I have the same G50-45, A8-6410.
Sometimes. mine gets warm. But heat should not be a big problem, as most hardware is rated for quite high termperatures. Look up Linus Tech Tips' YouTube video on Hardware HEAT.
Sure, there is a possibility you "fried" something, but here is something else you can try in the mean-time:
(1) CAREFULLY remove the BATTERY from the laptop. Set it on some old NEWSPAPER and leave it out of the way for awhile.
(2) PLUG IN your A/C ADAPTOR and plug into any regular Mains (but perhaps preferably an Extension cord with a fused breaker in case of electric storm, or whatnot).
(3) Try to operate the Laptop normally.
If the Laptop WORKS (turns on, boots up, gets into Windows, and you can run a program, check some email, maybe play a game), then it's more likely an issue with the BATTERY.
** SHUT DOWN THE LAPTOP again, at this point. **
(4) Have a look at the NEWSPAPER where the battery is. If there are any damp / wet spots, YOUR BATTERY IS LEAKING and should be DISPOSED OF CAREFULLY .. AND you should Wash Your Hands, to make sure you don't have any battery-acid residue.
If the battery is NOT leaking or punctured or damaged in any way, REPLACE the battery in the laptop.
THIS time, when you power it on, keep an eye on the LEDs at the front.
If your PERIPHERAL VISION catches "normal" boot-up logos etc, then maybe you need to keep a better eye on how warm the laptop gets..
What you are LOOKING FOR .. especially if the laptop does NOT boot properly with the battery in .. is IF THE LEDs BLINK at all, and HOW MANY TIMES. IF you think you "missed" it, just watch for a few minutes -- one (or more) of the LEDs may BLINK a certain number of times. It might be 3-fast, 1 short pause, 2-fast... or 3-fast.... long pause.... 3-fast again (just means only 3 blinks, silly!)
This would be a BLINK CODE for DIAGNOSTIC purposes.
If the laptop does NOT BOOT with the battery in, AND you get a BLINK CODE, you'll need to look this up in a Lenovo Diagnostic supplementary manual for this laptop family (G-series, or maybe G50-series), or get on the phone to your local Lenovo and ask them.
If battery-in FAILS, but battery-out-AC-only WORKS, then more likely than not it's purely a battery issue.
You COULD double-check the CONTACTS between the battery and the laptop battery compartment.
If battery-in WORKS, but you experience further SHUT DOWN after some more / prolonged usage, then you MAY want to either get help or do it yourself -- TAKE THE LAPTOP APART, get a BRUSH-ENDED ATTACHMENT for the vacuum cleaner, and CLEAR OuT the HEAT SINK and FAN area!! Heat management can only be effective if the FAN and HEATSINK are as clear of excess dust / dirt as possible.
Try also to keep the laptop on a COOL surface during use. The more intensively you use the laptop('s processor) e.g. high graphics games, etc, the warmer it will get. Soft material eg your lap in tracksuit bottoms, or on your bed, will NOT LET THE LAPTOP "BREATHE", and heat will build up too much, causing a shut-down.
IF you live somewhere generally WARMER than average, recommend you look into getting a LAPTOP COOLING STAND or the like. Having extra cool air being brought into the underside of the laptop will help keep the operating temperature down, and reduce over-heat shutdowns. Just remember to BE CAREFUL that more blowing means possibly faster build-up of fluff / dust / dirt in the FAN and HEATSINK, and you will want to clean these out as regularly as you need, to maintain a good operating temperature.
GOOD LUCK!