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Referring to the image below, the arm involved in the exercise is bent to 90 degrees throughout the exercise, as if doing hammer curls, only that the arm stays in that position throughout this exercise. The arm rotates on the shoulder to move the dumbbell towards chest and then away from it multiple times. There is not vertical movement; all movement is horizontal.

It seems to me the muscles moving the dumbbell towards the chest are a combination of shoulder and chest, and for moving away are the antagonist shoulder muscles and back muscles. From this it therefore seems that the amount of dumbbell weight mostly affects the bicep holding it up, not so much the muscles involved in rotating the dumbbell.

So, what is this exercise called, its purpose, and is the weight (when using a dumbbell) much of an influence on anything else besides the bicep?

Update

This exercise makes sense if using horizontal resistance, for example when using cable as shown below, therefore putting the resistance on the shoulder, chest or back, depending on the side you are pulling from. But with a dumbbell, it seems completely redundant...

It is a form of external rotation and works on the rotator cuff. Related exercises covers the topic in more detail, but I'll emphasize using light weights as the muscles involved are comparatively small.

Using a horizontal cable to put all of the load against the direction of motion, or lying down so that gravity accomplishes the same thing is probably more effective. Nonetheless, I have had a trainer have me perform both variations that you have shown and moving the dumbbell from side to side still works the rotator cuff even if the biceps are maintaining the vertical position of the dumbbell.

Your intuition that the muscles moving the barbell left and right are not working against gravity is correct - these are the internal and external rotators of the shoulder, and the only resistance they will have is the momentum of the dumbbell, not its weight. The bicep and anterior deltoid will be isometrically loaded to hold the dumbbell up. This is very distinct from the same movement performed with a cable, or with a dumbbell while lying on the ground, in which case either the internal or external shoulder rotators will experience resistance from gravity.

So, I'd call it an isometric bicep hold with added, cyclic shoulder rotation.

I couldn't guess what its intended purpose is, as that would require insight into the mind of the person who came up with it, however I would suggest that the only thing it would be useful for would be to allow you to identify that the person performing or prescribing the exercise doesn't know what they're doing.