In movies you often see archers running around or on horseback with a quiver with arrows on their backs. I'm wondering if this is an accurate depiction of how archers kept their quivers?
If so, how were the arrows kept in position while running/moving so violently?
In addition to what /u/Hanrohan wrote, many horse archers, especially from the East, like Mongols, Tartars or Turks used quivers that were made of semi-rigid leather and held arrows tight between two layers of leather, usually in one, or someone in two rows. See here: https://kitabhona.org.ua/libwar_bow/kolchan1-1/1-4.jpg (Ottoman Empire, 2nd half of 16th century, property of German National Museum in Nürnberg).
Please take note that only a small part of the arrow is sandwiched between layers of leather what make the arrow easier to pull with quiver hanging on the lower back or hip of the archer.
Sometimes more elaborate constructions was used, with a wooden or leather divider holding the arrows what prevented them from rattling (and thus reducing chance of damage) and made reaching for arrows easier (archers were reaching for arrows while watching their targets or general surroundings). Such quiver can be seen here: https://kitabhona.org.ua/libwar_bow/kolchan1-1/1-2.jpg. (Ottoman Empire, 16th century, property of Art and History Museum in Vienna).
Just to show how the quivers were most possibly worn, please consult the contemporary drawing of an Ottoman archer with a good depiction of a quiver (made by Melchior Lorck in 1575): https://upload.wikimedia.org/wikipedia/commons/3/33/Melchior_Lorck_-_A_bearded_archer-2.jpg
And here are two photos of modern archers using traditional bow and quiver (note how the arrow stays in place even it is the last one and how the second archer hold the arrow she just pulled from quiver):
Photo #1 - http://www.alamy.com/stock-photo-ottoman-archer-riding-and-shooting-on-horseback-150390445.html Photo #2 - http://www.alamy.com/ottoman-archer-riding-and-shooting-on-horseback-image177833904.html
Its a good question with a simple answer, they didn't carry a quiver in the traditional 'robin hood' sense. Instead of an open cone, which you drop the arrows in, it was basically a bag, sometimes with some form of divider or structuring element, like a wooden framework, to keep the quiver rigid. Tied up tight, it would bundle the arrows together without them falling out. Here's an image of a (reconstruction) of an archers wargear. The bag can be seen above the arrows.
Most longbow men in the hundred years war would carry only about 20 arrows a piece, whilst the vast majority of the ammunition would be part of the armies baggage train, where arrows were stored in sheaves, (think like the cliché bundles of wheat).
As for the moving about, it was generally considered that the superior position in a battle was to be on the defensive, rather than the offensive, which suited English archers, who usually were moved as little as possible, because otherwise they weren't firing.
In addition, if they had to run away, most archers would ditch their arrows and bows, as captured English archers did not generally eceive a warm welcome.
Sources: The Hundred years war - Robin Neillands
The methods used by archers to carry their arrows are very diverse. Usually, a quiver or arrow case or arrow bag is used. Usually, the quiver is worn at the hip. The quiver can be short or long, open or closed. Arrows can be carried either point up or point down.
Methods without quivers:
Carry the arrows in the bow hand. You can't carry very many arrows like this; I've seen 6 carried in the bow hand, but 2-3 is easier.
Worn through belt. Only suitable for a few arrows. This rabbit has at least 3 worn through the belt. This human archer has 3.
Carry in the drawing hand. This is perhaps more useful as a way of having one or two immediately available arrows ready, while keeping most in a quiver or arrow case.
Archers on foot:
Closed quiver, point up. Point-up carry is most common for closed quivers.
Long open quiver, point up. This quiver would be worn at the hip.
Mounted archers:
Closed quiver, point up. The hinged lid of the quiver is clearly visible.
For transporting arrows, a variety of cases were used. These usually have lids, like the example from Cameroon. Note that the case is longer than the arrows. Some cases like this have long lids, and arrows can be removed more easily when shooting, but cases like this don't allow easy one-by-one removal - they're for transport rather than shooting from. Cases like this were used in Europe, Africa, and Asia. This example would be worn over the shoulder using the strap.
Quivers/bags/cases often have spacers or similar, to keep the arrows in place.
Sometimes you see quivers worn on the back:
