Does MySQL updated index if index key is not updated?
It is logical to think that the current databases are smart enough to not update the index which data have not changed.

But this logic should be straight supported by Mysql manual. And I've really found this statement on page which describes Speed of UPDATE Statements:


  An update statement is optimized like a SELECT query with the
  additional overhead of a write. The speed of the write depends on the
  amount of data being updated and the number of indexes that are
  updated. Indexes that are not changed do not get updated.