I heard many times phrases like "Смотри у меня" or "Смотри мне". As I understand they communicate a warning to a person, like "I am watching you", however their literal meaning is confusing. Please explain what makes them grammatically correct.
This is so called дательный риторический (rhetorical dative) or дательный экспрессивный (expressive dative) which is paralleled by the Latin dativus ethicus (ethic dative).
Historically, one of the dative's functions was to emphasise personal interest or involvement:
Что ми шумить, что ми звенить давечя рано предъ зорями? // What noise is that (I wonder), what peal is that (I wonder), just now early before the dawn?
It was widely used in Latin and Ancient Greek, as well as Old Russian, but there is no direct parallel in English for that construct.
If you squint hard enough, this is somewhat close to the English rhetoric construct in "We're almost there, soldier. Do you think you could stay awake another five minutes for me?", when it's not necessarily being done for you, but this is how you express your involvement in the process.
This usage is fossilized in several Russian expressions:
- Пойду-ка я себе домой
- Тоже мне герой
- Здесь вам не равнина
Another one of dative's functions was expressing possession, and some of the ethic dative expressions in Russian have since been modelled using a Russian way to express possession: the proximal possession using у plus the genitive:
- Кто это тут у нас? // Who's there?
- Он у нас чемпион // He's a champion
- Ты у меня не спи // Don't you sleep
All those sentences emphasize somewhat condescending interest or affection.
Смотри мне (у меня), смотрите мне (у меня)
Малый академичесаий словарь determines it as an interjection ( imperative interjection) which derived from the verb ( смотреть) Imperative interjections express wills, orders, warnings, threats ,etc. (Розенталь).
Смотрите, не опоздайте! (warning) Смотри у меня! (Threat) .
In the past interjections were often used with personal pronouns in the dative.
увы мне грешному, ох мне скверному
како суть прельщени; увы нам; увы мне
The remnants of this can be seen in folklore, so it was an ordinary thing.
ахти мне горевати; охти мне молоды , ах мне; исполать тебе,
So the meaning is approximately the following: (у меня , мне is a person who expresses his wish)
Behave yourself, I warn you, otherwise... Смотри мне (у меня).
It's something like "you'd [better] watch it" in English. It's indeed a warning that might even sound as a threat. You are asking about what justifies using of "у меня" or "мне" and, well, you can think of "смотри мне" as a short form of something like "смотри не натвори мне глупостей" and about "смотри у меня" you can think of as a short form of something like "смотри, будь у меня осторожным".
This, of course, might seem to be just a сheap trick, since it looks like it doesn't answer the question, because we still can ask "but what
у меня means in this context"? However if we will now try to translate both expressions this can give you a hint.
"у меня" is roughly an equivalent to "I got"
Like in phrases "у меня есть машина" or "у меня зазвонил телефон". So you can think of "смотри мне" as of shortened form of "look not to got me into trouble".
"мне" is roughly an equivalent to "me"
Like in "не досаждай мне", so you can think of "смотри мне" as of something like "look not to cause me troubles".
But! I was saying this here, on Russian SE dozens of time and I will repeat it now - just learn the phrases as they are, just accept them the way they are used as if it was an atomic, non-divisible entity. May be at some point in time you'll start to see things deeper, understanding what "будучи" in "будучи уверенным" means, or "мне" in "тоже мне" - but in my experience at first you just need to memorize.