No, its never there. I have seen the movie million times and in both cases it is not there. He is only wearing his brown jacket that gets torned at the sleeve.
Um, maybe he rolled it up and put it in his pack? I really have no idea, I'm just trying to come up with something. I know Aragorn wouldn't have lost his beautiful elf cloak. What PJ might have done with it, though, I can't say. :)
No, the cloak is there both times. The cloak isn't fastened properly (It's caught on his left shoulder) so you can't see the Leaf of Lothlorien, but you can see the cloak floating in the water beneath his left arm. Once he's on the bank the camera angle (from beside his left shoulder) allows you to see the brooch.
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