving better or exhibits the same issues.
I believe the issue you have described can be most likely resolved with a new motherboard, especially if the power supply route does not yield positive results. Bad news is, a p3 motherboard is rather hard to get, and hard to replace, and rather pricey. I would sooner consider getting another notebook pc before replacing the board.