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In everyday understanding, centrifugal force (from Latin centrum "center" and fugere "to flee") represents the effects of inertia that arise in connection with rotation and which are experienced as an outward force away from the center of rotation. In Newtonian mechanics the term centrifugal force is used to refer to one of two distinct, but equally valid concepts: a reaction force corresponding to a centripetal force or an inertial force (also called a "fictitious" force) observed in a non-inertial reference frame. The term is also used in Lagrangian mechanics to describe certain terms in the generalized force that are due to the choice of coordinate system.

Reactive centrifugal force

A reactive centrifugal force is the reaction force to a centripetal force. A mass undergoing curved motion, such as circular motion, constantly accelerates toward the axis of rotation. This centripetal acceleration is provided by a centripetal force, which is exerted on the mass by some other object. In accordance with Newton's Third Law of Motion, the mass exerts an equal and opposite force on the object. This is the "real" or "reactive" centrifugal force: it is directed away from the center of rotation, and is exerted by the rotating mass on the object that originates the centripetal acceleration.^[1]^[2]^[3]

The concept of the reactive centrifugal force is used often in mechanical engineering sources that deal with internal stresses in rotating solid bodies.^[4] Newton's reactive centrifugal force still appears in some sources, and often is referred to as the centrifugal force rather than as the reactive centrifugal force.^[5]^[6]^[7]^[8]^[9]^[10]^[11]^[12]^[13]

Fictitious centrifugal force

Nowadays, centrifugal force is most commonly introduced as a force associated with describing motion in a non-inertial reference frame, and referred to as a fictitious or inertial force (a description that must be understood as a technical usage of these words that means only that the force is not present in a stationary or inertial frame).^[14]^[15] There are three contexts in which the concept of the fictitious force arises when describing motion using Newtonian mechanics.^[16] In the first context, the motion is described relative to a rotating reference frame about a fixed axis at the origin of the coordinate system. For observations made in the rotating frame, all objects appear to be under the influence of a radially outward force that is proportional to the distance from the axis of rotation and to the rate of rotation of the frame. The second context is similar, and describes the motion using an accelerated local reference frame attached to a moving body, for example, the frame of passengers in a car as it rounds a corner.^[16] In this case, rotation is again involved, this time about the center of curvature of the path of the moving body. In both these contexts, the centrifugal force is zero when the rate of rotation of the reference frame is zero, independent of the motions of objects in the frame.^[17]

The third context is related to the use of generalized coordinates as is done in the Lagrangian formulation of mechanics, discussed below. Here the term "centrifugal force" is an abbreviated substitute for "generalized centrifugal force", which in general has little connection with the Newtonian concept of centrifugal force.

If objects are seen as moving from a rotating frame, this movement results in another fictitious force, the Coriolis force; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced. Together, these three fictitious forces allow for the creation of correct equations of motion in a rotating reference frame.^[17]

Reactive vs. fictitious force

The table below compares various facets of the "reactive force" and "fictitious force" concepts of centrifugal force.

	Reactive centrifugal force	Fictitious centrifugal force
Reference frame	Any	Rotating frames
Exerted by	Bodies moving in circular paths	Acts as if emanating from the rotation axis, but no real source
Exerted upon	The object(s) causing the curved motion, not upon the body in curved motion	All bodies, moving or not; if moving, Coriolis force also is present
Direction	Opposite to the centripetal force causing curved path	Away from rotation axis, regardless of path of body
Analysis	Kinematic: related to centripetal force	Kinetic: included as force in Newton's laws of motion

Lagrangian formulation of centrifugal force

Lagrangian mechanics formulates mechanics in terms of generalized coordinates {q_k}, which can be as simple as the usual polar coordinates (r, θ) or a much more extensive list of variables.^[18]^[19] Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler-Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dq_k/dt)²} are called centrifugal forces.^[20]^[21]

The Lagrangian approach to polar coordinates that treats (r, θ ) as generalized coordinates, $({\dot {r}},\ {\dot {\theta }})$ as generalized velocities and $({\ddot {r}},\ {\ddot {\theta }})$ as generalized accelerations, is outlined in another article, and found in many sources.^[22]^[23]^[24] For the particular case of single-body motion found using the generalized coordinates (r, θ ) in a central force, the Euler-Lagrange equations are the same equations found using Newton's second law in a co-rotating frame. For example, the radial equation is:

\mu {\ddot {r}}=\mu r{\dot {\theta }}^{2}-{\frac {dU}{dr}}\ ,

where U(r) is the central force potential. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete radial acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.^[25]

This approach is especially useful for deriving the Kepler orbits in celestial mechanics. When the problem of two bodies mutually attracted by gravitation is considered, it can be transformed using Jacobi coordinates to an equivalent one-body problem attracted to the barycenter of the two bodies. The centrifugal force then appears as a radially outward inverse cube law term in the planetary orbital equation.^[26]^[27]^[28] The radial equation then becomes:

\mu {\ddot {r}}=-k/r^{2}+{\frac {\ell ^{2}}{\mu r^{3}}}\ ,

where the variable r is the radial distance from the barycenter to the equivalent single body, ℓ is the (fixed) angular momentum, μ is the reduced mass, and k is a parameter related to the force of gravity. The solutions of this equation yield orbits that are either elliptical, parabolic, or hyperbolic, depending on the initial energy and angular momentum. The solution is not unique until the values of r and dr / dt are specified at some particular time t.^[22]

