Mechanics of Planar Particle Motion - Fictitious Forces in Polar Coordinates - Polar Coordinates in An Inertial Frame of Reference

Polar Coordinates in An Inertial Frame of Reference

Below, the acceleration of a particle is derived as seen in an inertial frame using polar coordinates. There are no "state-of-motion" fictitious forces in an inertial frame, by definition. Following that presentation, the contrasting terminology of "coordinate" fictitious forces is presented and critiqued on the basis of the non-vectorial transformation behavior of these "forces".

In an inertial frame, let be the position vector of a moving particle. Its Cartesian components (x, y) are:

with polar coordinates r and θ depending on time t.

Unit vectors are defined in the radially outward direction :

and in the direction at right angles to :

These unit vectors vary in direction with time:

and:

Using these derivatives, the first and second derivatives of position are:

where dot-overmarkings indicate time differentiation. With this form for the acceleration, in an inertial frame of reference Newton's second law expressed in polar coordinates is:

where F is the net real force on the particle. No fictitious forces appear because all fictitious forces are zero by definition in an inertial frame.

From a mathematical standpoint, however, it sometimes is handy to put only the second-order derivatives on the right side of this equation; that is we write the above equation by rearrangement of terms as:

where a "coordinate" version of the "acceleration" is introduced:

consisting of only second-order time derivatives of the coordinates r and θ. The terms moved to the force-side of the equation are now treated as extra "fictitious forces" and, confusingly, the resulting forces also are called the "centrifugal" and "Coriolis" force.

These newly defined "forces" are non-zero in an inertial frame, and so certainly are not the same as the previously identified fictitious forces that are zero in an inertial frame and non-zero only in a non-inertial frame. In this article, these newly defined forces are called the "coordinate" centrifugal force and the "coordinate" Coriolis force to separate them from the "state-of-motion" forces.