Try dissecting it and see if there are any loose connections

does it make a popping sound when it turns off? if so, combined w/ the fact it does so when you open it (thus moving the hinge), i think you may have a loose or possibly damaged screen ribbom cable. the 3ds' top screen has a ribbon cable that runs through the hinge and [cont.]--

connects to the motherboard housed in the bottom half of the console. this ribbon cable is also connected to the speakers iirc. if its loose or damaged, it can cause the 3ds to turn off randomly or even not turn on at all, and its most easy to identify bc the speakers--

will make a pop sound, and the screen may also flash, before the system forcefully shuts off. im not sure what model of 3ds you have, but most of them aren't too hard to open up and check the ribbon cable connection on the motherboard/bottom half of the console. --

i recommend looking around on yt to find video guides on this kinda thing, the website ifixit also has some detailed step-by-step teardown guides. i would first check the ribbon cable's port to see if its loose, just unplug it and replug it then reassemble and test if it fixed.--

if it didn't fix, there's a chance the actual cable inside of the hinge may be damaged, in which case it'd be best to send it into a repair shop for a replacement/professional look since its extremely finnicky to get into the hinge and inspect it/replace it without potentially--

damaging it further, or damaging your console's hinge. (unless you're already experienced w/ tech stuff) i would recommend backing up your sd card's data and game saves asap, as this problem can definitely worsen and then you may not get a chance to. :( damn i yapped lmao