The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference,^[29] but in the case of motion in a central potential the result is the same as the Newtonian centrifugal force derived in a co-rotating frame.^[16] The Lagrangian use of "centrifugal force" in other, more general cases, however, has only a limited connection to the Newtonian definition. Although the two formulations of mechanics must lead to the same equations given the same choice of variables, the connection between them may be obscure,^[30] and the same terminology employs different meanings. In particular, "generalized forces" (often referred to without the adjective "generalized") in most cases are not Newtonian forces, and do not transform as vectors. Unlike the Newtonian centrifugal force, the Lagrangian centrifugal force may be non-zero even in an inertial frame of reference.

Centrifugal force and absolute rotation

Can absolute rotation be detected? In other words, can one decide whether an observed object is rotating or if it is you, the observer that is rotating? Newton suggested two experiments to resolve this problem. One is the effect of centrifugal force upon the shape of the surface of water rotating in a bucket. The second is the effect of centrifugal force upon the tension in a string joining two spheres rotating about their center of mass. A related third suggestion was that rotation of a sphere (such as a planet) could be detected from its shape (or "figure"), which is formed as a balance between containment by gravitational attraction and dispersal by centrifugal force.

Rotating bucket

Newton suggested the shape of the surface of the water indicates the presence or absence of absolute rotation relative to the fixed stars: rotating water has a curved surface, still water has a flat surface. Because rotating water has a concave surface, if the surface you see is concave, and the water does not seem to you to be rotating, then you are rotating with the water.

Centrifugal force is needed to explain the concavity of the water in a co-rotating frame of reference (one that rotates with the water) because the water appears stationary in this frame, and so should have a flat surface. Thus, observers looking at the stationary water need the centrifugal force to explain why the water surface is concave and not flat. The centrifugal force pushes the water toward the sides of the bucket, where it piles up deeper and deeper, Pile-up is arrested when any further climb costs as much work against gravity as is the energy gained from the greater centrifugal force at larger radius.

If you need a centrifugal force to explain what you see, then you are rotating. Newton's conclusion was that rotation is absolute.^[31]

Other thinkers suggest that pure logic implies only relative rotation makes sense. For example, Bishop Berkeley and Ernst Mach (among others) suggested that it is relative rotation with respect to the fixed stars that matters, and rotation of the fixed stars relative to an object has the same effect as rotation of the object with respect to the fixed stars.^[32] Newton's arguments do not settle this issue; his arguments may be viewed, however, as establishing centrifugal force as a basis for an operational definition of what we actually mean by absolute rotation.^[33]

Rotating spheres

Newton also proposed another experiment to measure one's rate of rotation: using the tension in a cord joining two spheres rotating about their center of mass. Non-zero tension in the string indicates rotation of the spheres, whether or not the observer thinks they are rotating. This experiment is simpler than the bucket experiment in principle, because it need not involve gravity.

Beyond a simple "yes or no" answer to rotation, one may actually calculate one's rotation. To do that, one takes one's measured rate of rotation of the spheres and computes the tension appropriate to this observed rate. This calculated tension then is compared to the measured tension. If the two agree, one is in a stationary (non-rotating) frame. If the two do not agree, to obtain agreement, one must include a centrifugal force in the tension calculation; for example, if the spheres appear to be stationary, but the tension is non-zero, the entire tension is due to centrifugal force. From the necessary centrifugal force, one can determine one's speed of rotation; for example, if the calculated tension is greater than measured, one is rotating in the sense opposite to the spheres, and the larger the discrepancy the faster this rotation.

The centrifugal force is not simply cerebral, but actually is experienced by the rotating observer.^[34] That is, forces experienced by the rotating observer are equally real, whether their origin is fundamental or simply in the rotation of the observer.

Figure of the Earth

In a similar fashion, if we did not know the Earth rotates about its axis, we could infer this rotation from the centrifugal force needed to account for the bulging observed at its equator.^[35]^[36]

In his Principia, Newton proposed the shape of the rotating Earth was that of a homogeneous ellipsoid formed by an equilibrium between the gravitational force holding it together and the centrifugal force pulling it apart. The Earth's surface is an equipotential, that is, no work is done moving upon the Earth's surface, either against gravity or against centrifugal force. Based upon this equilibrium, Newton determined a flattening expressed by the ratio of diameters: 230 to 229.^[37]^[38] A modern measurement of the Earth's oblateness leads to an equatorial radius of 6378.14 km and a polar radius of 6356.77 km,^[39] about 1/10% less oblate than Newton's estimate.^[40] A theoretical determination of the precise extent of oblateness in response to a centrifugal force requires an understanding of the make-up of the planet, not only today but during its formation.^[41]^[42]

History of conceptions of centrifugal and centripetal forces

Christiaan Huygens coined the term "centrifugal force" (vis centrifuga) in his 1673 Horologium Oscillatorium on pendulums, and Newton coined the term "centripetal force" (vis centripita) in his discussions of gravity in his 1684 De Motu Corpurum.^[43] Gottfried Leibniz as part of his 'solar vortex theory' conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts. The inverse cube law centrifugal force appears in an equation representing planetary orbits, including non-circular ones, as Leibniz described in his 1689 Tentamen de motuum coelestium causis.^[44] Leibniz's equation is still used today to solve planetary orbital problems, although the 'solar vortex theory' is no longer used as its basis.^[45]

