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A non-inertial 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 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. [3][4]

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

Contents

Avoiding fictitious forces in calculations

Non-inertial frames can be avoided. Of course, measurements with respect to non-inertial reference frames can be transformed to an 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, 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 "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

Detection of a non-inertial frame: need for fictitious forces

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.[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 Earth-bound (non-inertial) 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 non-inertial 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 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:[14][15]

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

Fictitious forces in curvilinear coordinates

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 so-called "forces" are non-zero in all frames of reference, inertial or non-inertial, 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 non-inertial frames. These so-called "forces" are defined by determining the acceleration of a particle within the curvilinear coordinate system, and then separating the simple double-time 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.

References and notes

  1. ^ Emil Tocaci, Clive William Kilmister (1984). Relativistic Mechanics, Time, and Inertia. Springer. p. 251. ISBN 9027717699. http://books.google.com/books?id=7dVRL51JRI0C&pg=PA251&dq=%22non-inertial+frame%22&lr=&as_brr=0&sig=ACfU3U2Nu9ibcmQrKNrnhgvDYD7i007DmA.  
  2. ^ Wolfgang Rindler (1977). Essential Relativity. Birkhäuser. p. 25. ISBN 354007970X. http://books.google.com/books?id=0J_dwCmQThgC&pg=PT43&dq=laws+physics+%22simplest+form%22&lr=&as_brr=0&sig=ACfU3U357CbPUxoReKNgKU9J7bUzEoHTzQ.  , Ludwik Marian Celnikier (1993). Basics of Space Flight. Atlantica Séguier Frontières. p. 286. ISBN 2863321323. http://books.google.com/books?id=u2kf5uuaC6oC&pg=PA286&dq=laws+physics+%22simplest+form%22&lr=&as_brr=0&sig=ACfU3U3cW7wsafRC_dG_sX1XUziOmlfL7Q.  
  3. ^ Lawrence E. Goodman & William H. Warner (2001). Dynamics (Reprint of 1963 edition ed.). Courier Dover Publications. ISBN 048642006X. http://books.google.com/books?id=2z0ue1xk7gUC&printsec=frontcover&dq=Goodman+and+Warner&lr=&as_brr=0&sig=ACfU3U1pnp7jgg-HLvhEKN6nql1ahCRzcQ#PPA358,M1.  
  4. ^ Albert Shadowitz (1988). Special relativity (Reprint of 1968 edition ed.). Courier Dover Publications. p. 4. ISBN 0486657434. http://books.google.com/books?id=1axfJqUT6R0C&pg=PA4&dq=%22inertial+force%22&lr=&as_brr=0&sig=ACfU3U1UisQ10N7FZP4L4dJM5iKgco6AdA.  
  5. ^ M. Alonso & E.J. Finn (1992). Fundamental university physics. , Addison-Wesley. ISBN 0201565188. http://books.google.com/books?id=c5UAAAAACAAJ&dq=isbn=0201565188&lr=&as_brr=0.  
  6. ^ “The inertial frame equations have to account for VΩ and this very large centripetal force explicitly, and yet our interest is almost always the small relative motion of the atmosphere and ocean, V' , since it is the relative motion that transports heat and mass over the Earth. … To say it a little differently—it is the relative velocity that we measure when [we] observe from Earth’s surface, and it is the relative velocity that we seek for most any practical purposes.” MIT essays by James F. Price, Woods Hole Oceanographic Institution (2006). See in particular §4.3, p. 34 in the Coriolis lecture
  7. ^ Peter Ryder (2007). Classical Mechanics. Aachen Shaker. pp. 78–79. ISBN 978-3-8322-6003-3. http://books.google.com/books?id=j1Y5FfdQHsQC&pg=PA80&dq=real+%22fictitious+force%22&lr=&as_brr=0&sig=ACfU3U1HwBagOANaT7-ZWTdsJ0khhnrz-Q#PPA79,M1.  
  8. ^ Giuliano Toraldo di Francia (1981). The Investigation of the Physical World. CUP Archive. p. 115. ISBN 052129925X. http://books.google.com/books?id=cFQ7AAAAIAAJ&pg=PA46&dq=laws+physics+%22simplest+form%22&lr=&as_brr=0&sig=ACfU3U0zI1ZXjyB3G6Z3AI3zM_Z2YfYN6g#PPA115,M1.  
  9. ^ Raymond A. Serway (1990). Physics for scientists & engineers (3rd Edition ed.). Saunders College Publishing. p. 135. ISBN 0030313589. http://books.google.com/books?lr=&as_brr=0&q=%22fictitious+forces+do+not+exist+when+the+motion+is+observed+in+an+inertial+frame.+The+fictitious+forces+are+used+only+in+an+accelerating%22&btnG=Search+Books.  
  10. ^ a b V. I. Arnol'd (1989). Mathematical Methods of Classical Mechanics. Springer. p. 129. ISBN 978-0-387-96890-2. http://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=.  
  11. ^ Milton A. Rothman (1989). Discovering the Natural Laws: The Experimental Basis of Physics. Courier Dover Publications. p. 23. ISBN 0486261786. http://books.google.com/books?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22&lr=&as_brr=0&source=gbs_search_s&cad=5&sig=ACfU3U33YE3keeD7lDVtQvt-ltW87Lsq2Q.  
  12. ^ Sidney Borowitz & Lawrence A. Bornstein (1968). A Contemporary View of Elementary Physics. McGraw-Hill. p. 138. http://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=The+effect+of+his+being+in+the+noninertial+frame+is+to+require+the+observer+to&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=.  
  13. ^ Leonard Meirovitch (2004). Methods of analytical Dynamics (Reprint of 1970 edition ed.). Courier Dover Publications. p. 4. ISBN 0486432394. http://books.google.com/books?id=GfCil84YTm4C&pg=PA4&dq=%22in+accelerated+systems,+we+must%22&lr=&as_brr=0&sig=ACfU3U0UrA5jcOx4pB9QIlyA7BQiXwAV5Q.  
  14. ^ Harald Iro (2002). A Modern Approach to Classical Mechanics. World Scientific. p. 180. ISBN 9812382135. http://books.google.com/books?id=-L5ckgdxA5YC&pg=PA179&dq=force+%22non-inertial+%22&lr=&as_brr=0&sig=ACfU3U3fCW68SLm1zalPZWi0LvLK4DrXYg#PPA180,M1.  
  15. ^ 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.

See also

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