To expound on my "nay," your PS is NOT more casual than another piece of writing. It may have a different style than that thesis you're writing, but casual isn't where you want to take it.
The PS is first and foremost a way to show adcoms about yourself, soemthign unique, something that doesn't come out elsewhere in the app. BUT it's also a writing sample. So much of being a lawyer is writing and drafting briefs and memoranda. They need some assurance that they're not accepting someone that can barely string two sentences together.