Huygens, who was, along with Leibniz, a neo-Cartesian and critic of Newton, concluded after a long correspondence that Leibniz's writings on celestial mechanics made no sense, and that his invocation of a harmonic vortex was logically redundant, because Leibniz's radial equation of motion follows trivially from Newton's laws. Even the most ardant modern defenders of the cogency of Leibniz's ideas acknowledge that his harmonic vortex as the basis of centrifugal force was dynamically superfluous.^[46]

There is evidence that Isaac Newton originally conceived of a similar approach to centrifugal force as Leibniz, though he seems to have changed his position at some point. When Leibniz produced his equation for planetary orbits in which the centrifugal force appeared as an outward inverse cube law force in the radial direction:^[27]

{\ddot {r}}=-k/r^{2}+l^{2}/r^{3}

.

Newton himself appears to have previously supported an approach similar to that of Leibniz.^[47] Later, Newton in his Principia crucially limited the description of the dynamics of planetary motion to a frame of reference in which the point of attraction is fixed. In this description, Leibniz's centrifugal force was not needed and was replaced by only continually inward forces toward the fixed point.^[48] Newton objected to this Liebniz's equation on the grounds that it allowed for the centrifugal force to have a different value from the centripetal force, arguing on the basis of his third law of motion, that the centrifugal force and the centripetal force must constitute an equal and opposite action-reaction pair. In this however, Newton was mistaken, as the reactive centrifugal force which is required by the third law of motion is a completely separate concept from the centrifugal force of Leibniz's equation.^[49]

It wasn't until the latter half of the 18th century that the modern "fictitious force" understanding of the centrifugal force as a pseudo-force artifact of rotating reference frames took shape.^[50] In a 1746 memoir by Daniel Bernoulli, the "idea that the centrifugal force is fictitious emerges unmistakably."^[51] Bernoulli, in seeking to describe the motion of an object relative to an arbitrary point, showed that the magnitude of the centrifugal force depended on which arbitrary point was chosen. In other words, the centrifugal force depended on the reference frame of the observer, as opposed to other forces which depended only on the properties of the objects involved in the problem and were independent of the frame. Also in the second half of the 18th century, Joseph Louis Lagrange in his Mécanique Analytique explicitly stated that the centrifugal force depends on the rotation of a system of perpendicular axes.^[51] In 1835, Gaspard-Gustave Coriolis analyzed arbitrary motion in rotating systems, specifically in relation to waterwheels. He coined the phrase "compound centrifugal force" for a term which bore a similar mathematical expression to that of centrifugal force, albeit that it was multiplied by a factor of two.^[52] The force in question was perpendicular to both the velocity of an object relative to a rotating frame of reference and the axis of rotation of the frame. Compound centrifugal force eventually came to be known as the Coriolis Force.^[53]^[54]

In part I of his 1861 paper On Physical Lines of Force, James Clerk Maxwell used the concept of centrifugal force in order to explain magnetic repulsion. He considered that magnetic lines of force are comprised of molecular vortices aligned along their mutual axes of rotation. When two magnets repel each other, the magnetic lines of force in the space between the like poles spread outwards and away from each other. Maxwell considered that the repulsion is due to centrifugal force acting in the equatorial plane of the molecular vortices.

The modern interpretation is that centrifugal force in a rotating reference frame is a pseudo-force that appears in equations of motion in rotating frames of reference, to explain effects of inertia as seen in such frames.^[55] Leibniz's centrifugal force may be understood as an application of this conception, as a result of his viewing the motion of a planet along the radius vector, that is, from the standpoint of a special reference frame rotating with the planet.^[56]^[48]^[49] Leibniz introduced the notions of vis viva (kinetic energy)^[57] and action,^[58] which eventually found full expression in the Lagrangian formulation of mechanics. In deriving Leibniz's radial equation from the Lagrangian standpoint, a rotating reference frame is not used explicitly,^[59] but the result is equivalent to that found using Newtonian vector mechanics in a co-rotating reference frame.^[60]^[61]^[62]

