Systems of Reference in Classical Dynamics

In terms of any given system of rectilinear inertial coordinates, the classical equations of motion for a particle of mass m are given by Newton’s law

where X and F are column vectors and dots signify derivatives with respect to time. Letting x and f denote the force and position vectors expressed in terms of a rotating system of spatially rectilinear coordinates (with the same spatial origin), it follows that x and f are related to X and F by

where M is a time-dependent rotation matrix. Inserting these expressions into equation (1) and multiplying through by M^-1, we get

Thus, multiplying out the right hand side, we have the equation of motion in terms of the rotating coordinates

The second term on the right side is called the Coriolis acceleration, and the third term represents the combination of the centripetal acceleration and the Euler acceleration. (The Euler component vanishes if the rotation rate of the coordinate system is constant.) To illustrate with a simple example, consider the case of a rectilinear coordinate system rotating about the “z” axis of a Cartesian inertial coordinate system. The rotation matrix M and its inverse are of the form

The derivatives of M are

Inserting these expressions into equation (2), we get

The main diagonal components of the right hand matrix represent the centripetal acceleration, and the off-diagonal components represent the Euler acceleration. Letting x,y denote the components of the vector x, and letting f_x,f_y denote the components of f, this can be written in component form as

All the accelerations and forces in this example are in the xy plane, because we’ve assumed the particle has no motion in the z direction, which is also the axis of rotation. These equations show that the Coriolis acceleration, comprised of the second terms, is perpendicular to the velocity of the particle, with magnitude 2mωr where r is the distance from the origin. Also, the centripetal acceleration, comprised of the third terms, points directly toward the origin (i.e., the axis of rotation) with magnitude mω²r. Lastly, the Euler acceleration, comprised of the fourth terms, is perpendicular to the radial direction, and has magnitude mr(dω/dt). This is illustrated for a typical case in the figure below.

For an alternative derivation, we can make use of the usual vector operations to express the total second derivative of the absolute position vector r. Here the expression “absolute position vector” refers to the position vector with respect to any system of inertial coordinates. Now, the absolute velocity of a particle at any given space coordinates can be expressed as the sum of two parts: (1) the relative velocity of the particle with respect to a given coordinate system, and (2) the absolute velocity of the given coordinate system at those coordinates. Noting that the absolute velocity of the coordinate point r, given that the coordinate system is rotating with angular speed ω, is v = ω x r, so the total derivative of r with respect to time can be written as

where partial derivative denotes that it is evaluated with respect to the rotating coordinate system. The total derivative is itself a vector, and the same formula applies to the differentiation of any vector, so we can immediately evaluate the second total derivative as

Noting that the cross product is distributive, and that the chain rule applies to derivatives of cross products, this gives

Again, all partial derivatives denote differentiation with respect to the arbitrary coordinates. Multiplying through by the mass m, and equating the result to the force vector, the equation of motion is

The terms of this expression correspond exactly to the terms of equation (4), as can be seen by considering the case in which ω coincides with the z axis and the particle motion is confined to the xy plane. As before, we see that the Coriolis term is perpendicular to the relative velocity vector, the centripetal term points in the radial direction, and the Euler term is perpendicular to the radial direction. The magnitudes of these vectors also agree with the expressions derived previously.

Of course, we need not think in terms of a rotating coordinate system in order to derive these extra acceleration terms. Often we find these same terms derived in the context of stationary polar coordinates. Consider the standard three-dimensional polar coordinates r,θ,ϕ, where r is the radial coordinate, θ is the “latitude” (with θ = 0 at the North pole), and ϕ is the longitude. Letting u_r, u_θ, and u_ϕ denote unit vectors in the three coordinate directions respectively. It follows that the position and velocity vectors of a particle are

The acceleration of the particle in terms of these stationary polar coordinates is

Now, if all the particles of interest are moving on or near a sphere (such as the Earth) that is rotating about the North-South polar axis, we can split up their motions into two parts, one due to the rotation of the sphere, and another due to their individual motions relative to the sphere. Letting Ω denote the angular speed of the sphere, we can express the longitudinal derivative of each particle as

where ψ is the longitude relative to lines drawn on the sphere. Inserting this into the expression for the absolute acceleration, we get

The angular velocity vector of the sphere and the velocity vector of a particle relative to the sphere are

so we have

Similarly we have

Therefore the total acceleration of a particle can be written in the form

This shows that the familiar expressions for the Euler, centripetal, and Coriolis accelerations are consistent with the stationary reference frame coordinates, provided we account fully for the motions of the particles. These “extra” terms are just the result of partitioning the angular speed (about a certain axis) into two parts, and segregating the terms involving one of those parts.

The first derivation presented above was restricted to rectilinear coordinate systems, so that the position coordinates X would be related to the rotating position coordinates x by a linear transformation. This is why we excluded spatially curvilinear coordinate systems. This restriction can be removed by working with the velocity vector V = dX/dt instead of the position vector X, since the velocity transformation is linear for arbitrary curvilinear coordinate systems. In what follows, all vectors and matrices include time as well as space components. We re-write (1) as

where V is the velocity column vector of a particle of mass m in terms of coordinates X, and by evaluating the total derivatives of these coordinates as functions of the arbitrary coordinates x, we get the linear relation

where C is a 4x4 square matrix whose components are the partial derivatives of X^j with respect to x^k. Inserting these expressions into (5), evaluating the derivative, and multiplying through by the inverse of C, we get

The second term on the right side consists of the Christoffel symbols, as discussed in the note on Curved Coordinates and Fictitious Forces.

