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":^[11][12]

We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted ${\displaystyle {\mathfrak {R}}}$, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame ${\displaystyle {\mathfrak {R}}}$ by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame ${\displaystyle {\mathfrak {R}}}$, can be considered to give a physical realization of ${\displaystyle {\mathfrak {R}}}$. In a frame ${\displaystyle {\mathfrak {R}}}$, coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.

— Jean Salençon, Stephen Lyle. (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity p. 9

In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…the notion of frame of reference has reappeared as a structure distinct from a coordinate system.

— John D. Norton: General Covariance and the Foundations of General Relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7.

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^[13] or as seen from a rotating frame.^[14] A time-dependent description of observations does not change the frame of reference in which the observations are made and recorded.

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 when using curvilinear coordinates. For example, in polar coordinates the acceleration is given by (see below for details):

$${\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}\ ,}$$

which contains not just double time derivatives of the coordinates but added terms. This example employs polar coordinates, but more generally the added terms depend upon which coordinate system is chosen (that is, polar, elliptic, or any other system). Sometimes these coordinate-system dependent terms also are referred to as "fictitious forces", introducing a second meaning for "fictitious forces", despite the fact that these terms do not have the vector transformation properties expected of forces. For example, see Shankar^[21] and Hildebrand.^[22] 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, the inertial forces of Newtonian mechanics, 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.^[23]

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.

Lagrangian approach

See also: Lagrangian mechanics

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".^{[24][25][26][27]} 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 = 3N - m independent generalized coordinates {q_k}.^[28]

In classical mechanics, the Lagrangian is defined as the kinetic energy, ${\displaystyle T}$, of the system minus its potential energy, ${\displaystyle U}$.^[29] In symbols,

$${\displaystyle L=T-U.}$$

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:^[30]

Definition:

$${\displaystyle L({\boldsymbol {q}},\ {\boldsymbol {\dot {q}}},\ t)=T-U}$$

is the Lagrange function or Lagrangian, q_i are the generalized coordinates, ${\displaystyle {\dot {q_{i}}}}$ are generalized velocities,

• ${\displaystyle \partial L/\partial {\dot {q_{i}}}}$   are generalized momenta,
• ${\displaystyle \partial L/\partial q_{i}}$   are generalized forces,
• ${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}-{\frac {\partial L}{\partial q_{i}}}=0}$   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 ^[22] shows in polar coordinates with the q_k = (r, θ) the "generalized momenta" are:

$${\displaystyle p_{r}=m{\dot {r}}\ ,\ p_{\theta }=mr^{2}{\dot {\theta }}\ ,}$$

leading, for example, to the generalized force:

$${\displaystyle {\frac {d}{dt}}p_{r}=Q_{r}+mr{\dot {\theta }}^{2}\ ,}$$

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 ${\displaystyle mr{\dot {\theta }}^{2}\ .}$ 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 McQuarrie^[31] Hildebrand,^[22] and von Schwerin.^[32] Below is an example of this usage as employed in the design of robotic manipulators:^[33][34][35]

In the above [Lagrange-Euler] equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in ${\displaystyle \mathbf {\dot {q}} }$ where the coefficients may depend on ${\displaystyle \mathbf {q} }$. These are further classified into two types. Terms involving a product of the type ${\displaystyle {{\dot {q}}_{i}}^{2}}$ are called centrifugal forces while those involving a product of the type ${\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}$ for i ≠ j are called Coriolis forces. The third type is functions of ${\displaystyle \mathbf {q} }$ only and are called gravitational forces.

— Shuzhi S. Ge, Tong Heng Lee & Christopher John Harris: Adaptive Neural Network Control of Robotic Manipulators, pp. 47-48

For a robot manipulator, the equations may be written in a form using Christoffel symbols Γ_ijk (discussed further below) as:^[36][37]

$${\displaystyle \sum _{j=1}^{n}\ M_{ij}({\boldsymbol {q}}){\ddot {q}}_{j}+\sum _{j,k=1}^{n}\Gamma _{ijk}{\dot {q}}_{j}{\dot {q}}_{k}+{\frac {\partial V}{\partial q_{i}}}=\Upsilon _{i}\ ;i=1,\dots ,n\ ,}$$

where M is the "manipulator inertia matrix" and V is the potential energy due to gravity (for example), and ${\displaystyle \Upsilon _{i}}$ are the generalized forces on joint i. The terms involving Christoffel symbols therefore determine the "generalized centrifugal" and "generalized Coriolis" terms.