References

^ Mook & Vargish 1987, p. 47.
^ Signell 2002, "Acceleration and force in circular motion", §5b, p. 7.
^ Mohanty 2004, p. 121.
^ Roche 2001, "Introducing motion in a circle". Retrieved 2009-05-07.
^ Edward Albert Bowser (1920). An elementary treatise on analytic mechanics: with numerous examples (25th ed.). D. Van Nostrand Company. p. 357.
^ Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond. Rutgers University Press. p. 126. ISBN 9780813529080.
^ Ervin Sidney Ferry (2008). A Brief Course in Elementary Dynamics. BiblioBazaar. pp. 87–88. ISBN 9780554609843.
^ Willis Ernest Johnson (2009). Mathematical Geography. BiblioBazaar. p. 15–16. ISBN 9781103199587.
^ Eugene A. Avallone, Theodore Baumeister, Ali Sadegh, Lionel Simeon Marks (2006). Marks' standard handbook for mechanical engineers (11 ed.). McGraw-Hill Professional. p. 15. ISBN 0071428674.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Richard Cammack, Anthony Donald Smith, Teresa K. Attwood, Peter Campbell (2006). Oxford dictionary of biochemistry and molecular biology (2 ed.). Oxford University Press. p. 109. ISBN 0198529171.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Joseph A. Angelo (2007). Robotics: a reference guide to the new technology. Greenwood Press. p. 267. ISBN 1573563374.
^ P. Grimshaw, A. Lees, N. Fowler, A. Burden (2006). Sport and exercise biomechanics. Routledge. p. 176. ISBN 185996284X.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Joel Dorman Steele (2008). Popular Physics (Reprint ed.). READ books. p. 31. ISBN 1408691345.
^ Takwale & Puranik 1980, p. 248.
^ Jacobson 1980, p. 80.
^ ^a ^b ^c Bini, D. et al. (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D 6 (1)
^ ^a ^b Fetter & Walecka 2003, pp. 38-39.
^ For an introduction see, for example, Cornelius Lanczos (1986). The variational principles of mechanics (Reprint of 1970 University of Toronto ed.). Dover. p. 1. ISBN 0486650677.
^ For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics of Multibody Systems (2 ed.). Cambridge University Press. p. 90 ff. ISBN 0521544114.
^ Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3540692533.
^ Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. p. pp. 47-48. ISBN 981023452X. In the above Euler-Lagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in ${\boldsymbol {\dot {q}}}$ where the coefficients may depend on ${\boldsymbol {q}}$ . These are further classified into two types. Terms involving a product of the type ${{\dot {q}}_{i}}^{2}$ are called centrifugal forces while those involving a product of the type ${\dot {q}}_{i}{\dot {q}}_{j}$ for i ≠ j are called Coriolis forces. The third type is functions of ${\boldsymbol {q}}$ only and are called gravitational forces. {{cite book}}: |page= has extra text (help)CS1 maint: multiple names: authors list (link)
^ ^a ^b See, for example, Eq. 8.20 in John R Taylor (2005). op. cit. pp. 299 ff. ISBN 189138922X. Cite error: The named reference "Taylor299" was defined multiple times with different content (see the help page).
^ Francis Begnaud Hildebrand (1992). Methods of Applied Mathematics (Reprint of 1965 2nd ed.). Courier Dover Publications. p. 156. ISBN 0486670023.
^ V. B. Bhatia (1997). Classical Mechanics: With Introduction to Nonlinear Oscillations and Chaos. Alpha Science Int'l Ltd. p. 82. ISBN 8173191050.
^ Henry M. Stommel and Dennis W. Moore (1989). An Introduction to the Coriolis Force Columbia University Press. pp 36-38
^ See Eq. 8.37 in John R Taylor (2005). Classical Mechanics. University Science Books. p. 306. ISBN 189138922X.
^ ^a ^b Linton 2004, p. 285.
^ Herbert Goldstein 'Classical Mechanics', equation 3-12
^ See Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. pp. 40–41. ISBN 0521358833. for an explanation of how the Newtonian centrifugal force generates a potential term in the Lagrangian.
^ For example, while the Newtonian picture uses a single frame of reference, the Lagrangian generalized coordinates may refer simultaneously to several different frames, making the connection to the Newtonian picture complex.Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris. op. cit.. p. 136. ISBN 981023452X.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Max Born and Günther Leibfried. Einstein's Theory of Relativity. Courier Dover Publications. p. 78–79. ISBN 0486607690.
^ BK Ridley (1995). Time, Space, and Things (3 ed.). Cambridge University Press. p. 146. ISBN 0521484863.
^ Rather than justifying a causal link between rotation and centrifugal effects, Newton's arguments may be viewed as defining "absolute rotation" by stating a procedure for its detection and measurement involving centrifugal force. See Robert Disalle (2002). I. Bernard Cohen & George E. Smith (ed.). The Cambridge Companion to Newton. Cambridge University Press. pp. 44–45. ISBN 0521656966.
^ See cartoon: XKCD demonstrates the life and death importance of centrifugal force.
^ Archibald Tucker Ritchie (1850). The Dynamical Theory of the Formation of the Earth. Longman, Brown, Green and Longmans. p. 529.
^ John Clayton Taylor (2001). Hidden unity in nature's laws. Cambridge University Press. p. 26. ISBN 0521659388.
^ Isaac Newton: Principia (July 5, 1687) Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.
^ See the Principia on line at Andrew Motte Translation. Other sources and some notes are found at Philosophiæ Naturalis Principia Mathematica.
^ Charles D Brown (1998). Spacecraft mission design (2 ed.). American Institute of Aeronautics & Astronomy. p. 58. ISBN 1563472627.
^ This error is the difference in the estimated ratio of diameters. However, a more demanding measure of oblateness is the flattening, defined as f = (a−b)/a where a and b are the semimajor and semiminor axes. Using the cited numbers, the flattening of Newton's prediction differs by 23% from that of modern estimates.
^ Hugh Murray (1837). "Figure and constitution of the Earth deduced from the theory of gravitation". The Encyclopædia of Geography. Vol. vol. 1. Carey, Lea & Blanchard. pp. 124 ff. {{cite book}}: |volume= has extra text (help)
^ Alexander Winchell (1888). World-life; Or, Comparative Geology. SC Griggs & Co. p. 425.
^ Whiteside 2008, pp. 4-5.
^ Gillies 1995, p. 130.
^ Herbert Goldstein 'Classical Mechanics', equation 3-12
^ A. R. Hall, Philosophers at War, 2002, pp 150-151
^ Swetz et al. 1997, p. 268.
^ ^a ^b Linton 2004, pg. 264
^ ^a ^b Swetz et al. 1997, p. 269.
^ Wilson 1994, "Newton's Orbit Problem: A Historian's Response".
^ ^a ^b Meli 1990, "The Relativization of Centrifugal Force".
^ Dugas & Maddox 1988, p. 374.
^ Persson 1998, "How Do We Understand the Coriolis Force?".
^ Slate 1918, p. 137.
^ Steinmetz 2005, p. 49.
^ Aiton 1962, "The celestial mechanics of Leibniz in the light of Newtonian criticism".
^ Bertrand Russell (1992). A Critical Exposition of the Philosophy of Leibniz (Reprint of 1937 2nd ed.). Routledge. p. 96. ISBN 041508296X.
^ Wolfgang Lefèvre (2001). Between Leibniz, Newton, and Kant. Springer. p. 39. ISBN 0792371984.
^ Goldstein 2002, pp.74-77
^ Goldstein 2002, pg. 176
^ Taylor 2005, pp. 358-359
^ Whiting, J.S.S. (1983). "Motion in a central-force field" (PDF). Physics Education. 18 (6): pp. 256–257. ISSN 0031-9120. Retrieved May 7, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)
^ Rizzi & Ruggiero 2004, p. 272.
^ Rindler 2006, pp. 7-8.
^ Barbour & Pfister 1995, pp. 6-8.
^ Barbour & Pfister 1995, p. 69.
^ Eriksson 2008, p. 194.

