Centrifugal force (planar motion)

In classical mechanics, centrifugal force (from Latin "centrum" "center" and "fugere" "to flee") is one of the three so-called "inertial forces" or "fictitious forces" that enter the equations of motion when Newton's laws are formulated in a non-inertial reference frame. The other two fictitious forces are the Coriolis force and the Euler force. This article describes the centrifugal force that acts upon objects in planar motion when observed from non-inertial reference frames. "Fictitious forces" (also known as a "pseudo forces", "inertial forces" or "d'Alembert forces"), exist for observers in a non-inertial reference frames. See, for example, cite web
url=http://scienceworld.wolfram.com/physics/CentrifugalForce.html
publisher=scienceworld.wolfram.com
title=Centrifugal Force
accessdate=2008-05-18, cite web
url=http://www.britannica.com/eb/article-9022100/centrifugal-force
title=Centrifugal Force - Britannica online encyclopedia
accessdate=2008-05-18, cite book |title=Einstein's Theory of Relativity|author=Max Born & Günther Leibfried|url=http://books.google.com/books?id=Afeff9XNwgoC&pg=PA76&dq=%22inertial+forces%22&lr=&as_brr=0&sig=0kiN27BqUqHaZ9CkPdqLIjr-Nnw#PPA77,M1
page=pp.76-78|isbn=0486607690|publisher=Courier Dover Publications|location=New York|year=1962, [http://www-istp.gsfc.nasa.gov/stargaze/Sframes2.htm NASA: "Accelerated Frames of Reference: Inertial Forces"] , [http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm Science Joy Wagon: "Centrifugal force - the "false" force"] ] cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden, Tudor S. Ratiu |isbn=038798643X |year=1999 |publisher=Springer |page=p. 251 |url=http://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251&vq=Euler+force&dq=isbn:038798643X&source=gbs_search_s&sig=ACfU3U0DkJL1h3lGMMbXyKY15GtPpspHuQ] cite book |title=Classical Mechanics|author=John Robert Taylor|title=|title=Classical Mechanics |page=Chapter 9, pp. 327 ff |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn:189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA327,M1
isbn=189138922X|publisher=University Science Books|location=Sausalito CA|year=2004] cite book |title=Classical Dynamics of Particles and Systems |author=Stephen T. Thornton &Jerry B. Marion |page=Chapter 10 |year=2004 |isbn=0534408966 |publisher=Brook/Cole |location=Belmont CA |edition=5th Edition |url=http://worldcat.org/oclc/52806908&referer=brief_results]

Although frames rotating about a fixed axis are the most commonly analyzed (see Centrifugal force (rotating reference frame)), centrifugal forces also arise in more general non-inertial reference frames. The centrifugal forces considered here arise when observing a moving particle from several different non-inertial frames, for example, a "local" frame (one tied to the moving particle so it appears stationary), and a "co-rotating" frame (one with an arbitrarily located but fixed axis and a rate of rotation that makes the particle appear to have only radial motion and zero azimuthal motion).

Unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects.

The apparent motion that may be ascribed to centrifugal force is sometimes called the centrifugal effect. [cite web|url=http://dlmcn.com/circle.html|title=Centrifugal and Coriolis Effects|author=David McNaughton|accessdate=2008-05-18] [cite web|title=Centrifugal Effect|url=http://www.physics.sfsu.edu/~lwilliam/111/shm/tsld027.htm|author=Lynda Williams|accessdate=2008-05-18]

Analysis using fictitious forces

The connection between inertial frames and fictitious forces (also called "inertial forces" or "pseudo-forces"), is expressed, for example, by Arnol'd:cite book |title=Mathematical Methods of Classical Mechanics |page=p. 129 |author=V. I. Arnol'd |isbn=978-0-387-96890-2 |year=1989 |url=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=|publisher=Springer] A slightly different tack on the subject is provided by Iro:cite book |url= http://books.google.com/books?id=-L5ckgdxA5YC&pg=PA179&dq=force+%22non-inertial+%22&lr=&as_brr=0&sig=ACfU3U3fCW68SLm1zalPZWi0LvLK4DrXYg#PPA180,M1 |author = Harald Iroh |page=p. 180 |title=A Modern Approach to Classical Mechanics |isbn=9812382135 |publisher=World Scientific |year=2002]

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: cite book |title=Classical Mechanics |author=K.S. Rao |page=p. 162 |url=http://books.google.com/books?id=Al7LRzoQhxsC&pg=PA268&vq=by+adding+the+terms&dq=real+%22inertial+forces%22&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U0uHeQp1Yb_zTlGk50Hj40vcMVkHg#PPA162,M1
isbn=8173714363 |year=2003 |publisher=Orient Longman ] cite book |title=Analytical Mechanics |author =Louis N. Hand, Janet D. Finch |page=p. 267 |url=http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA267&vq=fictitious+forces&dq=Hand+inauthor:Finch&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U33emV_6eJZihu3M6IZKurSt85_eg
isbn=0521575729 |publisher=Cambridge University Press |year=1998 ] Because fictitious forces do not originate from other objects, there is no originating object to experience an associated reaction force: does not apply to fictitious forces.

Moving objects and observational frames of reference

Next, it is observed that time varying coordinates are used in both inertial and non-inertial frames of reference, so the use of time varying coordinates should not be confounded with a change of observer, but is only a change of the observer's choice of description. Elaboration of this point and some citations on the subject follow.

Frame of reference and coordinate system

The term frame of reference is used often in a very broad sense, but for the present discussion its meaning is restricted to refer to an observer's "state of motion", that is, to either an inertial frame of reference or a non-inertial frame of reference.

