How do we explain using the definite article for adjectives in "the beautiful"? Is it possible in Poetry? or is it possible in titling as in newspapers if I am not mistaken? I think it is equivalent to "The beautiful America", but I would like to know if there are any grammar rules I do not know of.
This is a standard "nickname/epithet" construction, well-known in forms like...
Because they're bestowed titles, such epithets don't have to adhere to conventional principles of grammar. But probably for all examples of "X the Y" (where X is a proper noun and Y is an adjective), you could parse it as...
"X who is known as the archetypal example of a Y [king/country/Viking invader/etc.]".
I don't think OP is quite correct in assuming America the beautiful is equivalent to the beautiful America. Firstly because that second form (which would rarely be encountered) tends to imply there might be other countries equally deserving of the descriptor beautiful. Secondly because it would often be used in contexts where the speaker/writer also refers to another "America" which is not, in fact, beautiful ("an ugly America", in that link).
Note that this doesn't apply to all descriptive epithets. For example, Jones is a common surname in Wales, so three people with that name in one village might be identified as Jones the Bread, Jones the Mill, Jones the Milk. In those cases, the "descriptive" element is a noun. But arguably they're a different type of epithet, used primarily for disambiguation rather than to provide additional information about a single "already-known" entity.
Largely agree with FumbleFingers. An additional thought:
Adjectives can be used as nouns, meaning the idea of that thing, or a person or object that is in some way representative of that thing.
For example, one might write, "How can we know the good?" Meaning, how can we understand this idea of "good", or how can we know what things qualify as "good".
It's a short step from there to characterizing a particular individual as an example of the adjective. "I think that Fred best meets our ideal of the Diligent."
And from there it's a short step to giving someone or something a title reflecting this characterization, "Ethelred the Unready", "America the Beautiful", and so forth, as FumbleFingers discusses.
And sure, in most cases I don't think such a title would be understood to mean that we believe that that example is the only example of the abstraction, or even necessarily the most or the best. Just that it is AN example. Hagar the Horrible is not the only horrible person in the world: even being of Norwegian ancestry myself, I must concede that there have been other horrible people in history. I don't suppose he's even the most horrible. Just that he is horrible.