I support this idea of these two meeting during training. I've been hoping to do a swordplay log soon.

Forged anew by Elven hands, Andúril was once the broken shards of sword Narsil once wielded by King Elendil. This was the blade Aragorn carried during his journeys south as member of the Fellowship. Unlike any common sword, Andúril possesses the faint etchings of Elvish runes upon its blade and sheath that speak of its glory. /totally did not copy and paste

It's also known as the Flame of the West!