A properly fitted bra controls the jiggle because of the elasticated bands and straps - Think of what happens when you wear fitted briefs rather than boxer shorts.
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Momentum will increase the bounce, from casual walking, power walking, jogging to running. If you are running, I would suggest a sports bra, so to keep them in place and to help support. It won't hurt your back as much (even if they don't hurt right now, they can later on).
Try a sports bra; they're specifically designed to reduce bouncing and chafing of your boobs when you run. It's not really a good idea to run bra-less, you might end up getting sores around and under your boobs. Even A-cups need bras to hold them in place when running.
Because running involves much more drastic acceleration along the axis perpendicular to your tits. Compression bras will mitigate this to a noticeable degree, but possibly at the cost of discomfort from the compression of the ribcage.
Because there's more movement, you're heavier on your feet when you run (usually) so it jiggles everything that will move. Yeah, a bra would help reduce the giggling. Try a sports bra if regular bras are uncomfortable for you.