Non-inertial reference frame

Non-inertial reference frame
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A non-inertial reference frame is a reference frame that is not an inertial reference frame.¹ As such, the laws of physics in such a frame do not take on their most simple form, as required by the special principle of relativity.² To explain the motion of bodies entirely within the viewpoint of non-inertial reference frames, fictitious forces (also called inertial forces, pseudo-forces 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 non-inertial frame. ³⁴

One might say that F $=$ m a holds in any coordinate system provided the term "force" is redefined to include the so-called "reversed effective forces" or "inertia forces".

– Lawrence E. Goodman, William H. Warner: Dynamics, p. 358

Of course, measurements with respect to non-inertial reference frames can be transformed to a convenient inertial frame, incorporating directly the acceleration of the non-inertial 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 and even a calculational viewpoint. As pointed out by Ryder for the case of rotating frames as used in meteorology:⁶

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 "non-existent") 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. 78-79

That a given frame is non-inertial 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.⁷ The rotation of the Earth causes the pendulum to change its plane of oscillation (fixed in space) with respect to its surroundings (moving with the Earth). The explanation of the apparent change in orientation from an Earth-bound (non-inertial) frame of reference 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 non-inertial is established by the presence of fictitious forces.⁸⁹¹⁰¹¹¹²

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:¹³

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

As stated by Iro:¹⁴¹⁵

An additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.

– Harald Iro in A Modern Approach to Classical Mechanics p. 180

References and notes

^ Emil Tocaci, Clive William Kilmister (1984). Relativistic Mechanics, Time, and Inertia. Springer, p. 251. ISBN 9027717699.
^ Wolfgang Rindler (1977). Essential Relativity. Birkhäuser, p. 25. ISBN 354007970X. , Ludwik Marian Celnikier (1993). Basics of Space Flight. Atlantica Séguier Frontières, p. 286. ISBN 2863321323.
^ Lawrence E. Goodman & William H. Warner (2001). Dynamics, Reprint of 1963 edition, Courier Dover Publications. ISBN 048642006X.
^ Albert Shadowitz (1988). Special relativity, Reprint of 1968 edition, Courier Dover Publications, p. 4. ISBN 0486657434.
^ M. Alonso & E.J. Finn (1992). Fundamental university physics. , Addison-Wesley. ISBN 0201565188.
^ Peter Ryder (2007). Classical Mechanics. Aachen Shaker, pp. 78-79. ISBN 978-3-8322-6003-3.
^ Giuliano Toraldo di Francia (1981). The Investigation of the Physical World. CUP Archive, p. 115. ISBN 052129925X.
^ Raymond A. Serway (1990). Physics for scientists & engineers, 3rd Edition, Saunders College Publishing, p. 135. ISBN 0030313589.
^ V. I. Arnol'd (1989). Mathematical Methods of Classical Mechanics. Springer, p. 129. ISBN 978-0-387-96890-2.
^ Milton A. Rothman (1989). Discovering the Natural Laws: The Experimental Basis of Physics. Courier Dover Publications, p. 23. ISBN 0486261786.
^ Sidney Borowitz & Lawrence A. Bornstein (1968). A Contemporary View of Elementary Physics. McGraw-Hill, 138.
^ Leonard Meirovitch (2004). Methods of analytical Dynamics, Reprint of 1970 edition, Courier Dover Publications, p. 4. ISBN 0486432394.
^ V. I. Arnol'd (1989). Mathematical Methods of Classical Mechanics. Springer, p. 129. ISBN 978-0-387-96890-2.
^ Harald Iro (2002). A Modern Approach to Classical Mechanics. World Scientific, p. 180. ISBN 9812382135.
^ In this connection, it may be noted that a change in coordinate system, for example , from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of fictitious forces, despite the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another.

References and notes

See also