The terms “fictitious force” and “inertial force” are sometimes conflated, but properly speaking they refer to two very distinct cases. An inertial force on an object is evaluated relative to the momentarily co-moving inertial coordinate system, so it represents the absolute acceleration of the object. In contrast, a fictitious force is evaluated relative to an arbitrarily selected system of reference, which may itself be accelerating, and hence the fictitious force on an object is not an inherent attribute of the object’s absolute state of motion. A given object in a given state of absolute motion has unique inertial forces, but it is subject to infinitely many different fictitious forces, corresponding to all possible systems of reference. It’s also worth noting that, in the context of these definitions, there is no such thing as a Coriolis inertial force, because by definition the inertial force on a body is evaluated in terms of a system of reference in which it is momentarily at rest. As a result, there are just two components of imaginary force, the normal and tangential components in the osculating plane. These correspond to the centrifugal and Euler forces respectively.

It’s also worth noting that much of the literature on dynamics does a poor job at defining fictitious forces and inertia forces, due to the fundamental failure to clearly define the basic concepts of general space-time coordinate systems and frames, and in particular inertial coordinate systems and frames. Most introductory texts tacitly assume rectilinear spatial coordinate systems (without, of course, even acknowledging the epistemological issues involved in determining things like “straightness”), and then gloss over the distinction between coordinate systems and frames of reference, and then make assertions such as “fictitious forces do not appear in inertial frames”. Such statements are hopelessly ambiguous and misguided, for many reasons. First, it hardly needs to be said that fictitious forces do not “appear in” any frame of reference. They are actually components of the absolute acceleration, which we may (if we choose) move to the “force side” of the equation of motion and call them forces. Second, we can also move some or all of the absolute acceleration over to the force side of the equations, regardless of whether the system of reference is inertial or not. Third, the fundamental definition of an inertial coordinate system (in classical mechanics) is one in terms of which the physical forces applied to a particle equal the second time derivatives of the space coordinates of the particle. Now, a frame is an equivalence class of mutually stationary coordinate systems, but as such it contains many difference spatial coordinates, such as Cartesian, polar, cylindrical, and so on. Force may equal the second time derivative in terms of some of these coordinate systems but not in terms of others, so in order to give meaning to the assertion about no fictitious forces in inertial frames we must stipulate that the “extra” terms in the expression for the absolute acceleration arising from curved space axes will not be brought over to the force side of the equation. This is a purely arbitrary stipulation, and we could just as well say the same about the terms arising due to curved time axes.

The figures below illustrate two circumstances in which inertial paths do not coincide with linear loci in terms of coordinate systems with curved axes (relative to inertially straight lines). In each figure, the coordinates X,Y,T are inertial. The figure on the left depicts an accelerating coordinate system x,y,t, as shown by the fact that its time axis is curved with respect to the inertial time axis. The figure on the right depicts a system x,y,t with a curvilinear spatial (y) coordinate axis.

In each case the blue vector represents the inertial trajectory of a free particle. The deviation of this inertial trajectory from the coordinate axis to which it is initially tangent represent a pseudo-acceleration, which (multiplied by the mass and negated) might be treated as a pseudo-force in the equation of motion expressed in terms of these coordinates. This illustrates why extra terms appear in the expression for the absolute acceleration whenever the space and/or time axes are not inertially straight. Thus neither of the coordinate systems depicted above can properly be called an inertial coordinate system. However, the right hand coordinate system is stationary with respect to an inertial coordinate system, and hence it is by definition a representative of an inertial frame. The concepts of “stationary” and “frame” treat the time axis differently than the space axes. Of course, the classical concept of a “frame” also entails the assumption of absolute simultaneity, which is found to be of questionable value in the context of special relativity, but even aside from this, the decision to speak in terms of “frames” represents a decision to treat any extra terms (in the expression for absolute acceleration) arising from curved space axes as acceleration terms, rather than moving them to the force side of the equation of motion and treating them as fictitious forces. Needless to say, we can also choose to treat the extra terms arising from curved time axes (i.e., acceleration) in exactly the same way, in which case we have no fictitious forces at all. The affine connection of Newtonian spacetime is perfectly adequate to unambiguously define absolute acceleration in terms of any coordinate system, regardless of whether the space and/or time axes are curved.

In view of this, it’s clear that statements such as “fictitious forces arise only in non-inertial reference frames” are specious – unless they are interpreted as arbitrary restrictions on the definition of “fictitious force”. Unfortunately, introductory texts on classical mechanics are rarely clear about the foundational aspects of the subject. They also tend to adopt the language of “presentism” when discussing space and frames as three-dimensional concepts, despite the fact that they immediately define inertial frames as those in which certain laws of motion are valid. Since motion involves time, a frame must be a four-dimensional concept. The failure to grasp this is largely responsible for the difficulty that many people have when they go on to consider special relativity. We might say that the main obstacle to understanding the mechanics of special relativity is never having properly understood the significance of system of reference in classical mechanics.

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