The introduction of generalized fictitious forces often is done without notification and without specifying the word "generalized". This use of terminology can lead to confusion because 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

Position vector r, always points radially from the origin.

Velocity vector v, always tangent to the path of motion.

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. The setup is not restricted to 2D space, but a plane in any higher dimension.

Below, the acceleration of a particle is derived as seen in an inertial frame using polar coordinates. There are no "state-of-motion" fictitious forces in an inertial frame, by definition. Following that presentation, the contrasting terminology of "coordinate" fictitious forces is presented and critiqued on the basis of the non-vectorial transformation behavior of these "forces".

In an inertial frame, let ${\displaystyle \mathbf {r} }$ be the position vector of a moving particle. Its Cartesian components (x, y) are:

$${\displaystyle \mathbf {r} =(r\cos \theta ,\ r\sin \theta )\ ,}$$

with polar coordinates r and θ depending on time t.

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

$${\displaystyle {\hat {\boldsymbol {r}}}={\frac {\partial \mathbf {r} }{\partial r}}=(\cos \theta ,\ \sin \theta )}$$

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

$${\displaystyle {\hat {\boldsymbol {\theta }}}={\frac {\partial ^{2}{\mathbf {r} }}{\partial r\,\partial \theta }}=(-\sin \theta \ ,\cos \theta )\ .}$$

These unit vectors vary in direction with time:

$${\displaystyle {\frac {d}{dt}}{\hat {\boldsymbol {r}}}=(-\sin \theta ,\ \cos \theta ){\frac {d\theta }{dt}}={\frac {d\theta }{dt}}{\hat {\boldsymbol {\theta }}},}$$

and:

$${\displaystyle {\frac {d}{dt}}{\hat {\boldsymbol {\theta }}}=(-\cos \theta ,\ -\sin \theta ){\frac {d\theta }{dt}}=-{\frac {d\theta }{dt}}{\hat {\boldsymbol {r}}}.}$$

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

$${\displaystyle {\boldsymbol {v}}={\frac {d\mathbf {r} }{dt}}={\dot {r}}{\hat {\boldsymbol {r}}}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}}}$$

and

$${\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\boldsymbol {r}}}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}\ ,}$$

where dot-overmarkings indicate time differentiation. With this form for the acceleration ${\displaystyle {\boldsymbol {a}}}$, in an inertial frame of reference Newton's second law expressed in polar coordinates is:

$${\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}=m({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\boldsymbol {r}}}+m(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}\ ,}$$

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:

$${\displaystyle {\boldsymbol {F}}+mr{\dot {\theta }}^{2}{\hat {\mathbf {r} }}-m2{\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}=m{\tilde {\boldsymbol {a}}}=m{\ddot {r}}{\hat {\boldsymbol {r}}}+mr{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}\ ,}$$

where a "coordinate" version of the "acceleration" is introduced:

$${\displaystyle {\tilde {\boldsymbol {a}}}={\ddot {r}}{\hat {\boldsymbol {r}}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}\ ,}$$

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.^[38] 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.

Figure 2: Two coordinate systems differing by a displacement of origin. Radial motion with constant velocity v in one frame is not radial in the other frame. Angular rate ${\displaystyle {\dot {\theta }}=0}$, but ${\displaystyle {\dot {\theta }}'\neq 0\ .}$

Change of origin

Figure 2 shows the "centrifugal term" ${\displaystyle r{\dot {\theta }}^{2}}$ does not transform as a true force. Suppose in frame S a particle moves radially away from the origin at a constant velocity. See Figure 2. The force on the particle is zero by Newton's first law. Now we look at the same thing from frame S' , which is the same, but displaced in origin. In S' the particle still is in straight line motion at constant speed, so again the force is zero.

