Why does Naruto run like this?
It looks silly to me, and I'm not seeing the advantage.
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This isn't at all unique to Naruto. In fact, it's so common, it has its own TV tropes entry: Ninja Run.
It seems to be a stylistic choice in anime and manga. I suspect it originated in order to emphasize the line of action, which is a concept in art, especially comics and animation, to draw a character with their body following a specific line or curve in order to emphasize the concept that the character is moving by drawing the viewer's eye in the direction of the movement the artist is trying to depict. The arms splayed out behind emphasize the line of action that is parallel to the ground, as the characters are supposed to be running across the ground very quickly. This stylistic choice made more sense in the days when all anime had very few frames of animation, so the illusion of movement through actual animation was minimal.
You can see this in loads and loads of anime. If you see characters run like this - they are probably ninjas or samurais. In those shows you can often see that different characters run in different styles even if they are all skilled fighters. The running style is a result of the "fighting culture" the character is from.
As to why anime shows ninjas running like this, there is no distinct answer. It is considered that it should reflect some technique from ancient Japanese martial arts - there are different theories as to why anyone would do that and none of them is convincing enough.
But apparently it is a myth that ether ninja or samurais were train to run with their hands behind their backs. There is no historical evidence of that, so probably it is just a stylistic decision in anime and manga to add some mystic vibe to legendary Japanese martial arts.
When I was in Army training (20 years ago), I had a fellow soldier run with his hands clasped behind his back. He said it limited his breathing during the slower training runs.
This guy was what I'd call a true runner, as he normally ran sub 4 minute miles. He would normally run/sprint the 2 mile PT test in 7.something minutes.
I've also talked to other runners more recently and they've said that while running, your body should be perpendicular to the road. They went on to say that when running downhill, your arms should be trailing behind you to maintain balance.
The guy in the above anime is running bent over forward, so it may also be to maintain balance.
Ok, so this is long after this Question was asked and answered, but I think I just found the most real answer someone could get.
While randomly viewing YouTube videos, I found one where a lady talked to a real ninja and showed him the Naruto running style. He said it was accurate and explained that swinging your arms wastes energy, leaning forward puts your center of gravity in front of your feet, and your arms are there to help maintain balance.
This ninja also demonstrated (a little) this technique.
In a previous video, the same ninja mentioned that swinging the arms and torso would also swing the swords at the belt, which would get in the way while running, and demonstrated why he kept his right hand at his belt while running.
IIRC, in medieval Japan, samurai would run this way because they believed it to be more efficient, and it it allowed them to run while carrying weapons without waving them all over the place.
Then some European sailors turned up, they both noticed the differences in running styles, and soon some foot races were had and it was discovered that the upright posture with swinging arms was a more effective manner of running, so this running posture fell out of favour until it came to be associated with the ninja in Japanese popular culture.