I think you will find the elbow comes into it just as much, though perhaps more emphasis on the wrist. Holding your pick firmly but not too tightly is important. Heck, as with violin, there is a complete set of skills for mandolin, many of which you already have with the violin. For example, you will automatically fret the strings better than a guitar player will as it is done much more in the manner of a violin than guitar.
One important thing is to keep a steady up/down rhythm. You might start by striking a string only on the down stroke then later strike an occasional note on the up. Only down strokes may give you quarter notes, add the up and you've got eighth notes while picking at the same speed.
Don't think on it too hard at first. Just get a feel for a smooth up/down rhythm.
There are quite a few online lessons out there, especially good is "Mike Marshall's Mandolin School"
http://artistworks.com/mandolin-lessons-mike-marshall
John
Austin TX
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