Bibliography

Aiton, E.J. (1 March 1962). "The celestial mechanics of Leibniz in the light of Newtonian criticism". Annals of Science (Taylor & Francis) 18 (1): pp. 31-41. doi:10.1080/00033796200202682
Barbour, Julian B. and Herbert Pfister (1995). Mach's principle: from Newton's bucket to quantum gravity. Birkhäuser. ISBN 0817638237
Dugas, René and J. R. Maddox (1988). A History of Mechanics. Courier Dover Publications. ISBN 0486656322
Eriksson, Ingrid V. (2008). Science education in the 21st century. Nova Books. ISBN 1600219519
Fetter, Alexander L. and John Dirk Walecka (2003). Theoretical Mechanics of Particles and Continua (Reprint of McGraw-Hill 1980 ed.). Courier Dover Publications. ISBN 0486432610
Goldstein, Herbert (2002). Classical Mechanics. San Francisco : Addison Wesley. ISBN 0201316110
Gillies, Donald (1995). Revolutions in Mathematics. Oxford: University Press. ISBN 9780198514862
Jacobson, Mark Zachary (1980). Fundamentals of atmospheric modeling. Cambridge: University Press. ISBN 9780521637176
Klein, Felix & Arnold Sommerfeld (2008). The Theory of the Top, vol. 1 (New translation of 1910 ed.). ISBN 0817647201
Kobayashi, Yukio (2008). "Remarks on viewing situation in a rotating frame". European Journal of Physics 29 (3):pp. 599-606. ISSN 0143-0807
Linton, Christopher (2004). From Exodus to Einstein. Cambridge: University Press. ISBN 0521827507
Meli, Domenico Bertoloni (March 1990). "The Relativization of Centrifugal Force". Isis 81 (1): pp. 23-43. ISSN 0021-1753
Mohanty, A. K. (2004). Fluid Mechanics. PHI Learning Pvt. Ltd. ISBN 8120308948
Mook, Delo E. & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. ISBN 0691025207
Persson, Anders (July 1998). "How Do We Understand the Coriolis Force?". Bulletin of the American Meteorological Society 79 (7): pp. 1373–1385. ISSN 0003-0007
Rindler, Wolfgang (2006). Relativity. Oxford: University Press. ISBN 9780198567318
Rizzi, Guido and Matteo Luca Ruggiero (2004). Relativity in rotating frames. Springer. ISBN 9781402018053
Roche, John (September 2001). "Introducing motion in a circle". Physics Education 43 (5), pp. 399-405. ISSN 0031-9120
Signell, Peter (2002). "Acceleration and force in circular motion" Physnet. Michigan State University.
Slate, Frederick (1918). The Fundamental Equations of Dynamics and its Main Coordinate Systems Vectorially Treated and Illustrated from Rigid Dynamics. Berkeley, CA: University of California Press. ASIN B000ML76V8
Steinmetz, Charles Proteus (2005). Four Lectures on Relativity and Space. Kessinger Publishing. ISBN 1417925302
Swetz, Frank; et al. (1997). Learn from the Masters!. Mathematical Association of America. ISBN 0883857030
Takwale, R. G. and P. S. Puranik (1980). Introduction to classical mechanics. Tata McGraw-Hill. ISBN 9780070966178
Taylor, John R. (2005). Classical Mechanics. University Science Books. ISBN 189138922X
Whiteside, D. T. (2008). The Mathematical Papers of Isaac Newton, VI. Cambridge: University Press. ISBN 9780521045858
Wilson, Curtis (May 1994). "Newton's Orbit Problem: A Historian's Response". The College Mathematics Journal 25 (3): pp. 193-200. ISSN 0746-8342

[Mook-1] Mook & Vargish 1987, p. 47.

