Curvilinear coordinates are a coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate system with the same number of coordinates in which the coordinate lines are curved. In the twodimensional case, instead of Cartesian coordinates x and y, e.g., p and q are used: the level curves of p and q in the xyplane. It is a requirement that the transformation is locally invertible (a onetoone map) at each point. This means that one can convert a point given in one coordinate system to its curvilinear coordinates and back.
Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R^{3} (e.g., motion in the field of a point mass/charge), is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Also boundary conditions may enforce symmetry. One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere.
Many of the concepts in vector calculus, which are given in Cartesian or spherical polar coordinates, can be formulated in arbitrary curvilinear coordinates. This gives a certain economy of thought, as it is possible to derive general expressions—valid for any curvilinear coordinate system—for concepts as gradient, divergence, curl, and the Laplacian. Wellknown examples of curvilinear systems are polar coordinates for R^{2}, and cylinder and spherical polar coordinates for R^{3}.
The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. While a Cartesian coordinate surface is a plane, e.g., z = 0 defines the xy plane, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere in R^{3}—which obviously is curved.
In Cartesian coordinates, the position of a point P(x,y,z) is determined by the intersection of three mutually perpendicular planes, x = const, y = const, z = const. The coordinates x, y and z are related to three new quantities q_{1},q_{2}, and q_{3} by the equations:
The above equation system can be solved for the arguments q_{1}, q_{2}, and q_{3} with solutions in the form:
The transformation functions are such that there's a onetoone relationship between points in the "old" and "new" coordinates, that is, those functions are bijections, and fulfil the following requirements within their domains:
is not zero; that is, the transformation is invertible according to the inverse function theorem. The condition that the Jacobian determinant is not zero reflects the fact that three surfaces from different families intersect in one and only one point and thus determine the position of this point in a unique way.^{[1]}
A given point may be described by specifying either x, y, z or q_{1}, q_{2}, q_{3} while each of the inverse equations describes a surface in the new coordinates and the intersection of three such surfaces locates the point in the threedimensional space (Fig. 1). The surfaces q_{1} = const, q_{2} = const, q_{3} = const are called the coordinate surfaces; the space curves formed by their intersection in pairs are called the coordinate lines. The coordinate axes are determined by the tangents to the coordinate lines at the intersection of three surfaces. They are not in general fixed directions in space, as is true for simple Cartesian coordinates. The quantities (q_{1}, q_{2}, q_{3} ) are the curvilinear coordinates of a point P(q_{1}, q_{2}, q_{3} ).
In general, (q_{1}, q_{2} ... q_{n} ) are curvilinear coordinates in ndimensional space.
From a more general and abstract perspective, a curvilinear coordinate system is simply a coordinate patch on the differential manifold E^{n} (ndimensional Euclidian space) that is diffeomorphic to the Cartesian coordinate patch on the manifold.^{[2]} Note that two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in differential topology.
Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as Earth sciences, cartography, and physics (quantum physics, relativity, etc.). The curvilinear coordinates (q_{1}, q_{2}, q_{3}) in this system are, respectively, r (radial distance or polar radius, r ≥ 0), θ (zenith or latitude, 0 ≤ θ ≤ 180°), and φ (azimuth or longitude, 0 ≤ φ ≤ 360°). The direct relationship between Cartesian and spherical coordinates is given by:
Solving the above equation system for r, θ, and φ gives the inverse relations between spherical and Cartesian coordinates:
The respective spherical coordinate surfaces are derived in terms of Cartesian coordinates by fixing the spherical coordinates in the above inverse transformations to a constant value. Thus (Fig.2), r = const are concentric spherical surfaces centered at the origin, O, of the Cartesian coordinates, θ = const are circular conical surfaces with apex in O and axis the Oz axis, φ = const are halfplanes bounded by the Oz axis and perpendicular to the xOy Cartesian coordinate plane. Each spherical coordinate line is formed at the pairwise intersection of the surfaces, corresponding to the other two coordinates: r lines (radial distance) are beams Or at the intersection of the cones θ = const and the halfplanes φ = const; θ lines (meridians) are semicircles formed by the intersection of the spheres r = const and the halfplanes φ = const ; and φ lines (parallels) are circles in planes parallel to xOy at the intersection of the spheres r = const and the cones θ = const. The location of a point P(r,θ,φ) is determined by the point of intersection of the three coordinate surfaces, or, alternatively, by the point of intersection of the three coordinate lines. The θ and φ axes in P(r,θ,φ) are the mutually perpendicular (orthogonal) tangents to the meridian and parallel of this point, while the r axis is directed along the radial distance and is orthogonal to both θ and φ axes.