What if one used polar coordinates in the two frames? In frame S the radial motion is constant and there is no angular motion. Hence, the acceleration is:

$${\displaystyle {\boldsymbol {a}}=\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\boldsymbol {r}}}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\boldsymbol {\theta }}}=0\ ,}$$

and each term individually is zero because ${\displaystyle {\dot {\theta }}=0,\ {\ddot {\theta }}=0}$ and ${\displaystyle {\ddot {r}}=0\ }$. There is no force, including no ${\displaystyle r{\dot {\theta }}^{2}}$ "force" in frame S. In frame S' , however, we have:

$${\displaystyle {\boldsymbol {a}}'=\left({\ddot {r}}'-r'{\dot {\theta }}'^{2}\right){\hat {\boldsymbol {r}}}'+\left(r'{\ddot {\theta }}'+2{\dot {r}}'{\dot {\theta }}'\right){\hat {\boldsymbol {\theta }}}'.\ }$$

In this case the azimuthal term is zero, being the rate of change of angular momentum. To obtain zero acceleration in the radial direction, however, we require:

$${\displaystyle {\ddot {r}}'=r'{\dot {\theta }}'^{2}\ .}$$

The right-hand side is non-zero, inasmuch as neither ${\displaystyle r'}$ nor ${\displaystyle {\dot {\theta }}'}$ is zero. That is, we cannot obtain zero force (zero ${\displaystyle {\boldsymbol {a}}'}$) if we retain only ${\displaystyle {\ddot {r}}'}$ as the acceleration; we need both terms.

Despite the above facts, suppose one were to adopt polar coordinates, and wish to say that ${\displaystyle r{\dot {\theta }}^{2}}$ is "centrifugal force", and reinterpret ${\displaystyle {\ddot {r}}}$ as "acceleration" (without dwelling upon any possible justification). How does this decision fare when one considers that a proper formulation of physics is geometry and coordinate-independent? See the article on general covariance.^[39] To attempt to form a covariant expression, this so-called centrifugal "force" can be put into vector notation as:

$${\displaystyle {\boldsymbol {F_{\dot {\theta }}}}=-{\boldsymbol {\omega \times }}\left({\boldsymbol {\omega \times r}}\right)\ ,}$$

with:

$${\displaystyle {\boldsymbol {\omega }}={\dot {\theta }}{\boldsymbol {\hat {k}}}\ ,}$$

and

$${\displaystyle {\boldsymbol {{\hat {k}},}}}$$

a unit vector normal to the plane of motion. Unfortunately, although this expression formally looks like a vector, when an observer changes origin the value of ${\displaystyle {\dot {\theta }}}$ changes (see Figure 2), so observers in the same frame of reference standing on different street corners see different "forces" even though the actual events they witness are identical.

How can a physical force (be it fictitious or real) be zero in one frame S, but non-zero in another frame S' identical, but a few feet away? Even for exactly the same particle behavior the expression ${\displaystyle r{\dot {\theta }}^{2}}$ is different in every frame of reference, even for very trivial distinctions between frames. In short, if we take ${\displaystyle r{\dot {\theta }}^{2}}$ as "centrifugal force", it does not have a universal significance: it is unphysical.

Beyond this problem, the real impressed net force is zero. (There is no real impressed force in straight-line motion at constant speed). If one were to adopt polar coordinates, and wish to say that ${\displaystyle r{\dot {\theta }}^{2}}$ is "centrifugal force", and reinterpret ${\displaystyle {\ddot {r}}}$ as "acceleration", the oddity results in frame S' that straight-line motion at constant speed requires a net force in polar coordinates, but not in Cartesian coordinates. Moreover, this perplexity applies in frame S', but not in frame S.

The behavior of ${\displaystyle r{\dot {\theta }}^{2}}$ indicates that one must say that ${\displaystyle r{\dot {\theta }}^{2}}$ is not centrifugal force, but simply one of two terms in the acceleration. This view, that the acceleration is composed of two terms, is frame-independent: there is zero centrifugal force in any and every inertial frame. It also is coordinate-system independent, which means that one can use Cartesian, polar, or any other curvilinear system because they all produce zero.

Apart from the above physical arguments, of course, the derivation above, based upon application of the mathematical rules of differentiation, shows the radial acceleration does indeed consist of the two terms ${\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}}$.

That said, the next subsection shows there is a connection between these centrifugal and Coriolis terms and the fictitious forces that pertain to a particular rotating frame of reference (as distinct from an inertial frame).