[Signell-2] Signell 2002, "Acceleration and force in circular motion", §5b, p. 7.

[Mohanty-3] Mohanty 2004, p. 121.

[Roche-4] Roche 2001, "Introducing motion in a circle". Retrieved 2009-05-07.

[Bowser-5] Edward Albert Bowser (1920). An elementary treatise on analytic mechanics: with numerous examples (25th ed.). D. Van Nostrand Company. p. 357.

[Holton-6] Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond. Rutgers University Press. p. 126. ISBN 9780813529080.

[7] Ervin Sidney Ferry (2008). A Brief Course in Elementary Dynamics. BiblioBazaar. pp. 87–88. ISBN 9780554609843.

[Johnson-8] Willis Ernest Johnson (2009). Mathematical Geography. BiblioBazaar. p. 15–16. ISBN 9781103199587.

[Avallone-9] Eugene A. Avallone, Theodore Baumeister, Ali Sadegh, Lionel Simeon Marks (2006). Marks' standard handbook for mechanical engineers (11 ed.). McGraw-Hill Professional. p. 15. ISBN 0071428674.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Oxford-10] Richard Cammack, Anthony Donald Smith, Teresa K. Attwood, Peter Campbell (2006). Oxford dictionary of biochemistry and molecular biology (2 ed.). Oxford University Press. p. 109. ISBN 0198529171.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Angelo-11] Joseph A. Angelo (2007). Robotics: a reference guide to the new technology. Greenwood Press. p. 267. ISBN 1573563374.

[Lees-12] P. Grimshaw, A. Lees, N. Fowler, A. Burden (2006). Sport and exercise biomechanics. Routledge. p. 176. ISBN 185996284X.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Steele-13] Joel Dorman Steele (2008). Popular Physics (Reprint ed.). READ books. p. 31. ISBN 1408691345.

[14] Takwale & Puranik 1980, p. 248.

[15] Jacobson 1980, p. 80.

[Bini1997-16] Bini, D. et al. (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D 6 (1)

[Fetter-17] Fetter & Walecka 2003, pp. 38-39.

[Lanczos-18] For an introduction see, for example, Cornelius Lanczos (1986). The variational principles of mechanics (Reprint of 1970 University of Toronto ed.). Dover. p. 1. ISBN 0486650677.

[Shabana1-19] For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics of Multibody Systems (2 ed.). Cambridge University Press. p. 90 ff. ISBN 0521544114.

[Ott-20] Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3540692533.

[Ge-21] Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. p. pp. 47-48. ISBN 981023452X. In the above Euler-Lagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in ${\boldsymbol {\dot {q}}}$ where the coefficients may depend on ${\boldsymbol {q}}$ . These are further classified into two types. Terms involving a product of the type ${{\dot {q}}_{i}}^{2}$ are called centrifugal forces while those involving a product of the type ${\dot {q}}_{i}{\dot {q}}_{j}$ for i ≠ j are called Coriolis forces. The third type is functions of ${\boldsymbol {q}}$ only and are called gravitational forces. {{cite book}}: |page= has extra text (help)CS1 maint: multiple names: authors list (link)

[Taylor299-22] See, for example, Eq. 8.20 in John R Taylor (2005). op. cit. pp. 299 ff. ISBN 189138922X. Cite error: The named reference "Taylor299" was defined multiple times with different content (see the help page).

[Hildebrand-23] Francis Begnaud Hildebrand (1992). Methods of Applied Mathematics (Reprint of 1965 2nd ed.). Courier Dover Publications. p. 156. ISBN 0486670023.

[Bhatia-24] V. B. Bhatia (1997). Classical Mechanics: With Introduction to Nonlinear Oscillations and Chaos. Alpha Science Int'l Ltd. p. 82. ISBN 8173191050.

[25] Henry M. Stommel and Dennis W. Moore (1989). An Introduction to the Coriolis Force Columbia University Press. pp 36-38

[Taylor306-26] See Eq. 8.37 in John R Taylor (2005). Classical Mechanics. University Science Books. p. 306. ISBN 189138922X.

[Linton285-27] Linton 2004, p. 285.

[Goldstein3_12-28] Herbert Goldstein 'Classical Mechanics', equation 3-12

[Whittaker-29] See Edmond T Whittaker. A treatise on the analytical dynamics of particles and rigid bodies (Reprint of 1917 2nd ed.). Cambridge University Press. pp. 40–41. ISBN 0521358833. for an explanation of how the Newtonian centrifugal force generates a potential term in the Lagrangian.

[frame-30] For example, while the Newtonian picture uses a single frame of reference, the Lagrangian generalized coordinates may refer simultaneously to several different frames, making the connection to the Newtonian picture complex.Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris. op. cit.. p. 136. ISBN 981023452X.{{cite book}}: CS1 maint: multiple names: authors list (link)

[31] Max Born and Günther Leibfried. Einstein's Theory of Relativity. Courier Dover Publications. p. 78–79. ISBN 0486607690.

