Is there any reason why the same psalm is repeated?
Are these psalms composed by the same author?
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Psalm 14 is for the director of music, and is ascribed to David. Psalm 53 is also for the director of music and is “according to mahalath - a Maschil of David”. A ‘Maschil’ is a contemplative Psalm.
The Hebrew word ‘mahalath’ comes from a word meaning lyre. https://www.behindthename.com/name/mahalath
Although both psalms contain very similar words, the musical arrangement may have been different.
There is a clear lyrical difference towards the end of each psalm:
Psalm 14:5–6 states, “But there they are, overwhelmed with dread, / for God is present in the company of the righteous. / You evildoers frustrate the plans of the poor, / but the Lord is their refuge.” By contrast, Psalm 53:5 says, “But there they are, overwhelmed with dread, / where there was nothing to dread. / God scattered the bones of those who attacked you; / you put them to shame, for God despised them.”
What is the difference? Psalm 14 focuses more on God’s deliverance of the righteous, while Psalm 53 focuses more on God’s defeat of the wicked. It is possible that one of the songs is an adaptation of the earlier song, and the change in lyrics commemorates a specific event.
The third difference between the two psalms regards the use of God’s name. Psalm 14 uses “the Lord” (Yahweh) in verses 2, 4, 6, and 7. Psalm 53 uses “God” (Elohim) in all seven places where God is mentioned. Source: https://www.gotquestions.org/Psalms-14-53.html
Although the musical arrangements to both psalms may have been quite different, there is one significant similarity - both psalms are about the salvation of God. Both psalms end with these words: “Oh, that salvation for Israel would come out of Zion! / When the Lord restores his people, / let Jacob rejoice and Israel be glad!” (Psalm 14:7; cf. 53:6).
This article I found looks into the division of the psalms into five books: https://www.gotquestions.org/Psalms-five-books.html
I found this link mentioning their similarity. The writer suggests it may have been an editorial adaptation since they seem to have been written at different times based on the words each uses for God (Psalm 14 = Yahweh, Psalm 53 = Elohim). It could be similar to how some old hymns are currently being updated with modern language to appeal to modern culture.