The term coordinate system is used to differentiate between different possible choices for a set of variables to describe motion, choices available to any observer, regardless of their state of motion. Examples are Cartesian coordinates, polar coordinates and (more generally) curvilinear coordinates. Here are two quotes relating "state of motion" and "coordinate system":cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= p. 9 |author=Jean Salençon, Stephen Lyle |url=http://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22&lr=&as_brr=0&sig=ACfU3U1tEWQICZdsXeuLyfmH2PoLgZnMGA
isbn=3540414436 |year=2001 |publisher=Springer ] John D. Norton (1993). [http://www.pitt.edu/~jdnorton/papers/decades.pdf "General covariance and the foundations of general relativity: eight decades of dispute"] , "Rep. Prog. Phys.", 56, pp. 835-6.]

Time varying coordinate systems

In a general coordinate system, the basis vectors for the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. It may be noted that coordinate systems attached to both inertial frames and non-inertial frames can have basis vectors that vary in time, space or both, for example the description of a trajectory in polar coordinates as seen from an inertial frame.See Moore and Stommel, Chapter 2, p. 26, which deals with polar coordinates in an inertial frame of reference (what these authors call a "Newtonian frame of reference"), cite book |page=p. 26 |author=Henry Stommel & Dennis W. Moore |title=An Introduction to the Coriolis Force |url=http://books.google.com/books?id=-JQx_t3yGB4C&printsec=frontcover&dq=coriolis+inauthor:Stommel&lr=&as_brr=0&sig=ACfU3U0gX4wrzVzo7bwD7I8HJ_bd24e2Rg#PPA26,M1 |isbn=0231066368 |year=1989 |publisher=Columbia University Press] or as seen from a rotating frame.For example, Moore and Stommel point our that in a "rotating" polar coordinate system, the acceleration terms include reference to the rate of rotation of the "rotating frame". cite book |page=p. 55 |author=Henry Stommel & Dennis W. Moore |title=An Introduction to the Coriolis Force |url=http://books.google.com/books?id=-JQx_t3yGB4C&printsec=frontcover&dq=coriolis+inauthor:Stommel&lr=&as_brr=0&sig=ACfU3U0gX4wrzVzo7bwD7I8HJ_bd24e2Rg#PPA55,M1] A time-dependent "description" of observations does not change the frame of reference in which the observerations are made and recorded.

Centrifugal force in a local coordinate system

In discussion of a particle moving in a circular orbit,The term "particle" is used in mechanics to describe an object without reference to its orientation. The term rigid body is used when orientation is also a factor. Thus, the center of mass of a rigid body is a "particle". ] in an inertial frame of reference one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats, change perspective, and talk about fictitious centrifugal and Euler forces. But what underlies this switch is a change of observational frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play. That switch is unconscious, but real.

Suppose we sit on a particle in planar motion. What analysis underlies a switch of hats to introduce fictitious centrifugal and Euler forces?

To explore that question, begin in a stationary frame of reference. By using a coordinate system commonly used in planar motion, the so-called "local" coordinate system,Observational frames of reference and coordinate systems are independent ideas. A frame of reference is a physical notion related to the observer's state of motion. A coordinate system is a mathematical description, which can be chosen to suit the observations. A change to a coordinate system that moves in time affects the description of the particle motion, but does not change the observer's state of motion. For more discussion, see Frame of reference] as shown in Figure 1, it becomes easy to identify formulas for the centripetal inward force normal to the trajectory (in direction opposite to u_n in Figure 1), and the tangential force parallel to the trajectory (in direction u_t), as shown next.

To introduce the unit vectors of the local coordinate system shown in Figure 1, one approach is to begin in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 1, the arc length "s" is the distance the particle has traveled along its path in time "t". The path r ("t") with components "x"("t"), "y"("t") in Cartesian coordinates is described using arc length "s"("t") as:The article on curvature treats a more general case where the curve is parametrized by an arbitrary variable (denoted "t"), rather than by the arc length "s".] :$mathbf\{r\}(s)\; =\; left\; [\; x(s),\; y(s)\; ight]\; .$One way to look at the use of "s" is to think of the path of the particle as sitting in space, like the trail left by a skywriter, independent of time. Any position on this path is described by stating its distance "s" from some starting point on the path. Then an incremental displacement along the path "ds" is described by::$dmathbf\{r\}(s)\; =\; left\; [\; dx(s),\; dy(s)\; ight]\; =left\; [\; x\text{'}(s),\; y\text{'}(s)\; ight]\; ds\; ,$ where primes are introduced to denote derivatives with respect to "s". The magnitude of this displacement is "ds", showing that:cite book |title=Railroad Vehicle Dynamics: A Computational Approach |author=Ahmed A. Shabana, Khaled E. Zaazaa, Hiroyuki Sugiyama |page=p. 91 |url=http://books.google.com/books?id=YgIDSQT0FaUC&pg=RA1-PA207&dq=%22generalized+coordinate%22&lr=&as_brr=0&sig=ACfU3U2tosoLUEAUNkGu2x8TTtuxLfeLGQ#PRA1-PA91,M1
isbn=1420045814 |publisher=CRC Press |year=2007 ] :$left\; [\; x\text{'}(s)^2\; +\; y\text{'}(s)^2\; ight]\; =\; 1\; .$    anchor|Eq. 1(Eq. 1)This displacement is necessarily tangent to the curve at "s", showing that the unit vector tangent to the curve is::$mathbf\{u\}\_t(s)\; =\; left\; [\; x\text{'}(s),\; y\text{'}(s)\; ight]\; ,$while the outward unit vector normal to the curve is :$mathbf\{u\}\_n(s)\; =\; left\; [\; y\text{'}(s),\; -x\text{'}(s)\; ight]\; ,$Orthogonality can be verified by showing the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1.