[Ridley-32] BK Ridley (1995). Time, Space, and Things (3 ed.). Cambridge University Press. p. 146. ISBN 0521484863.

[Cohen-33] Rather than justifying a causal link between rotation and centrifugal effects, Newton's arguments may be viewed as defining "absolute rotation" by stating a procedure for its detection and measurement involving centrifugal force. See Robert Disalle (2002). I. Bernard Cohen & George E. Smith (ed.). The Cambridge Companion to Newton. Cambridge University Press. pp. 44–45. ISBN 0521656966.

[XKCD-34] See cartoon: XKCD demonstrates the life and death importance of centrifugal force.

[Ritchie-35] Archibald Tucker Ritchie (1850). The Dynamical Theory of the Formation of the Earth. Longman, Brown, Green and Longmans. p. 529.

[Taylor-36] John Clayton Taylor (2001). Hidden unity in nature's laws. Cambridge University Press. p. 26. ISBN 0521659388.

[Newton-37] Isaac Newton: Principia (July 5, 1687) Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.

[Principia-38] See the Principia on line at Andrew Motte Translation. Other sources and some notes are found at Philosophiæ Naturalis Principia Mathematica.

[Brown-39] Charles D Brown (1998). Spacecraft mission design (2 ed.). American Institute of Aeronautics & Astronomy. p. 58. ISBN 1563472627.

[Flatness-40] This error is the difference in the estimated ratio of diameters. However, a more demanding measure of oblateness is the flattening, defined as f = (a−b)/a where a and b are the semimajor and semiminor axes. Using the cited numbers, the flattening of Newton's prediction differs by 23% from that of modern estimates.

[Encyclopædia-41] Hugh Murray (1837). "Figure and constitution of the Earth deduced from the theory of gravitation". The Encyclopædia of Geography. Vol. vol. 1. Carey, Lea & Blanchard. pp. 124 ff. {{cite book}}: |volume= has extra text (help)

[Jupiter-42] Alexander Winchell (1888). World-life; Or, Comparative Geology. SC Griggs & Co. p. 425.

[43] Whiteside 2008, pp. 4-5.

[44] Gillies 1995, p. 130.

[45] Herbert Goldstein 'Classical Mechanics', equation 3-12

[46] A. R. Hall, Philosophers at War, 2002, pp 150-151

[47] Swetz et al. 1997, p. 268.

[Linton264-48] Linton 2004, pg. 264

[Swetz269-49] Swetz et al. 1997, p. 269.

[50] Wilson 1994, "Newton's Orbit Problem: A Historian's Response".

[Meli-51] Meli 1990, "The Relativization of Centrifugal Force".

[52] Dugas & Maddox 1988, p. 374.

[53] Persson 1998, "How Do We Understand the Coriolis Force?".

[54] Slate 1918, p. 137.

[55] Steinmetz 2005, p. 49.

[56] Aiton 1962, "The celestial mechanics of Leibniz in the light of Newtonian criticism".

[Russell-57] Bertrand Russell (1992). A Critical Exposition of the Philosophy of Leibniz (Reprint of 1937 2nd ed.). Routledge. p. 96. ISBN 041508296X.

[Lefèvre-58] Wolfgang Lefèvre (2001). Between Leibniz, Newton, and Kant. Springer. p. 39. ISBN 0792371984.

[59] Goldstein 2002, pp.74-77

[60] Goldstein 2002, pg. 176

[61] Taylor 2005, pp. 358-359

[62] Whiting, J.S.S. (1983). "Motion in a central-force field" (PDF). Physics Education. 18 (6): pp. 256–257. ISSN 0031-9120. Retrieved May 7, 2009. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)

[63] Rizzi & Ruggiero 2004, p. 272.

[64] Rindler 2006, pp. 7-8.

[65] Barbour & Pfister 1995, pp. 6-8.

[Barbour-66] Barbour & Pfister 1995, p. 69.

[Eriksson-67] Eriksson 2008, p. 194.

