places the problem might be.

1) The battery (unlikely since you tried a second battery);
2) The AC power adapter used for charging (and we assume the power outlet is working).
3) The camera's battery charger/power area.

Without a meter to check connections on the AC power adapter it is impossible to know if any power is getting out.

If the AC power adapter is plugged into the camcorder, will the camcorder turn on?

Just in case, link to the camcorder's manual (assuming you are talking about the JVC GZ-HD10

I could not find anything on a "GZ-HD&AG"...