Лікар каже, що після тренування можуть боліти м’язи, спина і плечі.

Що introduces a subordinate clause and means that.

So Лікар каже, що... is the normal way to say The doctor says that... In Ukrainian, this що is usually kept, whereas in English that is often optional.

Yes, Лікар говорить, що... is possible, but каже sounds especially natural when you are reporting the content of what someone says.

A useful rough distinction is:

• казати / каже = to say
• говорити / говорить = to speak, to talk, to say

In sentences with reported speech, каже, що... is very common.

After the preposition після you use the genitive case.

So тренування here is genitive singular. It just happens to look the same as the nominative singular, which is common with some neuter nouns.

• nominative: тренування
• genitive: тренування

So even though the form does not change, the case still does.

Yes, but it means something slightly different.

• після тренування = after the workout / after a workout
• після тренувань = after workouts / after training sessions

The singular is often used generically too, so після тренування can mean after exercise in a general sense.

Because можуть is a modal verb: can / may. After a modal verb, Ukrainian uses the infinitive.

So:

• болять = hurt / are hurting
• можуть боліти = may hurt / can hurt

If you removed можуть, then you would need a finite verb:

• М’язи, спина і плечі болять.

But with можуть, you say:

• М’язи, спина і плечі можуть боліти.

Because the subject is the whole list:

• м’язи
• спина
• плечі

Together they form a compound subject, and that makes the verb plural.

So even though спина is singular, the full subject is plural overall, which is why you get можуть.

They are in the nominative case because they are the subject of боліти.

This is important for English speakers, because English often makes pain feel object-like in meaning, but in Ukrainian the body part itself is the grammatical subject:

• спина болить = the back hurts
• плечі болять = the shoulders hurt

So here м’язи, спина і плечі are not objects; they are the things doing the hurting grammatically.

Not the same.

• боліти = to hurt, to ache, to be sore
• хворіти = to be ill, to be sick, to suffer from an illness
• заболіти = to start hurting

So for muscles, back, shoulders, head, stomach, and similar body-part pain, боліти is the normal verb.

Examples:

• У мене болить голова. = My head hurts.
• Після бігу можуть боліти ноги. = After running, your legs may hurt.

Because that is simply the natural way to refer to those body parts here.

• м’язи = muscles, normally plural
• плечі = shoulders, normally plural because there are two
• спина = back, normally singular

If you wanted to talk about one shoulder, you could say плече.

Ukrainian often leaves out possessives when they are obvious from context, especially with body parts.

In English, you might often say your muscles, back, and shoulders may hurt. In Ukrainian, that would be possible, but it is also very normal to say just:

• можуть боліти м’язи, спина і плечі

If needed, you could add a possessive for clarity:

• можуть боліти ваші м’язи, спина і плечі

But the version without ваші sounds natural and general.

Grammatically, можуть is present tense, but in this sentence it expresses a general possibility. In English, that is often translated with a future-like idea:

• may hurt after a workout
• can get sore after exercise

So it is not really about what is happening right now. It is a general statement about what can happen after training.

Yes. Ukrainian word order is fairly flexible, although some orders sound more neutral than others.

The given sentence is natural and neutral:

• Лікар каже, що після тренування можуть боліти м’язи, спина і плечі.

You could also say:

• Лікар каже, що м’язи, спина і плечі можуть боліти після тренування.

Both are correct. The difference is mostly about emphasis and flow, not basic meaning.

Because що introduces a subordinate clause, and Ukrainian normally separates that clause with a comma.

So:

• Лікар каже, що...

is punctuated correctly.

There is no comma before і here because м’язи, спина і плечі is just a simple list.