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@@ Line 1: / Line 1: @@
 {{wiktionary|centrifugal force}}
-In everyday understanding, '''centrifugal force''' (from [[Latin]] ''centrum'' "center" and ''fugere'' "to flee") represents the effects of [[inertia]] that arise in connection with rotation and which are experienced as an outward force away from the center of rotation.
+In everyday understanding, '''centrifugal force''' (from [[Latin]] ''centrum'' "center" and ''fugere'' "to flee") represents the effects of [[inertia]] that arise in connection with rotation and which are experienced as an outward force away from the center of rotation. In [[Newtonian mechanics]] the term centrifugal force is used to refer to one of two distinct, but equally valid concepts: a [[reaction (physics)|reaction]] force corresponding to a centripetal force or an [[inertial force]] (also called a "fictitious" force) observed in a non-inertial [[reference frame]]. The term is also used in [[Lagrangian mechanics]] to describe certain terms in the [[generalized force]] that are due to the choice of [[coordinate system]].
@@ Line 74: / Line 74: @@
 {{cite book |title=Popular Physics |year=2008  |author=Joel Dorman Steele |url=http://books.google.com/books?id=Fho7X_xzmQUC&pg=PA31 |page=31 |isbn=1408691345 |publisher=READ books |edition = Reprint }}</ref>
-== Centrifugal force in the inertial path ==
-Centrifugal force occurs naturally as an inertial force in the straight line inertial path. This fact is exposed when we express Newton's laws in curvilinear coordinates that are measured relative to the inertial frame of reference.<ref name=Shankar>{{cite book |title = Principles of quantum Mechanics | author=R. Shankar |url=http://books.google.com/books?id=2zypV5EbKuIC&pg=PA81&dq=Coriolis+%22polar+coordinates%22&lr=&as_brr=0 }}</ref>
 ==Fictitious centrifugal force==
@@ Line 155: / Line 151: @@
 :<math> \mu \ddot r = \mu r\dot\theta^2 - \frac {dU}{dr}          \ ,       </math>
 where ''U(r)'' is the central force potential. The left side is a "generalized force" and the first term on the right is the "generalized centrifugal force". However, the left side is not comparable to a Newtonian force, as it does not contain the complete radial acceleration, and likewise, therefore, the terms on the right-hand side are "generalized forces" and cannot be interpreted as Newtonian forces.<ref>Henry M. Stommel and Dennis W. Moore (1989). ''An Introduction to the Coriolis Force'' Columbia University Press. pp 36-38</ref>
+This approach is especially useful for deriving the [[Kepler orbit]]s in celestial mechanics. When the [[two body problem|problem of two bodies]] mutually attracted by gravitation is considered, it can be transformed using [[Jacobi coordinates]] to an equivalent one-body problem attracted to the [[barycenter]] of the two bodies. The centrifugal force then appears as a radially outward inverse cube law term in the planetary orbital equation.<ref name=Taylor306>See Eq. 8.37 in
+{{cite book |author=John R Taylor |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA306 |page=306 |title=Classical Mechanics |publisher=University Science Books |isbn=189138922X |year=2005}}
+</ref><ref name="Linton285">[[#Reference-idLinton2004|Linton 2004]], p. 285.</ref><ref name=Goldstein3_12> Herbert Goldstein 'Classical Mechanics', equation 3-12</ref> The radial equation then becomes:
+::<math>\mu \ddot r = -k/r^{2} + \frac{\ell^{2}}{\mu r^{3}} \ ,</math>
+where the variable ''r'' is the radial distance from the barycenter to the equivalent single body, ''ℓ'' is the (fixed) [[angular momentum]], μ is the [[reduced mass]], and ''k'' is a parameter related to the force of gravity. The solutions of this equation yield orbits that are either elliptical, parabolic, or hyperbolic, depending on the initial energy and angular momentum. The solution is not unique until the values of ''r'' and ''dr / dt'' are specified at some particular time ''t''.<ref name=Taylor299>
+See, for example, Eq. 8.20 in {{cite book |author=John R Taylor |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA299 |pages=299 ''ff'' |title=op. cit. |isbn=189138922X |year=2005}}
+</ref>
 The Lagrangian centrifugal force is derived without explicit use of a rotating frame of reference,<ref name=Whittaker>
@@ Line 166: / Line 170: @@
 </ref> and the same terminology employs different meanings. In particular, "generalized forces" (often referred to without the adjective "generalized") in most cases are ''not'' Newtonian forces, and do not transform as vectors. Unlike the Newtonian centrifugal force, the Lagrangian centrifugal force may be non-zero even in an [[inertial frame]] of reference.
-== Centrifugal force in planetary orbits ==
-When the [[two body problem|problem of two bodies]] mutually attracted by gravitation is considered, it can be transformed using [[Jacobi coordinates]] to an equivalent one-body problem attracted to the [[barycenter]] of the two bodies. The centrifugal force then appears as a radially outward inverse cube law term in the planetary orbital equation.<ref name=Taylor306>See Eq. 8.37 in
-{{cite book |author=John R Taylor |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA306 |page=306 |title=Classical Mechanics |publisher=University Science Books |isbn=189138922X |year=2005}}
-</ref><ref name="Linton285">[[#Reference-idLinton2004|Linton 2004]], p. 285.</ref><ref name=Goldstein3_12> Herbert Goldstein 'Classical Mechanics', equation 3-12</ref> The radial equation then becomes:
-::<math>\mu \ddot r = -k/r^{2} + \frac{\ell^{2}}{\mu r^{3}} \ ,</math>
-where the variable ''r'' is the radial distance from the barycenter to the equivalent single body, ''ℓ'' is the (fixed) [[angular momentum]], μ is the [[reduced mass]], and ''k'' is a parameter related to the force of gravity. The above equation is a second order differential equation in the radial distance, and it solves to yield a conic section. The two arbitrary constants are the [[eccentricity]] and the [[semi-latus rectum]] of the conic section. The value of these two constants will be determined by the initial energy and the initial angular momentum, and the value of the eccentricity will then determine whether the orbit is an ellipse, a parabola, or a hyperbola.
-The solution is not unique until the values of ''r'' and ''dr / dt'' are specified at some particular time ''t''.<ref name=Taylor299>
-See, for example, Eq. 8.20 in {{cite book |author=John R Taylor |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA299 |pages=299 ''ff'' |title=op. cit. |isbn=189138922X |year=2005}}
-</ref>.
 == Centrifugal force and absolute rotation ==