Why? Mint is derived from Ubuntu, so there's very little you'll be getting in Mint that you don't have in Ubuntu. If you're looking for new desktop environments, you can add most of those as packages within Ubuntu and switch back and forth without rebooting. (You can even install Mint's flagship Cinnamon on Ubuntu -- at least in theory [I've never tried it].) If you just want to get a taste for something different, installing it in a virtual machine (using VirtualBox, VMware, or the like) is likely to be easier and safer than dual-booting.
If you really must dual-boot, the first thing to know is that you'll need another /boot partition. (That's your current /dev/sda2.) In fact, your current /boot is a little on the small side -- that's a known Ubuntu bug. Thus, you may need to shrink your LVM partition to make room for an extra /boot partition, and perhaps to replace the one you've got now.
Another complication you may run into is boot loaders. Both Ubuntu and Mint will try to install GRUB to the EFI System Partition (ESP; your /dev/sda1) in the EFI/ubuntu directory, so they'll overwrite each other. There are ways around this:
- You can live with it, and hope that whichever distribution ends up controlling correctly identifies the other and creates suitable GRUB entries. This might or might not work, though, so you might be forced to do it some other way....
- You can create multiple ESPs and then play games with partition type codes (using
parted, GParted, gdisk, etc.) -- ensure that when you install, only one ESP has the proper ESP type code (a "boot flag" in parted or GParted, or type EF00 in gdisk), and change type codes as necessary to get each distribution to install to its own separate ESP.
- You can abandon GRUB 2, or at least the pre-packaged GRUB 2 that each distribution provides, in favor of something else. My own rEFInd is likely to be the easiest to manage, but there are other options, too -- see my page on the subject for details.
Dual-booting Ubuntu and Arch seems more worthwhile than dual-booting Ubuntu and Mint, since Arch isn't based on Ubuntu; but you'll still have to create a /boot partition for Arch. The last I checked, the Arch people liked gummiboot, but gummiboot has been assimilated into the systemd collective; I believe it's now called systemd-boot. OTOH, it's easier to switch to whatever boot loader you like when using Arch, so you're not really limited to either GRUB 2 or gummiboot/systemd-boot. In any event, with Arch I'd either avoid GRUB 2 or hope that Ubuntu's GRUB 2 can correctly detect Arch.
