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In the sentence "Ножка у сту́ла сломалась" (the chair leg broke), why do we need the у? The genitive case on "стул" already lets you know that it's the chair's leg that's broken. More generally, when and why is у used to indicate posession in Russian rather than simply using the genitive case?
By the way, I'm aware that "сту́ла" is already in the genitive, I'm asking why у is there as well.
First, try to recall how Russian typically expresses possession:
The same у + Genitive can introduce a sentence about something related to or possessed by a person or an object (always without "есть"):
You can interpret it as "The mouse I have isn't working", but essentially it is understood as if you just used a possessive construction ("моя мышка", "окна дома", "брат моей жены"). To my ear, it is a tiny bit less formal, but still quite neutral in style — I am sure I would not use in a legal document and probably in a scientific article but would not hesitate if I need it anywhere else.
Since Russian has a flexible word order, you can emphasize different words using both word order an intonation. So several options are valid:
This device has its niche because it may be useful from a narrative point of view. Rather than stating facts about something belonging to something, you inform your listener that the sentence is about what happened to the object in question, what is interesting about that object, what the situation is.
"У меня мама заболела" is a bit more personal (it means "My mother is ill"), since it focuses on "your situation" ("As for me, my mother is ill"). "Моя мама заболела" is just the most straightforward way: anyone could replace "my mother" in teh sentence, and it would keep the same overall tone.
Another example. Say, someone's dear pet died, and their friend noticed they are sad. Then "У меня собака умерла" is a far more natural thing to say than "Моя собака умерла": after all, you are conmmenting on your situation, not on what your dog did.
UPD: Also, look at the first part of Anixx's answer. "Ножка стула" can truly mean just any leg of a chair or chair leg, including legs severed from the chair. Or those freshly made and not yet attached to any chair. So, it may mean the leg that is intended for use with a chair but not currently associated with any specific chair (similarly to how bath rug may be different from rug in the bathroom).
"У стула ножка/Ножка у стула ...." explicitly means the leg that the chair currently has — or at least had (has) at the moments you are referring to. An example of such use: «У моего кресла ролик заедает. Вот, я принёс — можешь глянуть?» (My armchair's castor gets stuck. Here, I brought it — could you please take a look? → here you mean that the castor jams when it is still attached).
- Ножка стула = chair leg (designed to serve as a leg in a chair, not necessary currently a part of a chair)
- Ножка у стула = leg of a/the chair (currently serving as part of a/the chair, attached)
If the legs are separated from the chair (for instance the chair in a disassembled state), in this case you would use "ножка стула" but not "ножка у стула".
Я заказал новую ножку стула в магазине, чтобы заменить сломанную = I have ordered a chair leg from a store so to replace the broken one.
If you are speaking about legs belonging to two or more chairs, you would use "ножка стула", while if you are speaking about one chair you say "ножка у стула".
Смотри, одна ножка стула синяя, а другая красная = Look, one chair leg is blue, the other is red. (legs of different chairs or the same chair or detached from chairs)
Смотри, одна ножка у стула синяя, а другая красная = Look, one leg of a/the chair is blue, but the other is red, (legs of the same chair)
Смотри, у одного стула ножка синяя, а у другого красная = Look, the leg of one chair is blue, but of the other one is red. (legs of different chairs)
Another difference is that I would say it conveys a coloring somewhat similar to indefinite article in English. "Ножка у стула сломалась" largerly means you introduce a new chair, not discussed or implyed before, while "ножка стула сломалась" kinda implies that is is known about which chair you are speaking. If it is the chair which you are sitting in at the moment, you can use the both.
Note also the following. If you want to underline that it is the leg of the chair is broken, you should use у:
Что случилось? "What happened?"
Ножка у стула сломалась. "A leg of a/the chair has broken" or "A chair's leg has broken"
If you want to underline that the leg is broken, you should not use the у:
Что c ножкой стула? "What happened to the chair's leg?"
Ножка стула сломалась. "The chair's leg has broken"
I concur with Shady_arc's analysis but would like to elaborate on the final statement which I think summarises the quintessence of the matter, namely: you are commenting on your situation, not on what your dog did.
First, consider a simple example:
Let us develop one of Shady_arc's examples to try and demonstrate the difference in the emphasis of each sentence:
Now I'll give my mouse to Andrey.
which is almost equivalent to the first example, it still describes the condition of my mouse. However,
Similarly:
So the original sentence indeed describes the chair which has been rendered unusable due to a broken leg, rather than the broken chair leg which may or may not have rendered some chair unusable.
Interestingly, a search for "My dog died" returns over 3 million hits in Google, as opposed to "I lost my dog" - 1 million. In Russian, on the other hand, the construct "У меня что-то случилось с предметом (человеком)" will be by far more widespread than the construct "С моим предметом (человеком) что-то случилось". I'll refrain from drawing any conclusions on possessiveness of English vs egocentricity of Russian. However, I could suggest this final illustration which might shed some additional light on the matter:
I hope this helps.
I cannot give exact reference to rule, but in this case there is no difference, both are correct. Here у means that chair has a leg. Probably, without у will be a bit more correct, I'm not sure.
Russian is my native language, but I really never thought about this problem before. You can use both variants.
To me the variant "Ножка у сту́ла сломалась" sounds more informal than "Ножка сту́ла сломалась". It almost gives animate qualities to the стул. Compare, for example, "Нога у меня болит" (and other sad examples with mice and dogs given by others on this page). You would not be able to say "Нога меня болит" though. The preposition "У", in the sense of possession, seems to imply an animate subject.
