Analysis Using Fictitious Forces
The appearance of fictitious forces normally is associated with use of a non-inertial frame of reference, and their absence with use of an inertial frame of reference. The connection between inertial frames and fictitious forces (also called inertial forces or pseudo-forces), is expressed, for example, by Arnol'd:
The equations of motion in an non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.
— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129
A slightly different tack on the subject is provided by Iro:
An additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.
— H Iro in A Modern Approach to Classical Mechanics p. 180
Fictitious forces do not appear in the equations of motion in an inertial frame of reference: in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, however, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces and to pretend you are in an inertial frame.
Treat the fictitious forces like real forces, and pretend you are in an inertial frame.
— Louis N. Hand, Janet D. Finch Analytical Mechanics, p. 267
It should be mentioned that "treating the fictitious forces like real forces" means, in particular, that fictitious forces as seen in a particular non-inertial frame transform as vectors under coordinate transformations made within that frame, that is, like real forces.
Read more about this topic: Mechanics Of Planar Particle Motion
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