As an aside, notice that the use of unit vectors that are not aligned along the Cartesian "xy"-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with "s" to describe the path, but still observe the motion from the inertial frame.

Using the tangent vector, the angle of the tangent to the curve, say θ, is given by::$sin\; heta\; =frac\{y\text{'}(s)\}\{sqrt\{x\text{'}(s)^2+y\text{'}(s)^2\; =\; y\text{'}(s)\; ;$   and   $cos\; heta\; =frac\{x\text{'}(s)\}\{sqrt\{x\text{'}(s)^2+y\text{'}(s)^2\; =\; x\text{'}(s)\; .$The radius of curvature is introduced completely formally (without need for geometric interpretation) as::$frac\{1\}\{\; ho\}\; =\; frac\{d\; heta\}\{ds\}\; .$The derivative of θ can be found from that for sin θ::$frac\{d\; sin\; heta\}\{ds\}\; =\; cos\; heta\; frac\; \{d\; heta\}\{ds\}\; =\; frac\{1\}\{\; ho\}\; cos\; heta$::$=\; frac\{1\}\{\; ho\}\; x\text{'}(s)\; .$Now::$frac\{d\; sin\; heta\; \}\{ds\}\; =\; frac\{d\}\{ds\}\; frac\{y\text{'}(s)\}\{sqrt\{x\text{'}(s)^2+y\text{'}(s)^2$  $=\; frac\{y"(s)x\text{'}(s)^2-y\text{'}(s)x\text{'}(s)x"(s)\}\; \{left(x\text{'}(s)^2+y\text{'}(s)^2\; ight)^\{3/2\; ,$in which the denominator is unity according to Eq. 1. With this formula for the derivative of the sine, the radius of curvature becomes::$frac\; \{d\; heta\}\{ds\}\; =\; frac\{1\}\{\; ho\}\; =\; y"(s)x\text{'}(s)\; -\; y\text{'}(s)x"(s)$ $=frac\{y"(s)\}\{x\text{'}(s)\}\; =\; -frac\{x"(s)\}\{y\text{'}(s)\}\; ,$where the equivalence of the forms stems from differentiation of Eq. 1::$x\text{'}(s)x"(s)\; +\; y\text{'}(s)y"(s)\; =\; 0\; .$ Having set up the description of any position on the path in terms of its associated value for "s", and having found the properties of the path in terms of this description, motion of the particle is introduced by stating the particle position at any time "t" as the corresponding value "s (t)".

Using the above results for the path properties in terms of "s", the acceleration in the inertial reference frame as described in terms of the components normal and tangential to the path of the particle can be found in terms of the function "s"("t") and its various time derivatives (as before, "primes" indicate differentiation with respect to "s")::$mathbf\{a\}(s)\; =\; frac\{d\}\{dt\}mathbf\{v\}(s)$  $=\; frac\{d\}\{dt\}left\; [frac\{ds\}\{dt\}\; left(\; x\text{'}(s),\; y\text{'}(s)\; ight)\; ight]$::$=\; left(frac\{d^2s\}\{dt^2\}\; ight)mathbf\{u\}\_t(s)\; +left(frac\{ds\}\{dt\}\; ight)\; ^2\; left(x"(s),\; y"(s)\; ight)$::$=\; left(frac\{d^2s\}\{dt^2\}\; ight)mathbf\{u\}\_t(s)\; -\; left(frac\{ds\}\{dt\}\; ight)\; ^2\; frac\{1\}\{\; ho\}\; mathbf\{u\}\_n(s)\; ,$as can be verified by taking the dot product with the unit vectors u_t("s") and u_n("s"). This result for acceleration is the same as that for circular motion based on the radius ρ. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force.

Next, we change observational frames. Sitting on the particle, we adopt a non-inertial frame where the particle is at rest (zero velocity). This frame has a continuously changing origin, which at time "t" is the center of curvature (the center of the osculating circle in Figure 1) of the path at time "t", and whose rate of rotation is the angular rate of motion of the particle about that origin at time "t". This non-inertial frame also employs unit vectors normal to the trajectory and parallel to it.

The angular velocity of this frame is the angular velocity of the particle about the center of curvature at time "t". The centripetal force of the inertial frame is interpreted in the non-inertial frame where the body is at rest as a force necessary to overcome the centrifugal force. Likewise, the force causing any acceleration of speed along the path seen in the inertial frame becomes the force necessary to overcome the Euler force in the non-inertial frame where the particle is at rest. There is zero Coriolis force in the frame, because the particle has zero velocity in this frame. For a pilot in an airplane, for example, these fictitious forces are a matter of direct experience.However, the pilot also will experience Coriolis force, because the pilot is not a "particle". When the pilot's head moves, for example, the head has a velocity in the non-inertial frame, and becomes subject to Coriolis force. This force causes pilot disorientation in a turn. See Coriolis effect (perception), cite book |title=Space biology and medicine |author=Arnauld E. Nicogossian |page=p. 337 |url=http://books.google.com/books?id=aO6zut2K7lsC&pg=PA337&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U2ODlvCKri-JbJfB-OdyhXyhozbnw
isbn=1563471809 |year=1996 |publisher=American Institute of Aeronautics and Astronautics, Inc |location=Reston, VA, and cite book |title=Fundamentals of Space Medicine |author=Gilles Clément |page=p. 41 |url=http://books.google.com/books?id=Neura4O-taIC&pg=PA41&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U16LpzGILe3QVGIeOl5tyYFDAKLrA
isbn=1402015984 |year=2003 |publisher=Springer.] However, these fictitious forces cannot be related to a simple observational frame of reference other than the particle itself, unless it is in a particularly simple path, like a circle.