Figure 3: Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S' . The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S' at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle θ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

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).^[40] See Figure 3. 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, one can 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:

$${\displaystyle \theta '=\theta -\Omega t\ .}$$

Consequently,

$${\displaystyle {\dot {\theta }}'={\dot {\theta }}-\Omega \ .}$$

Plugging this result into the acceleration using the unit vectors of the previous section:

$${\displaystyle {\begin{aligned}{\frac {d^{2}\mathbf {r} }{dt^{2}}}&=\left[{\ddot {r}}-r\left({\dot {\theta }}'+\Omega \right)^{2}\right]{\hat {\mathbf {r} }}+\left[r{\ddot {\theta }}'+2{\dot {r}}\left({\dot {\theta }}'+\Omega \right)\right]{\hat {\boldsymbol {\theta }}}\\&=({\ddot {r}}-r{\dot {\theta }}'^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}'+2{\dot {r}}{\dot {\theta }}'){\hat {\boldsymbol {\theta }}}-\left(2r\Omega {\dot {\theta }}'+r\Omega ^{2}\right){\hat {\mathbf {r} }}+\left(2{\dot {r}}\Omega \right){\hat {\boldsymbol {\theta }}}\ .\end{aligned}}}$$

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:^[41]

$${\displaystyle -\left(2r\Omega {\dot {\theta }}'+r\Omega ^{2}\right){\hat {\mathbf {r} }}+\left(2{\dot {r}}\Omega \right){\hat {\boldsymbol {\theta }}}}$$

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 ${\displaystyle 2r\Omega {\dot {\theta }}'}$ is the radial component of the Coriolis force per unit mass, where ${\displaystyle r{\dot {\theta }}'}$ is the tangential component of the particle velocity as seen in the rotating frame. The term ${\displaystyle -\left(2{\dot {r}}\Omega \right){\hat {\boldsymbol {\theta }}}}$ 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.^[42]

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:

$${\displaystyle {\boldsymbol {F}}+mr{\dot {\theta }}'^{2}{\hat {\mathbf {r} }}-m2{\dot {r}}{\dot {\theta }}'{\hat {\boldsymbol {\theta }}}+m\left(2r\Omega {\dot {\theta }}'+r\Omega ^{2}\right){\hat {\mathbf {r} }}-m\left(2{\dot {r}}\Omega \right){\hat {\boldsymbol {\theta }}}=m{\ddot {r}}{\hat {\mathbf {r} }}+mr{\ddot {\theta }}'\ {\hat {\boldsymbol {\theta }}}=m{\tilde {{\boldsymbol {a}}.}}}$$

If one was to choose, for convenience, to treat ${\displaystyle {\tilde {\boldsymbol {a}}}}$ as "acceleration", then the terms ${\displaystyle (mr{\dot {\theta }}'^{2}{\hat {\mathbf {r} }}-m2{\dot {r}}{\dot {\theta }}'{\hat {\boldsymbol {\theta }}})}$ 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, the fictitious sources 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.

"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:

$${\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}=m\sum _{k=1}^{d}\left({\dot {v}}_{k}\ +\sum _{j=1}^{d}\sum _{i=1}^{d}v_{j}{\Gamma ^{k}}_{ij}{\dot {q}}_{i}\right){\boldsymbol {e_{k}}}\ ,}$$

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 rewritten, taking the second summation to the force-side of the equation as:

$${\displaystyle {\boldsymbol {F}}-m\sum _{j=1}^{d}\sum _{i=1}^{d}v_{j}{\Gamma ^{k}}_{ij}{\dot {q}}_{i}{\boldsymbol {e_{k}}}=m{\tilde {\boldsymbol {a}}}\ ,}$$

with the convention that the "acceleration" ${\displaystyle {\tilde {\boldsymbol {a}}}}$ is now:

$${\displaystyle {\tilde {\boldsymbol {a}}}=\sum _{k=1}^{d}{\dot {v}}_{k}{\boldsymbol {e_{k}}}\ .}$$

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. Moreover, these "forces" do not transform under coordinate transformations as vectors. Thus, the designation of the terms of the summation as "fictitious forces" uses this terminology for contributions that are completely different from any real force, and 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.^[50]

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.^[51] 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. From the standpoint of a Lagrangian formulation, they can be called generalized fictitious forces. See Hildebrand,^[22] for example.

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.^[35][52]