Inertial coordinate frames

Central to Newton-Euler mechanics is the idea that points in $ {\cal W}$ are expressed using an inertial coordinate frame. Imagine locating the origin and axes of $ {\cal W}$ somewhere in our universe. They need to be fixed in a way that does not interfere with our observations of the basic laws of motion. Imagine that we are playing racquetball in an indoor court and want to model the motion of the ball as it bounces from wall to wall. If the coordinate frame is rigidly attached to the ball, it will appear that the ball never moves; however, the walls, earth, and the rest of the universe will appear to spin wildly around the ball (imagine we have camera that points along some axis of the ball frame - you could quickly become ill trying to follow the movie). If the coordinate frame is fixed with respect to the court, then sensible measurements of the ball positions would result (the movie would also be easier to watch). For all practical purposes, we can consider this fixed coordinate frame to be inertial. Note, however, that the ball will dance around wildly if the coordinate frame is instead fixed with respect to the sun. The rotation and revolution of the earth would cause the ball to move at incredible speeds. In reality, inertial frames do not exist; nevertheless, it is a reasonable assumption for earth-based mechanical systems that an inertial frame may be fixed to the earth.

The properties that inertial frames should technically possess are 1) the laws of motions appear the same in any inertial frame, and 2) any frame that moves at constant speed without rotation with respect to an inertial frame is itself inertial. As an example of the second condition, suppose that the racquetball experiment is performed inside of a big truck that is driving along a highway. Ignoring vibrations, if the truck moves at constant speed on a straight stretch of road, then an inertial coordinate frame can be fixed to the truck itself, and the ball will appear to bounce as if the court was not moving. If, however, the road curves or the truck changes its speed, the ball will not bounce the right way. If we still believe that the frame attached to the truck is inertial, then the laws of motion will appear strange. The inertial frame must be attached to the earth in this case to correctly model the behavior of the truck and ball together.

Steven M LaValle 2012-04-20