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What is the main purpose behind him doing this?

Too keep his arms connected to his body during the swing.

why not the right side as well?

They have a drill for that as well, but you use a towel. Vijay is obviously working on his left side connectivity.

You can also use a head cover.........

I think the drill with the towel under the right arm has been ruled out by most pros...this used to be the way to feel keeping your arms connected, but after Nicklaus started using something under the left arm instead of the right supposedly everyone started looking at the drill better and discovered that the towel or whatever it is under your right arm doesn't do a whole lot of anything. I don't know, just what I read in one of the golf mags.

I remember reading that VJ at one time used a tennis ball under his left arm but injured himself and went to the glove.

I've had to use one of them earlier thi year... and one of those elastic bungy cords wrapped around my elbows to fix my flying elbow! I looked really weird walking around when i couldn't move my arms lol

Are you trying to keep the glove in place all the way to finish or is it supposed to fall out at a certain point in the swing?

If you're doing the drill right the glove will stay under your left arm the whole time. According to Golf Magazine most high handicappers might only want to try the half swing with this drill as they will not be able to get a full swing.

I agree with every one here. Note that the left arm will move across your chest some and back but you should not lose the glove except at the very end if your left elbow comes up. This is very good practice for curing a chicken wing block shot.