Have to witness a crime to write a ticket. i.e. can he write a ticket even if he was not there to witness the offense and has no witnesses to the offense?

If he didn't witness the crime, and nobody else witnessed the crime, then how did he determine that there was a crime?

If he finds evidence of a crime (such as a piece of your car at the scene of a crash), he has grounds to write a ticket for leaving the scene. But how does he know who to give it to, if multiple people drive the car?