The surfaces described by the inverse transformations are smooth functions within their defined domains. The Jacobian (functional determinant) of the inverse transformations is:
Coordinates are used to define location or distribution of physical quantities which are scalars, vectors, or tensors. Scalars are expressed as points and their location is defined by specifying their coordinates with the use of coordinate lines or coordinate surfaces. Vectors are objects that possess two characteristics: magnitude and direction.
To define a vector in terms of coordinates, an additional coordinateassociated structure, called basis, is needed. A basis in threedimensional space is a set of three linearly independent vectors {e_{1}, e_{2}, e_{3}}, called basis vectors. Each basis vector is associated with a coordinate in the respective dimension. Any vector can be represented as a sum of vectors A_{n}e_{n} formed by multiplication of a basis vector by a scalar coefficient, called component. Each vector, then, has exactly one component in each dimension and can be represented by the vector sum: A = A_{1}e_{1} + A_{2}e_{2} + A_{3}e_{3}, where A_{n} and e_{n} are the respective components and basis vectors. A requirement for the coordinate system and its basis is that A_{1}e_{1} + A_{2}e_{2} + A_{3}e_{3} ≠ 0 when at least one of the A_{n} ≠ 0. This condition is called linear independence. Linear independence implies that there cannot exist bases with basis vectors of zero magnitude because the latter will give zeromagnitude vectors when multiplied by any component. Noncoplanar vectors are linearly independent, and any triple of noncoplanar vectors can serve as a basis in three dimensions.
For the general curvilinear coordinates, basis vectors and components vary from point to point. If vector A whose origin is in point P (q_{1}, q_{2}, q_{3} ) is moved to point P' (q'_{1}, q'_{2}, q'_{3} ) in such a way that its direction and orientation are preserved, then the moved vector will be expressed by new components A'_{n} and basis vectors e^{'}_{n}. Therefore, the vector sum that describes vector A in the new location is composed of different vectors, although the sum itself remains the same. A coordinate basis whose basis vectors change their direction and/or magnitude from point to point is called local basis. All bases associated with curvilinear coordinates are necessarily local. Global bases, that is, bases composed of basis vectors that are the same in all points can be associated only with linear coordinates. A more exact, though seldom used, expression for such vector sums with local basis vectors is , where the dependence of both components and basis vector on location is made explicit (n is the number of dimensions). Local bases are composed of vectors with arbitrary order, magnitude, and direction and magnitude/direction vary in different points in space.
Basis vectors are usually associated with a coordinate system by two methods:
In the first case (axiscollinear), basis vectors transform like covariant vectors while in the second case (normal to coordinate surfaces), basis vectors transform like contravariant vectors. Those two types of basis vectors are distinguished by the position of their indices: covariant vectors are designated with lower indices while contravariant vectors are designated with upper indices. Thus, depending on the method by which they are built, for a general curvilinear coordinate system there are two sets of basis vectors for every point: {e_{1}, e_{2}, e_{3}} is the covariant basis, and {e^{1}, e^{2}, e^{3}} is the contravariant basis.
A key property of the vector and tensor representation in terms of indexed components and basis vectors is invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner), and these operations are inverse to one another according to the transformation rules. This means that in terms, in which an index occurs two times, one of the indices in the pair must be upper and the other index must be lower. Thus in the above vector sums, basis vectors with lower indices are multiplied by components with upper indices or vice versa, so that a given vector can be represented in two ways: A = A^{1}e_{1} + A^{2}e_{2} + A^{3}e_{3} = A_{1}e^{1} + A_{2}e^{2} + A_{3}e^{3}. Upon coordinate change, a vector transforms in the same way as its components. Therefore, a vector is covariant or contravariant if, respectively, its components are covariant or contravariant. From the above vector sums, it can be seen that contravariant vectors are represented with covariant basis vectors, and covariant vectors are represented with contravariant basis vectors. This is reflected in the Einstein summation convention according to which in the vector sums and the basis vectors and the summation symbols are omitted, leaving only A^{i} and A_{i} which represent, respectively, a contravariant and a covariant vector.
As stated above, contravariant vectors are vectors with contravariant components whose location is determined using covariant basis vectors that are built along the coordinate axes. In analogy to the other coordinate elements, transformation of the covariant basis of general curvilinear coordinates is described starting from the Cartesian coordinate system whose basis is called standard basis. The standard basis is a global basis that is composed of 3 mutually orthogonal vectors {i, j, k} of unit length, that is, the magnitude of each basis vector equals 1. Regardless of the method of building the basis (axiscollinear or normal to coordinate surfaces), in the Cartesian system the result is a single set of basis vectors, namely, the standard basis. To avoid misunderstanding, in this section the standard basis will be thought of as built along the coordinate axes.
Consider the onedimensional curve shown in Fig. 3. At point P, taken as an origin, x is one of the Cartesian coordinates, and q_{1} is one of the curvilinear coordinates (Fig. 3). The local basis vector is e_{1} and it is built on the q_{1} axis which is a tangent to q_{1} coordinate line at the point P. The axis q_{1} and thus the vector e_{1} form an angle α with the Cartesian x axis and the Cartesian basis vector i.