That said, from a qualitative standpoint, the path of an airplane can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius. See article discussing turning an airplane.

Next, reference frames rotating about a fixed axis are discussed in more detail.

Centrifugal force in polar coordinates

Description of particle motion often is simpler in non-Cartesian coordinate systems, for example, polar coordinates. When equations of motion are expressed in terms of any curvilinear coordinate system, extra terms appear that represent how the basis vectors change as the coordinates change.

Two terminologies

In a purely mathematical treatment, regardless of the frame that the coordinate system is associated with (inertial or non-inertial), extra terms appear in the acceleration of an observed particle. For example, in polar coordinates the acceleration is given by (see below for details):::which contains not just double time derivatives of the coordinates but added terms. This example employs polar coordinates, but the added terms depend upon which coordinate system is chosen (that is, polar, elliptic, or whatever).Sometimes these coordinate-system dependent terms also are referred to as "fictitious forces", introducing a second meaning for this term. According to this terminology, fictitious forces are determined in part by the coordinate system itself, regardless of the frame it is attached to, that is, regardless of whether the coordinate system is attached to an inertial or a non-inertial frame of reference. In contrast, the fictitious forces defined in terms of the "state of motion of the observer" vanish in inertial frames of reference. To distinguish these two terminologies, the fictitious forces that vanish in an inertial frame of reference are called in this article the "state-of-motion" fictitious forces and those that originate in the interpretation of time derivatives in particular coordinate systems are called "coordinate" fictitious forces.

Assuming it is clear that "state of motion" and "coordinate system" are "different", it follows that the dependence of centrifugal force (as in this article) upon "state of motion" and its independence from "coordinate system", which contrasts with the "coordinate" version with exactly the opposite dependencies, indicates that two different ideas are referred to by the terminology "fictitious force". The present article emphasizes one of these two ideas ("state-of-motion"), although the other also is described.

Below, polar coordinates are introduced for use in (first) an inertial frame of reference and then (second) in a rotating frame of reference. The two different uses of the term "fictitious force" are pointed out. First, however, follows a brief digression to explain further how the "coordinate" terminology for fictitious force has arisen.

A Lagrangian approach

To motivate the introduction of "coordinate" inertial forces by more than a reference to "mathematical convenience", what follows is a digression to show these forces correspond to what are called by some authors "generalized" fictitious forces or "generalized inertia forces".cite book |title=Advanced Dynamics |author= Donald T. Greenwood |isbn=0521826128 |publisher=Cambridge University Press |year=2003 |page=p. 77 |url=http://books.google.com/books?id=r2CSj1A-zWQC&pg=PA237&dq=%22generalized+inertia+force%22&lr=&as_brr=0&sig=ACfU3U3MK5xDZDKq3p-tuf-vNz-li-0Flw#PPA77,M1 ] cite book |title=Fundamentals of Multibody Dynamics: Theory and Applications |author=Farid M. L. Amirouche |page=p. 207 |url=http://books.google.com/books?id=_nlEcQYldeIC&pg=PA497&dq=%22generalized+inertial+force%22&lr=&as_brr=0&sig=ACfU3U1WZwSvXxTsX2BG4NwmgqAx5aa0ew#PPA207,M1 |isbn=0817642366 |publisher=Springer |year=2006 ] cite book |author= Harold Josephs, Ronald L. Huston |title=Dynamics of Mechanical Systems |page=p. 377 |url=http://books.google.com/books?id=vZeeO7J68EIC&pg=PA377&dq=%22generalized+inertia+force%22&lr=&as_brr=0&sig=ACfU3U0BRjNDkpj5MtRcyJCcTbuzu-N0Nw#PPA377,M1 |isbn=0849305934 |publisher=CRC Press |year=2002 ] cite book |title=Computational Dynamics |page=p. 217 |url=http://books.google.com/books?id=KUJkG13VF98C&pg=PA217&dq=%22generalized+inertia+force%22&lr=&as_brr=0&sig=ACfU3U37S59yoqvW8V-FZQ4PVAIB3CzGJQ |author= Ahmed A. Shabana |isbn=0471371440 |publisher=Wiley |year=2001 ] These forces are introduced via the Lagrangian mechanics approach to mechanics based upon describing a system by "generalized coordinates" usually denoted as {"q_k"}. The only requirement on these coordinates is that they are necessary and sufficient to uniquely characterize the state of the system: they need not be (although they could be) the coordinates of the particles in the system. Instead, they could be the angles and extensions of links in a robot arm, for instance. If a mechanical system consists of "N" particles and there are "m" independent kinematical conditions imposed, it is possible to characterize the system uniquely by "n" = 3"N - m" independent generalized coordinates {"q_k"}.cite book |title=The Variational Principles of Mechanics |author =Cornelius Lanczos |page=p. 10 |isbn=0-486-65067-7 |year=1986 |url=http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4&dq=isbn:0486650677&sig=ACfU3U2R5sLjGS22S-h8Z_j9RiPJnKcKZg#PPA7,M1 |publisher=Dover Publications |edition=Reprint of 4rth Edition of 1970 ]

In classical mechanics, the Lagrangian is defined as the kinetic energy, $T$, of the system minus its potential energy, $U$.cite book |title=The Variational Principles of Mechanics |author =Cornelius Lanczos |page=pp. 112-113 |isbn=0-486-65067-7 |year=1986 |url=http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4&dq=isbn:0486650677&sig=ACfU3U2R5sLjGS22S-h8Z_j9RiPJnKcKZg#PPA7,M1 |publisher=Dover Publications |edition=Reprint of 4rth Edition of 1970 ] In symbols,

:$L\; =\; T\; -\; U.quad$

Under conditions that are given in Lagrangian mechanics, if the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation, a particular family of partial differential equations.

