During summer 2023 I applied for a PhD position in mathematics at a European university. I got invited for an interview but did not quite make it to being offered a position due to a number of factors.
A professor of the university emailed me and proposed that I apply again for a new PhD position at the same university.
My question is the following: During the period between the two applications nothing changed in my CV, and the letter of motivation I sent during the first application adresses the new position as well, since it is on the same mathematical field. Should I attach the same letter adapted for the new position, changing the name of the project and bureaucratic numbers? Is this a bad idea? Should I change its structure? Should I write a brand new one?