Arwen vs. Eowyn. Well, let's think. (I'm approaching this like a
Grudge Match.
Okay, so we can't count on the Babe Factor, because they're both hot. (I'm not gay, I just think that if I want to look like someone, they're hot, and I'd like to look like either.)
On top of this, they're both got the Rage (TM). Eowyn has it aplenty, being a lonely girl. Arwen will soon have it, as soon as she learns there's any competition for Aragorn. 0 points awarded.
Let's talk about skill now. We can see that Arwen can clearly ride a horse well. I'll assume any horse, since she stole the one we saw her on. (This also speaks for the possibility of her fighting dirty, although that might not matter if this is a catfight.) Eowyn can of course ride well, being the most noble lady (and therefore the most priviledged, of the land of the horses. She's also Kick-A**, which I think should count as a category. If she can kill the King of the Nazgul, she can take on a sissy Elf. She's got the sword skills.
However, Arwen might too. We see her with that sword at the Ford. But we never see her use it.
On top of this, we have the Book vs. Movie factor. Eowyn does all this great stuff
in the book. Arwen doesn't. Their characters were designed by Tolkien to do certain things. Eowyn was designed to go work off her righteous Rage by killing people. Arwen spends her time sewing and singing.
I vote that Arwen gets a good beating, at least until Aragorn is torn apart in their tug-of-war. They'll soon be calling each other "sister", and proceed to start a feminist revolution. Aragorn is left in the mud, although he can't really get much dirtier anyhow.
Tie.