Here are some definitions:cite book |title= Mathematical Methods of Classical Mechanics |author=Vladimir Igorevich Arnolʹd |url=http://books.google.com/books?id=Pd8-s6rOt_cC&pg=PA129&dq=%22additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally%22&lr=&as_brr=0&sig=ACfU3U3k_JqejV-asMO-2LKNELQKDVjY7Q#PPA60,M1 |page=p. 60 |isbn=0387968903 |publisher=Springer |year=1989]

:Definition::: :is the "Lagrange function" or "lagrangian", "q_i" are the "generalized coordinates", $dot\{q\_i\}$ are "generalized velocities",::$partial\; L\; /\; partial\; dot\{q\_i\}$ :are "generalized momenta", ::$partial\; L/partial\; q\_i$:are "generalized forces",::$frac\{d\}\{dt\}\; frac\{partial\; L\}\{partial\; dot\{q\_i\; -frac\; \{partial\; L\}\{partial\; q\_i\}\; =\; 0$:are "Lagrange's equations".

It is not the purpose here to outline how Lagrangian mechanics works. The interested reader can look at other articles explaining this approach. For the moment, the goal is simply to show that the Lagrangian approach can lead to "generalized fictitious forces" that "do not vanish in inertial frames". What is pertinent here is that in the case of a single particle, the Lagrangian approach can be arranged to capture exactly the "coordinate" fictitious forces just introduced.

To proceed, consider a single particle, and introduce the generalized coordinates as {"q_k"} = ("r, θ"). Then Hildebrand cite book |title=Methods of Applied Mathematics |author=Francis Begnaud Hildebrand |page=pp. 156-157 |isbn=0486670023 |publisher=Courier Dover Publications |year=1965 |url=http://books.google.com/books?id=17EZkWPz_eQC&pg=PA156&dq=absence+fictitious+force&lr=&as_brr=0&sig=ACfU3U1rrR7AnDqhMl7XJkkOEMJLr8co2Q ] shows in polar coordinates with the "q_k" = "(r, θ)" the "generalized momenta" are:::$p\_r\; =\; m\; dot\; r\; ,\; p\_\{\; heta\}\; =\; mr^2\; dot\{\; heta\}\; ,$leading, for example, to the generalized force:::$frac\{d\}\{dt\}p\_r\; =Q\_r\; +\; mr\{dot\{\; heta^2\; ,$with "Q_r" the impressed radial force. The connection between "generalized forces" and Newtonian forces varies with the choice of coordinates. This Lagrangian formulation introduces exactly the "coordinate" form of fictitious forces mentioned above that allows "fictitious" (generalized) forces in inertial frames, for example, the term $mr\; \{\; dot\{\; heta^2\; .$ Careful reading of Hildebrand shows he doesn't discuss the role of "inertial frames of reference", and in fact, says " [The] presence or absence [of inertia forces] depends, not upon the particular problem at hand but "upon the coordinate system chosen"." By coordinate system presumably is meant the choice of {"q_k"}. Later he says "If "accelerations" associated with generalized coordinates are to be of prime interest (as is usually the case), the [nonaccelerational] terms may be conveniently transferred to the right … and considered as additional (generalized) inertia forces. Such inertia forces are often said to be of the "Coriolis" type."

In short, the emphasis of some authors upon coordinates and their derivatives and their introduction of (generalized) fictitious forces that do not vanish in inertial frames of reference is an outgrowth of the use of generalized coordinates in Lagrangian mechanics. For example, see McQuarriecite book |author=Donald Allan McQuarrie |title=Statistical Mechanics |page=pp.5-6 |url=http://books.google.com/books?id=itcpPnDnJM0C&pg=PA5&dq=centrifugal+%22polar+coordinates%22&lr=&as_brr=0&sig=ACfU3U3i8xqTmx894j9QhQagOncOyBPWWA#PPA5,M1
isbn=1891389157 |year=2000 |publisher=University Science Books ] and Hildebrand.cite book |title=Methods of Applied Mathematics |author=Francis Begnaud Hildebrand |page=pp. 156-157 |isbn=0486670023 |publisher=Courier Dover Publications |year=1965 |url=http://books.google.com/books?id=17EZkWPz_eQC&pg=PA156&dq=absence+fictitious+force&lr=&as_brr=0&sig=ACfU3U1rrR7AnDqhMl7XJkkOEMJLr8co2Q ] Below is an example of this usage as employed in the design of robotic manipulators:cite book |title=Automatic Differentiation of Algorithms: From Simulation to Optimization |page=p. 131 |author=George F. Corliss, Christele Faure, Andreas Griewank, Laurent Hascoet (editors) |url=http://books.google.com/books?id=z3nZ62xY43QC&pg=PA131&dq=centrifugal+generalized&lr=&as_brr=0&sig=ACfU3U2x3lfNrJyiJ4121VgAzh4bmb_xPA |isbn=0387953051 |publisher=Springer |year=2002 ] cite book |title=Advances in Computational Multibody Systems |page = p. 322 |author= Jorge A. C. Ambrósio (editor) |isbn=1402033923 |publisher=Springer |year=2003 |url=http://books.google.com/books?id=Jv6e3oxkI6YC&pg=PA322&dq=centrifugal+generalized&lr=&as_brr=0&sig=ACfU3U3Ho22GpT7PLsFsmy--WsgdxauPgg ] cite book |title=Adaptive Neural Network Control of Robotic Manipulators |author=Shuzhi S. Ge, Tong Heng Lee, Christopher John Harris |isbn=981023452X |publisher=World Scientific |year=1998 |page=pp. 47-48 |url=http://books.google.com/books?id=cdBENqlY_ucC&printsec=frontcover&dq=CHristoffel+centrifugal&lr=&as_brr=0#PPA47,M1 ] The introduction of "generalized" fictitious forces often is done without notification and without specifying the word "generalized". This sloppy use of terminology leads to endless confusion because these "generalized" fictitious forces, unlike the standard "state-of-motion" fictitious forces, do not vanish in inertial frames of reference.

