Learn Hardcore German: Nach dem langen Training bin ich körperlich müde, aber innerlich ruhig. - After the long training I am physically tired but calm inside.

There is a definite article (dem).
The noun is in the dative singular (because of nach).
With a definite article in the dative singular (all genders), the adjective takes -en.

So:

Nominative: das lange Training
Accusative: das lange Training
Dative: dem langen Training
Genitive: des langen Trainings

That is why it must be dem langen Training, not dem lange Training.

Why does the sentence say bin ich and not ich bin?

German main clauses follow the verb‑second (V2) rule: the finite verb is in the second position of the clause.

In your sentence, the first position is taken by the prepositional phrase:

Nach dem langen Training → first position
bin → finite verb → must come second
ich körperlich müde, aber innerlich ruhig → rest of the clause

So we get:

Nach dem langen Training bin ich …

If you put the subject first instead, then the verb follows it:

Ich bin nach dem langen Training körperlich müde, aber innerlich ruhig.

Both orders are correct; the version with Nach dem langen Training at the beginning emphasizes the time frame.

Why is Training capitalized?

In German, all nouns are capitalized, no matter where they appear in the sentence.

Training is a noun (it names an activity or event), so it must be written with a capital T: das Training.

Adjectives (lang, körperlich, innerlich) and verbs (bin) stay lower‑case unless they are at the start of a sentence or part of a proper name.

In English we just say “after long training”; why does German need dem (the article) here?

German uses definite and indefinite articles more often than English, especially with specific events or concrete instances.

Nach dem langen Training suggests a particular training session that both speaker and listener know about (e.g. “after the long training [we just had / we talked about]”).

You can say something more general in German too:

Nach langem Training bin ich … (no article, still dative: langem Training)

This sounds more like “after long periods of training / after training for a long time” in general, not one specific session.

In everyday speech, for a specific workout or practice, nach dem langen Training is the natural choice.

Why is there a comma before aber?

In German, when aber connects two main clauses, you must use a comma.

The sentence really consists of two (almost) full clauses:

(Nach dem langen Training) bin ich körperlich müde
(Nach dem langen Training) (bin ich) innerlich ruhig

In the second clause, bin ich is simply left out because it is understood, but grammatically it is still a main clause.

Because aber links these two main clauses, the comma before aber is obligatory:

…, bin ich körperlich müde, aber innerlich ruhig.

What exactly is going on in körperlich müde? Why not körperliche müde?

Here we have a predicate after the verb sein (bin):

ich bin körperlich müde

In this structure:

müde is a predicate adjective describing the subject ich.
körperlich modifies müde; it works like an adverb (physically tired).

Predicate adjectives in German do not take endings:

Ich bin müde.
Ich bin sehr müde.
Ich bin heute müde.
Ich bin körperlich müde.

You only add endings when the adjective stands directly before a noun:

die körperliche Müdigkeit (attributive adjective, with ending)
ein langes Training (adjective with article + noun)

So körperliche müde is wrong in this sentence; you need körperlich müde.

And what about innerlich ruhig – are these both adjectives, or is one an adverb?

Grammatically, it works the same way as körperlich müde:

ruhig is the predicate adjective after sein (understood bin ich).
innerlich modifies ruhig and behaves like an adverb, expressing on the inside / emotionally / mentally.

Again, because this is a predicate, no endings are used:

Ich bin ruhig.
Ich bin innerlich ruhig.

If you had a noun, then you would add endings:

eine innere Ruhe (an inner calm)
die innere Ruhe nach dem Training (the inner calm after the training)

Can I say Nachdem ich lange trainiert habe, bin ich körperlich müde, aber innerlich ruhig instead of Nach dem langen Training …?

Yes, that is grammatical and natural, but there is a structural difference:

Nach dem langen Training …
uses nach + noun (preposition + dative). It focuses on the event (“after the long training session”).
Nachdem ich lange trainiert habe, …
uses nachdem + clause (subordinating conjunction). It focuses on the action / process of having trained for a long time.

Both mean roughly the same in this context. The version with nachdem can feel a bit more descriptive of the activity itself, while nach dem Training feels more like referring to a definite session on the schedule.

Can I move innerlich and körperlich around in the sentence, or does the order have to stay like this?

German word order is fairly flexible here, but there are some tendencies:

The original:

Nach dem langen Training bin ich körperlich müde, aber innerlich ruhig.

Possible variants:

Nach dem langen Training bin ich innerlich ruhig, aber körperlich müde.
– still correct, but this now emphasizes the inner calm first and then contrasts it with physical tiredness.
Ich bin nach dem langen Training körperlich müde, aber innerlich ruhig.
– more neutral word order, starting with ich.
Ich bin körperlich müde, aber innerlich ruhig nach dem langen Training.
– grammatically OK, but nach dem langen Training at the end sounds a bit marked or poetic; in everyday speech you would usually put the time expression earlier.

Within the phrases körperlich müde and innerlich ruhig, however, the order is fixed:

körperlich müde ✅
müde körperlich ❌ (unidiomatic / wrong here)

Adverbs like innerlich, körperlich normally come before the adjectives they modify in such predicate constructions.