Learn Hardcore Finnish: Päivän lopussa olen väsynyt.

The inessive -ssa / -ssä regularly corresponds to English in or at a place or time.
So päivän lopussa is literally in the day’s end, used idiomatically as at the end of the day.

Why doesn’t Finnish use a preposition like at or of in this expression?

Finnish usually expresses what English does with prepositions by using case endings on nouns instead.

English: at the end of the day
Finnish: päivän (day + genitive) lopussa (end + inessive)

So:

of is expressed by the genitive (päivä → päivän)
at is expressed by the inessive (loppu → lopussa)

No separate words for at or of are needed; the endings handle that job.

Is Päivän lopussa a fixed phrase, or can I use the same pattern with other words?

It is not just a fixed phrase; it’s a productive pattern genitive + loppu + case.

You can say, for example:

Viikon lopussa – at the end of the week
Kuukauden lopussa – at the end of the month
Vuoden lopussa – at the end of the year
Tunnin lopussa – at the end of the lesson / hour

The meaning and structure are the same as in päivän lopussa.

What is the difference between päivän lopussa and päivän lopulla?

Both are correct, but there is a nuance:

Päivän lopussa (inessive -ssa)
- Focuses on the very end point of the day.
- Roughly: at the end of the day, when it has essentially ended or is just about to end.
Päivän lopulla (adessive -lla)
- Means towards the end of the day, not necessarily the very last moment.
- More like later in the day / towards the day’s end.

In everyday speech, people sometimes use them almost interchangeably, but lopussa is sharper, lopulla is looser/approximate.

Could I say Päivän lopuksi olen väsynyt instead? What would that mean?

You can say Päivän lopuksi olen väsynyt, but it has a slightly different nuance.

lopuksi comes from loppu
- -ksi (translative case) and often means finally / as a conclusion / by the end.
päivän lopuksi is more like finally, by the end of the day or as a result of the day.

Päivän lopussa olen väsynyt is a neutral way to say At the end of the day, I am tired (a state at that time).
Päivän lopuksi olen väsynyt feels more like summing up: By the time the day is over, I end up tired.

Could I move päivän lopussa to the end: Olen väsynyt päivän lopussa? Is that still correct?

Yes, Olen väsynyt päivän lopussa is grammatically correct.

The difference is mainly about information structure / emphasis:

Päivän lopussa olen väsynyt
- Starts with the time frame.
- Typical if you’re answering when you are tired, or setting the scene.
Olen väsynyt päivän lopussa
- Starts with I am tired.
- The time phrase feels more like an extra comment at the end.

Both are fine; Finnish word order is fairly flexible, but initial position often gives something a bit more emphasis.

Why is väsynyt in this form? Why not something like väsynyttä or väsynytnä?

Väsynyt is the nominative singular form of the adjective meaning tired.

With the verb olla (olen = I am), an adjective describing the subject is typically in the nominative:

Minä olen väsynyt. – I am tired.
Hän on iloinen. – He/She is happy.

You would not use the partitive väsynyttä here. That form appears in other structures (for example with some verbs, or when describing something in partitive).

Väsynytnä is the essive case and would mean being in a tired state, as in:

Hän tuli kotiin väsynytnä. – He/She came home (while) tired.

That is a different construction. For a simple I am tired, you use nominative väsynyt.

Why is there no minä in the sentence? Can I say Minä olen väsynyt?

The subject pronoun minä is optional in Finnish, because the personal ending on the verb (-n in olen) already shows the subject:

olen = I am
olet = you (sg.) are
on = he/she/it is

So:

Päivän lopussa olen väsynyt. – perfectly normal and natural.
Päivän lopussa minä olen väsynyt. – also correct, but minä adds emphasis (I am the one who is tired).

Using minä is often like stressing I in English.

Is päivän more like the day or a day in English? How do I know?

Finnish does not have articles (a / an / the), so päivän by itself can correspond to either of the day or of a day / of the day in general, depending on context.

In many general statements like this one, you would naturally translate it as:

At the end of the day I am tired.

If the wider context made one specific day clear (for example, a particular workday you are telling a story about), it could mean at the end of that day.
The form päivän itself does not show definiteness; you infer that from context.

How is this sentence different from using väsyttää, like Päivän lopussa minua väsyttää?

Both sentences are natural but have different grammar and a slightly different feel:

Päivän lopussa olen väsynyt.
- Literally: At the end of the day I am tired.
- Uses olla
  - adjective.
- Focuses on your state at that time.
Päivän lopussa minua väsyttää.
- Literally: At the end of the day, it tires me / I feel tired.
- väsyttää is an impersonal verb that means to make someone feel tired.
- minua is the object in partitive, literally me but functioning as I feel.

In everyday speech, both are used to express feeling tired, but:

olen väsynyt = I am tired (as a state).
minua väsyttää = I feel tired / I am being made tired (by something).

They often translate the same in English, but the Finnish structures differ.