Polar coordinates in an inertial frame of reference

Below, the "state-of-motion" fictitious forces defined for non-inertial frames are related to the "coordinate" versions for polar coordinates ("r", "θ"). In an inertial frame, let $mathbf\{r\}$ be the position vector of a moving particle. Its Cartesian components ("x", "y") are::$mathbf\{r\}\; =(rcos\; heta,\; rsin\; heta)\; ,$

with "r" and θ depending on time "t".

Unit vectors are defined in the radially outward direction $mathbf\{r\}$: :

and in the direction at right angles to $mathbf\{r\}$::

These unit vectors vary in direction with time::and::

Using these derivatives, the first and second derivatives of position are:

:

:where dot-overmarkings indicate time differentiation. With this form for the acceleration , in an inertial frame of reference Newton's second law expressed in polar coordinates is::where "F" is the net real force on the particle. No fictitious forces appear because all fictitious forces are zero by definition in an inertial frame.

From a mathematical standpoint, however, it sometimes is handy to put only the second-order derivatives on the right side of this equation; that is we write the above equation by rearrangement of terms as::where a "coordinate" version of the "acceleration" is introduced::consisting of only second-order time derivatives of the coordinates "r" and θ. The terms moved to the force-side of the equation are now treated as "extra" "fictitious forces" and, confusingly, the resulting forces also are called the "centrifugal" and "Coriolis" force.

These newly defined "forces" are non-zero in an "inertial frame", and so certainly are not the same as the previously identified fictitious forces that are zero in an inertial frame and non-zero only in a non-inertial frame.For a treatment using these terms as fictitious forces, see cite book |title=An Introduction to the Coriolis Force |author=Henry Stommel, Dennis W. Moore |page=p. 36 |url=http://books.google.com/books?id=-JQx_t3yGB4C&pg=PA36&dq=%22acceleration+terms+on+the+righthand%22&lr=&as_brr=0&sig=ACfU3U2TxfwVuND0hVjSKzIEdo9w81diQQ
isbn=0231066368] In this article, these newly defined forces are called the "coordinate" centrifugal force and the "coordinate" Coriolis force to separate them from the "state-of-motion" forces.

Co-rotating frame

In the case of planar motion of a particle, the "coordinate" centrifugal and Coriolis acceleration terms found above to be non-zero in an inertial frame can be shown to be the negatives of the "state-of-motion" centrifugal and Coriolis terms that appear in a very particular non-inertial "co-rotating" frame (see next subsection).For the following discussion, see cite book |url=http://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=shares+the+same+origin+O+and+is+rotating&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= |author=John R Taylor |title=Classical Mechanics |quote=At the chosen instant "t₀", the frame "S' " and the particle are rotating at the same rate....In the inertial frame, the forces are simpler (no "fictitious" forces) but the accelerations are more complicated.; in the rotating frame, it is the other way round. |page=§9.10, pp. 358-359 |isbn=1-891389-22-X |publisher=University Science Books |year=2005] See Figure 2. To define a co-rotating frame, first an origin is selected from which the distance "r(t)" to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment "t", the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, "dθ/dt". The co-rotating frame applies only for a moment, and must be continuously re-selected as the particle moves. For more detail, see Polar coordinates, centrifugal and Coriolis terms.

Polar coordinates in a rotating frame of reference

Next, the same approach is used to find the fictitious forces of a (non-inertial) rotating frame. For example, if a rotating polar coordinate system is adopted for use in a rotating frame of observation, both rotating at the same constant counterclockwise rate Ω, we find the equations of motion in this frame as follows: the radial coordinate in the rotating frame is taken as "r", but the angle θ' in the rotating frame changes with time::$heta\; \text{'}\; =\; heta\; -\; Omega\; t\; .$Consequently,:$dot\; heta\; \text{'}\; =\; dot\; heta\; -\; Omega\; .$Plugging this result into the acceleration using the unit vectors of the previous section::::The leading two terms are the same form as those in the inertial frame, and they are the only terms if the frame is "not" rotating, that is, if Ω=0. However, in this rotating frame we have the extra terms:cite book |title=An Introduction to the Coriolis Force |author=Henry Stommel & Dennis W. Moore |year=1989 |isbn=0231066368 |url=http://books.google.com/books?id=-JQx_t3yGB4C&pg=PA55&dq=%22an+additional+centrifugal+%22force%22%22&lr=&as_brr=0&sig=ACfU3U2rCWI1cSoAXXliJk_WyJxH8fQiSA |page=p. 55 ]

: The radial term Ω² "r" is the centrifugal force per unit mass due to the system's rotation at rate Ω and the radial term $2\; r\; Omega\; dot\; heta\; \text{'}$ is the radial component of the Coriolis force per unit mass, where $r\; dot\; heta\; \text{'}$ is the tangential component of the particle velocity as seen in the rotating frame. The term is the so-called "azimuthal" component of the Coriolis force per unit mass. In fact, these extra terms can be used to "measure" Ω and provide a test to see whether or not the frame is rotating, just as explained in the example of rotating identical spheres. If the particle's motion can be described by the observer using Newton's laws of motion "without" these Ω-dependent terms, the observer is in an inertial frame of reference where Ω=0.

