I am having some trouble understanding why accusative personal pronouns are added into a sentence that already has a nominative personal pronoun, for example.
Wie fühlst du dich?
Könnt ihr euch erinnern?
In example 1, there is dich and in example 2, there is euch. But to me, the sentences would make sense without dich and euch. They would be:
Wie fühlst du?
Könnt ihr erinnern?
Are these sentences wrong (without dich and euch)? And if so, why are they wrong?
In this case, the German translations of "to remember" and "to feel" are reflexive:
to remember [sth.] – sich [an etwas] erinnern
to feel [well] – sich [gut] fühlen
So that in fact, dich and euch are reflexive pronouns (which in turn behave like accusative personal pronouns). As a consequence, you need those pronouns for the verbs to mean what you want them to mean.
It would be similar to taking away the "myself" in
I hurt myself while sewing. → I hurt while sewing.
Furthermore, erinnern and fühlen are always transitive, meaning you need some kind of object. merely "Du fühlst" or "Ich erinnere" are not natural (exceptions will either fall under "artificial" or poetic licence).
Here are examples of non-reflexive use:
jemanden an etwas erinnern – to remind sb. of sth.
das Fell der Katze fühlen – to feel the cat's fur
I guess that's part of learning a language. It's not you who decide whether the grammar makes sense, but in cases, you have to learn rules which will contradict your intuition. That said, the property of verbs being reflexive it's not something preserved after you translate. To sum things up, you have to learn when a verb is reflexive.
If you write just
Wie fühlst du?
and you mean "how do you feel?" it's wrong. But maybe you are asking after the possible detection methods, then the non-reflexive version has a meaning. In that case it's even a scientific question, one of the partial answers being dank der Haut, die mit dem Nervensystem so und so verbunden ist, und so weiter und so fort.
It always helps visualizing what the words really mean or where they come from. And if you do that, it is relatively easy to see that "erinnern" describes the process of "going into oneself" — in order to get something which is stored there, but not directly relating to the memories themselves. "Erinnern" is about handling yourself, not your memories, thus the reflexive pronoun. "Remember", on the other hand, does not combine "re" + "membrum", as I first thought, thus describing how we put parts together again, but derives from "re" and "memor" (not "reminiscor", apparently), the latter basically meaning "keeping something from being forgotten". So, "to remember" indicates the direct handling of thing of the past, thus the accusative.
With "to feel", it's not so black and white as it appears in your question. There is an intransitive ("How do you feel?") and a transitive version of "to feel", in which case you also say "I feel myself ... " in English. And that easily explains why you need a reflexive pronoun with "fühlen" — it is a transitive verb.
So, it always helps to dig a little deeper into the matter as opposed to just saying, it is like that in my mother tongue, so it should be the same in the other language. Unfortunately, the other way round, German native speakers (and French, Italian, etc., by the way) are confronted with the problem of "false friends" where English terms and expressions are just literally translated without thinking of the proper German equivalent.
In addition to what others said (From this answer):
However, an increasing number of native German speakers in Germany (and probably none yet in Austria and Switzerland) now additionally use erinnern in much the same way that remember is used in English:
Ich erinnere noch die Anfangszeit des Unternehmens.
The usage here is more like reminisce or remember, that's one verb that is slowly starting to be accepted without its reflexiv helping pronoun.
