A noninertial reference frame is a frame of reference that is not an inertial frame of reference.^{[1]} As such, the laws of physics in such a frame do not take on their most simple form, as required by the theory of special relativity.^{[2]} To explain the motion of bodies entirely within the viewpoint of noninertial reference frames, fictitious forces (also called inertial forces, pseudoforces and d'Alembert forces) must be introduced to account for the observed motion, such as the Coriolis force or the centrifugal force, as derived from the acceleration of the noninertial frame. ^{[3]}^{[4]}
“  One might say that F = m a holds in any coordinate system provided the term "force" is redefined to include the socalled "reversed effective forces" or "inertia forces". …Lawrence E. Goodman, William H. Warner: Dynamics, p. 358  ” 
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Noninertial frames can be avoided. Of course, measurements with respect to noninertial reference frames can be transformed to an inertial frame, incorporating directly the acceleration of the noninertial frame as that acceleration is seen from the inertial frame.^{[5]}. This approach avoids use of fictitious forces (it is based on an inertial frame, where fictitious forces are absent, by definition) but it may be less convenient from an intuitive, observational, and even a calculational viewpoint.^{[6]} As pointed out by Ryder for the case of rotating frames as used in meteorology:^{[7]}
“  A simple way of dealing with this problem is, of course, to transform all coordinates to an inertial system. This is, however, sometimes inconvenient. Suppose, for example, we wish to calculate the movement of air masses in the earth's atmosphere due to pressure gradients. We need the results relative to the rotating frame, the earth, so it is better to stay within this coordinate system if possible. This can be achieved by introducing fictitious (or "nonexistent") forces which enable us to apply Newton's Laws of Motion in the same way as in an inertial frame. …Peter Ryder: Classical Mechanics, pp. 7879  ” 
That a given frame is noninertial can be detected by its need for fictitious forces to explain observed motions. For example, the rotation of the Earth can be observed using a Foucault pendulum.^{[8]} The rotation of the Earth seemingly causes the pendulum to change its plane of oscillation (which plane actually is fixed in space) because the surroundings of the pendulum move with the Earth. As seen from an Earthbound (noninertial) frame of reference, the explanation of this apparent change in orientation requires the introduction of the fictitious Coriolis force.
Another famous example is that of the tension in the string between rotating spheres. In that case, prediction of the measured tension in the string based upon the motion of the spheres as observed from a rotating reference frame requires the rotating observers to introduce a fictitious centrifugal force .
In general, the identification of a frame as noninertial is established by the presence of fictitious forces.^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}
“  The effect of his being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations…. …Sidney Borowitz and Lawrence A Bornstein: A Contemporary View of Elementary Physics, p. 138  ” 
“  If we insist on treating mechanical phenomena in accelerated systems, we must introduce fictitious forces, such as centrifugal and Coriolis forces. These fictitious forces are strictly of a kinematical nature and appear when the motion is expressed in terms of rotating coordinate systems. …Leonard Meirovitch: Methods of Analytical Dynamics , p. 4  ” 
Arnol'd says:^{[10]}
“  The equations of motion in an noninertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the noninertial nature of a system. …V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129  ” 
As stated by Iro:^{[14]}^{[15]}
“  An additional force due to nonuniform relative motion of two reference frames is called a pseudoforce. …Harald Iro in A Modern Approach to Classical Mechanics p. 180  ” 
A different use of the term "fictitious force" often is used in curvilinear coordinates, particularly polar coordinates. To avoid confusion, this distracting ambiguity in terminologies is pointed out here. These socalled "forces" are nonzero in all frames of reference, inertial or noninertial, and do not transform as vectors under rotations and translations of the coordinates (as all Newtonian forces do, fictitious or otherwise).
This incompatible use of the term "fictitious force" is unrelated to noninertial frames. These socalled "forces" are defined by determining the acceleration of a particle within the curvilinear coordinate system, and then separating the simple doubletime derivatives of coordinates from the remaining terms. These remaining terms then are called "fictitious forces". More careful usage calls these terms "generalized fictitious forces" to indicate their connection to the generalized coordinates of Lagrangian mechanics. The application of Lagrangian methods to polar coordinates can be found here.