These "extra terms" in the acceleration of the particle are the "state of motion" fictitious forces for this rotating frame, the forces introduced by rotation of the frame at angular rate Ω.This derivation can be found in cite book |title=An Introduction to the Coriolis Force |author=Henry Stommel, Dennis W. Moore |page= Chapter III, pp. 54 "ff" |url=http://books.google.com/books?id=-JQx_t3yGB4C&printsec=frontcover&dq=coriolis+inauthor:Stommel&lr=&as_brr=0&sig=ACfU3U0gX4wrzVzo7bwD7I8HJ_bd24e2Rg#PPA54,M1]

In this rotating frame, what are the "coordinate" fictitious forces? As before, suppose we choose to put only the second-order time derivatives on the right side of Newton's law::  

If we choose for convenience to treat as some so-called "acceleration", then the terms are added to the so-called "fictitious force", which are not "state-of-motion" fictitious forces, but are actually components of force that persist even when Ω=0, that is, they persist even in an inertial frame of reference. Because these extra terms are added, the "coordinate" fictitious force is not the same as the "state-of-motion" fictitious force. Because of these extra terms, the "coordinate" fictitious force is not zero even in an inertial frame of reference.

Notice however, the case of a rotating frame that happens to have the same angular rate as the particle, so that Ω = "dθ/dt" at some particular moment (that is, the polar coordinates are set up in the instantaneous, non-inertial co-rotating frame of Figure 2). In this case, at this moment, "dθ'/dt = 0". In this co-rotating non-inertial frame at this moment the "coordinate" fictitious forces are only those due to the motion of the frame, that is, they are the same as the "state-of-motion" fictitious forces, as discussed in the remarks about the co-rotating frame of Figure 2 in the previous section.

Centrifugal force in curvilinear coordinates

Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the acceleration of a particle expressed in this coordinate system, as described below in more detail. Consider description of a particle motion from the viewpoint of an "inertial frame of reference" in curvilinear coordinates. Suppose the position of a point "P" in Cartesian coordinates is ("x", "y", "z") and in curvilinear coordinates is ("q₁", "q₂". "q₃"). Then functions exist that relate these descriptions::$x\; =\; x(q\_1,\; q\_2,\; q\_3)\; ;$ $q\_1\; =\; q\_1(x,\; y,\; z)\; ,$and so forth. (The number of dimensions may be larger than three.) An important aspect of such coordinate systems is the element of arc length that allows distances to be determined. If the curvilinear coordinates form an orthogonal coordinate system, the element of arc length "ds" is expressed as::$ds^2\; =\; sum\_\{k=1\}^\{d\}\; left(\; h\_\{k\}\; ight)^\{2\}\; left(\; dq\_\{k\}\; ight)^\{2\}\; ,$where the quantities "h_k" are called "scale factors".cite book |title=Methods of Mathematical Physics |author=PM Morse & H Feshbach |publisher=McGraw Hill |page=p. 25 |edition =First Edition |year=1953] A change "dq_k" in "q_k" causes a displacement "h_k dq_k" along the coordinate line for "q_k". At a point "P", we place unit vectors e_k each tangent to a coordinate line of a variable "q_k". Then any vector can be expressed in terms of these basis vectors, for example, from an inertial frame of reference, the position vector of a moving particle r located at time "t" at position "P" becomes::where "q_k" is the vector dot product of r and e_k.The velocity v of a particle at "P", can be expressed at "P" as:::: where "v_k" is the vector dot product of v and e_k, and over dots indicate time differentiation.The time derivatives of the basis vectors can be expressed in terms of the scale factors introduced above. for example::  or, in general,   in which the coefficients of the unit vectors are the Christoffel symbols for the coordinate system. The general notation and formulas for the Christoffel symbols are:cite book |title=Methods of Mathematical Physics |author=PM Morse & H Feshbach |publisher=McGraw Hill |page=pp. 47-48 |edition =First Edition |year=1953] cite book |title=Continuum mechanics |author=I-Shih Liu |page= Appendix A2 |url=http://books.google.com/books?id=-gWqM4uMV6wC&pg=PT307&dq=CHristoffel+centrifugal&lr=&as_brr=0&sig=ACfU3U1dNjYh273fApEPFhrIf5Y2r7_e4Q#PPT291,M1
isbn=3540430199 |publisher=Springer |year=2002] :  h_j}^2}frac{partial h_i}{partial q_j} ,and the symbol is zero when all the indices are different.Using relations like this one, :::which allows all the time derivatives to be evaluated. For example, for the velocity:: ::::::with the Γ-notation for the Christoffel symbols replacing the curly bracket notation.Using the same approach, the acceleration is then:::Looking at the relation for acceleration, the first summation contains the time derivatives of velocity, which would be associated with acceleration if these were Cartesian coordinates, and the second summation (the one with Christoffel symbols) contains terms related to the way the unit vectors change with time.For application of the Christoffel symbols formalism to a rotating coordinate system, see cite book |author=Ludwik Silberstein |title=The Theory of General Relativity and Gravitation |page=pp. 30-32 |url=http://books.google.com/books?lr=&as_brr=0&id=C1SwFdj_kGUC&dq=CHristoffel+centrifugal&jtp=29#PRA1-PA30,M1
publisher=D. Van Nostrand |year=1922 ]

="State-of-motion" versus "coordinate" fictitious forces=

Earlier in this article a distinction was introduced between two terminologies, the fictitious forces that vanish in an inertial frame of reference are called in this article the "state-of-motion" fictitious forces and those that originate from differentiation in a particular coordinate system are called "coordinate" fictitious forces. Using the expression for the acceleration above, Newton's law of motion in the inertial frame of reference becomes::where "F" is the net real force on the particle. No "state-of-motion" fictitious forces are present because the frame is inertial, and "state-of-motion" fictitious forces are zero in an inertial frame, by definition.

