У меня дрожали губы, когда я пил холодную воду на морозе.

Both are possible, but У меня дрожали губы is a very common and natural Russian way to talk about body parts and physical states.

Literally, it is something like At me, lips were trembling, but in natural English it means My lips were trembling.

Russian often avoids possessive words like мой / моя / мои when it is already obvious whose body part is being discussed. So with body parts, feelings, and physical conditions, у + genitive is extremely common.

• У меня дрожали губы = My lips were trembling
• У него болела голова = His head hurt
• У неё замёрзли руки = Her hands got cold

Мои губы дрожали is grammatically correct, but it sounds more marked or emphatic in many contexts.

Because the subject is губы, which is plural.

In the past tense, Russian verbs agree with the subject in number, and in the singular they also agree in gender.

• губа дрожала = the lip was trembling
• губы дрожали = the lips were trembling

So дрожали is the past plural form of дрожать.

Меня is in the genitive case because it follows у.

The pattern is:

• у + genitive

So:

• я → меня
• ты → тебя
• он → него
• она → неё

In this sentence, у меня shows whose lips were trembling.

Because губы is the grammatical subject of дрожали.

Even though English focuses on I in my lips were trembling, Russian structures this as:

• У меня = marks the person involved
• губы = the thing doing the action
• дрожали = agrees with губы

So губы stays in the nominative plural.

Because пил is imperfective, and that fits the idea of an ongoing action in the background.

The sentence means something like:

• when/while I was drinking cold water in the freezing cold

The focus is not on completing the action, but on the process happening at that time.

• я пил = I was drinking / I drank
• я выпил = I drank up / I finished drinking

In this sentence, пил is more natural because the trembling happened during the action.

If you said когда я выпил холодную воду, that would sound more like when I finished drinking the cold water, which changes the meaning.

Russian past tense singular forms show gender.

• я пил = I drank (said by a male speaker)
• я пила = I drank (said by a female speaker)

So the given sentence implies that the speaker is male.
If a woman says it, it becomes:

У меня дрожали губы, когда я пила холодную воду на морозе.

Because воду is the direct object of пил, so it is in the accusative case.

The noun вода is feminine singular:

• nominative: вода
• accusative: воду

The adjective must match it:

• nominative: холодная вода
• accusative: холодную воду

So:

• я пил холодную воду = I was drinking cold water

На морозе means in the cold, out in freezing weather, or in the frost.

This is a very common Russian expression. It does not literally mean standing on top of frost in the English sense. It is just the normal idiomatic way to say that something happens in freezing outdoor conditions.

• мороз = frost, freezing cold
• на морозе = out in the freezing cold

So когда я пил холодную воду на морозе means something like:

• when I was drinking cold water out in the freezing cold

Because Russian uses certain prepositions idiomatically with certain nouns, and мороз normally goes with на in this meaning.

So Russian says:

• на морозе = in the freezing cold

This is just the standard expression you need to learn as a set phrase.

Compare:

• на солнце = in the sun
• на холоде = in the cold
• на морозе = in the frost / freezing cold

It can be understood as either when or while, depending on how you translate it into English.

Because пил is imperfective, the idea is an ongoing action, so in English while often fits well:

• My lips were trembling while I was drinking cold water in the freezing cold.

But when is also perfectly possible:

• My lips were trembling when I was drinking cold water in the freezing cold.

So когда introduces the time clause, and the exact English wording depends on style.

Yes. Russian word order is flexible, though different orders can sound more or less natural depending on emphasis.

The original sentence is neutral and natural:

У меня дрожали губы, когда я пил холодную воду на морозе.

You could also say:

Когда я пил холодную воду на морозе, у меня дрожали губы.

This puts more focus first on the time/action situation.

You could also say:

Губы у меня дрожали, когда я пил холодную воду на морозе.

That is also possible, with a slightly different emphasis.

So the meaning stays basically the same, but the information structure changes.

Yes, дрожать is the normal verb for to tremble, to shake, or to shiver.

For lips, hands, voice, etc., it is very natural:

• дрожали губы = the lips were trembling
• дрожали руки = the hands were shaking
• дрожал голос = the voice was trembling

Depending on context, Russian might also use verbs like трястись, but дрожать is the most neutral and natural choice here.

Because Russian often leaves out possessive pronouns with body parts when the owner is already clear.

So instead of saying:

• мои губы

Russian often prefers:

• у меня губы
• у меня болит голова
• у меня замёрзли ноги

This is one of the most common patterns English speakers need to get used to. Russian often sounds less possessive than English in these situations.