"State-of-motion" Versus "coordinate" Fictitious Forces
Earlier in this article a distinction was introduced between two terminologies, the fictitious forces that vanish in an inertial frame of reference are called in this article the "state-of-motion" fictitious forces and those that originate from differentiation in a particular coordinate system are called "coordinate" fictitious forces. Using the expression for the acceleration above, Newton's law of motion in the inertial frame of reference becomes:
where F is the net real force on the particle. No "state-of-motion" fictitious forces are present because the frame is inertial, and "state-of-motion" fictitious forces are zero in an inertial frame, by definition.
The "coordinate" approach to Newton's law above is to retain the second-order time derivatives of the coordinates {qk} as the only terms on the right side of this equation, motivated more by mathematical convenience than by physics. To that end, the force law can be rewritten, taking the second summation to the force-side of the equation as:
with the convention that the "acceleration" is now:
In the expression above, the summation added to the force-side of the equation now is treated as if added "forces" were present. These summation terms are customarily called fictitious forces within this "coordinate" approach, although in this inertial frame of reference all "state-of-motion" fictitious forces are identically zero. Moreover, these "forces" do not transform under coordinate transformations as vectors. Thus, the designation of the terms of the summation as "fictitious forces" uses this terminology for contributions that are completely different from any real force, and from the "state-of-motion" fictitious forces. What adds to this confusion is that these "coordinate" fictitious forces are divided into two groups and given the same names as the "state-of-motion" fictitious forces, that is, they are divided into "centrifugal" and "Coriolis" terms, despite their inclusion of terms that are not the "state-of-motion" centrifugal and Coriolis terms. For example, these "coordinate" centrifugal and Coriolis terms can be nonzero even in an inertial frame of reference where the "state-of-motion" centrifugal force (the subject of this article) and Coriolis force always are zero.
If the frame is not inertial, for example, in a rotating frame of reference, the "state-of-motion" fictitious forces are included in the above "coordinate" fictitious force expression. Also, if the "acceleration" expressed in terms of first-order time derivatives of the velocity happens to result in terms that are not simply second-order derivatives of the coordinates {qk} in time, then these terms that are not second-order also are brought to the force-side of the equation and included with the fictitious forces. From the standpoint of a Lagrangian formulation, they can be called generalized fictitious forces. See Hildebrand, for example.
Formulation of dynamics in terms of Christoffel symbols and the "coordinate" version of fictitious forces is used often in the design of robots in connection with a Lagrangian formulation of the equations of motion.
Read more about this topic: Mechanics Of Planar Particle Motion, Fictitious Forces in Curvilinear Coordinates
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