The "coordinate" approach to Newton's law above is to retain the second-order time derivatives of the coordinates {"q_k"} as the only terms on the right side of this equation, motivated more by mathematical convenience than by physics. To that end, the force law can be re-written, taking the second summation to the force-side of the equation as:: , with the convention that the "acceleration" is now::In the expression above, the summation added to the force-side of the equation now is treated as if added forces were present. These summation terms are customarily called fictitious forces within this "coordinate" approach, although in this inertial frame of reference all "state-of-motion" fictitious forces are identically zero. Thus, the designation of the terms of the summation as "fictitious forces" uses this terminology for contributions that are completely different from the "state-of-motion" fictitious forces. What adds to this confusion is that these "coordinate" fictitious forces are divided into two groups and given the "same names" as the "state-of-motion" fictitious forces, that is, they are divided into "centrifugal" and "Coriolis" terms, despite their inclusion of terms that are not the "state-of-motion" centrifugal and Coriolis terms. For example, these "coordinate" centrifugal and Coriolis terms can be nonzero "even in an inertial frame of reference" where the "state-of-motion" centrifugal force (the subject of this article) and Coriolis force always are zero.For a more extensive criticism of lumping together the two types of fictitious force, see cite book |author=Ludwik Silberstein |title=The Theory of General Relativity and Gravitation |page=p. 29 |url=http://books.google.com/books?lr=&as_brr=0&id=C1SwFdj_kGUC&dq=CHristoffel+centrifugal&jtp=29#PRA1-PA29,M1
publisher=D. Van Nostrand |year=1922 ]

If the frame is not inertial, for example, in a rotating frame of reference, the "state-of-motion" fictitious forces are included in the above "coordinate" fictitious force expression.See Silberstein.] Also, if the "acceleration" expressed in terms of first-order time derivatives of the velocity happens to result in terms that are "not" simply second-order derivatives of the coordinates {"q_k"} in time, then these terms that are not second-order also are brought to the force-side of the equation and included with the fictitious forces.

Formulation of dynamics in terms of Christoffel symbols and the "coordinate" version of fictitious forces is used often in the design of robots in connection with a Lagrangian formulation of the equations of motion. See cite book |title= Control of robot manipulators in joint space |author=R. Kelly, V. Santibáñez, Antonio Loría |page= p. 72 |url=http://books.google.com/books?id=jfhy4ZUPuhYC&printsec=frontcover&dq=Control+of+robot+manipulators+inauthor:kelly&lr=&as_brr=0&sig=ACfU3U2W_sj1AIIM-BoXw957U3Nu1PO1Jw#PPA72,M1
isbn=1852339942 |year=2005 |publisher=Springer ]

Notes and references

Further reading

* [http://members.tripod.com/~gravitee/booki2.htm Newton's description in Principia]
* [http://www.infoplease.com/ce6/sci/A0811114.html Centrifugal reaction force] - Columbia electronic encyclopedia
* M. Alonso and E.J. Finn, "Fundamental university physics", Addison-Wesley
* [http://regentsprep.org/Regents/physics/phys06/bcentrif/default.htm Centripetal force] vs. [http://regentsprep.org/Regents/physics/phys06/bcentrif/centrif.htm Centrifugal force] - from an online Regents Exam physics tutorial by the Oswego City School District
* [http://www.npl.washington.edu/av/altvw55.html Centrifugal force acts inwards near a black hole]
* [http://hyperphysics.phy-astr.gsu.edu/HBASE/corf.html#cent Centrifugal force] at the HyperPhysics concepts site
* [http://www.netsurf.com/nss/misc/centrifugal.html A list of interesting links]

External links

* [http://mensch.org/physlets/merry.html Motion over a flat surface] Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view.
* [http://mensch.org/physlets/inosc.html Motion over a parabolic surface] Java physlet by Brian Fiedler (from School of Meteorology at the University of Oklahoma) illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view.
* [http://www.youtube.com/watch?v=49JwbrXcPjc Animation clip] showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
* [http://www.mathpages.com/home/kmath633/kmath633.htm Centripetal and Centrifugal Forces] at MathPages
* [http://www.bbc.co.uk/dna/h2g2/A597152 Centrifugal Force] at h2g2
* [http://www.wisegeek.com/what-is-a-centrifuge.htm What is a centrifuge?]
* [http://math.ucr.edu/home/baez/physics/General/Centrifugal/centri.html John Baez: "Does centrifugal force hold the Moon up?" ]
* [http://xkcd.com/123/ XKCD demonstrates the life and death importance of centrifugal force]

See also

* Calculating relative centrifugal force
* Circular motion
* Coriolis force
* Coriolis effect (perception)
* Centripetal force
* Equivalence principle
* Bucket argument
* Euler force - a force that appears when the frame angular rotation rate varies
* Folk physics
* Rotational motion
* Reactive centrifugal force - a force that occurs as reaction due to a centripetal force
* Lamm equation
* Frame of reference
* Inertial frame of reference
* Fictitious force – a force that can be made to vanish by changing frame of reference
* Lagrangian point - gravitationally stable points in rotating reference frames
*Orthogonal coordinates
* Frenet-Serret formulas
*Statics
*Kinetics (physics)
*Kinematics
*Applied mechanics
*Analytical mechanics
*Dynamics (physics)
*Classical mechanics
*D'Alembert's principle
*Centrifuge