It can be seen from triangle PAB that where e_{1} is the magnitude of the basis vector e_{1} (the scalar intercept PB) and i is the magnitude of the Cartesian basis vector i which is also the projection of e_{1} on the x axis (the scalar intercept PA). It follows, then, that
However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reason: By increasing the distance from P, the angle between the curved line q_{1} and Cartesian axis x increasingly deviates from α. At the distance PB the true angle is that which the tangent at point C forms with the x axis and the latter angle is clearly different from α. The angles that the q_{1} line and q_{1} axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P. Let point E is located very close to P, so close that the distance PE is infinitesimally small. Then PE measured on the q_{1} axis almost coincides with PE measured on the q_{1} line. At the same time, the ratio (PD being the projection of PE on the x axis) becomes almost exactly equal to cos α.
Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and dq_{1}. Then
Thus, the directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts.
If and are smooth (continuously differentiable) functions and, therefore, the transformation ratios can be written as
that is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to the other system.
From the foregoing discussion, it follows that the component (projection) of e_{1} on the x axis is
Therefore the projection of the normalized local basis vector (e_{1} = 1) can be made a vector directed along the x axis by multiplying it with the standard basis vector i.
Doing the same for the coordinates in the other 2 dimensions, e_{1} can be expressed as: . Similar equations hold for e_{2} and e_{3} so that the standard basis {i, j, k} is transformed to local (ordered and normalised) basis {e_{1}, e_{2}, e_{3}} by the following system of equations:
Vectors e_{1}, e_{2}, and e_{3} at the right hand side of the above equation system are unit vectors (magnitude = 1) directed along the 3 axes of the curvilinear coordinate system. However, basis vectors in general curvilinear system are not required to be of unit length: they can be of arbitrary magnitude and direction. It can easily be shown that the condition e_{1} = e_{2} = e_{3} = 1 is a result of the above transformation, and not an a priori requirement imposed on the curvilinear basis. Let the local basis {e_{1}, e_{2}, e_{3}} not be normalised, in effect, leaving the basis vectors with arbitrary magnitudes. Then, instead of e_{1}, e_{2}, and e_{3} in the right hand side, there will be , , and which are again unit vectors directed along the curvilinear coordinate axes.
By analogous reasoning, but this time projecting the standard basis on the curvilinear axes ( i = j = k = 1 according to the definition of standard basis), one can obtain the inverse transformation from local basis to standard basis:
The above systems of linear equations can be written in matrix form as and where x_{i} (i = 1,2,3) are the Cartesian coordinates x, y, z and i_{i} are the standard basis vectors i, j, k. The system matrices (that is, matrices composed of the coefficients in front of the unknowns) are, respectively, and . At the same time, those two matrices are the Jacobian matrices J_{ik} and J^{−1}_{ik} of the transformations of basis vectors from curvilinear to Cartesian coordinates and vice versa. In the second equation system (the inverse transformation), the unknowns are the curvilinear basis vectors which are subject to the condition that in each point of the curvilinear coordinate system there must exist one and only one set of basis vectors. This condition is satisfied iff (if and only if) the equation system has a single solution.
From linear algebra, it is known that a linear equation system has a single solution only if the determinant of its system matrix is nonzero. For the second equation system, the determinant of the system matrix is which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.
Another, very important, feature of the above transformations is the nature of the derivatives: in front of the Cartesian basis vectors stand derivatives of Cartesian coordinates while in front of the curvilinear basis vectors stand derivatives of curvililear coordinates. In general, the following definition holds:
Covariant vector is an object that in the system of coordinates x is defined by n ordered numbers or functions (components) a_{i}(x^{1}, x^{2}, x^{3}) and in system q it is defined by n ordered components ā_{i}(q^{1}, q^{2}, q^{3}) which are connected with a_{i} (x^{1}, x^{2}, x^{3}) in each point of space by the transformation: .
– Mnemonic: Coordinates covary with the vector.
This definition is so general that it applies to covariance in the very abstract sense, and includes not only basis vectors, but also all vectors, components, tensors, pseudovectors, and pseudotensors (in the last two there is an additional sign flip). It also serves to define tensors in one of their most usual treatments.
The partial derivative coefficients through which vector transformation is achieved are called also scale factors or Lamé coefficients (named after Gabriel Lamé): . However, the h_{ik} designation is very rarely used, being largely replaced with √g_{ik}, the components of the metric tensor.
Let be an arbitrary basis for threedimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector can be expressed as
The components v^{k} are the contravariant components of the vector .
The reciprocal basis is defined by the relation
where is the Kronecker delta.
The vector can also be expressed in terms of the reciprocal basis:
The components v_{k} are the covariant components of the vector .
From these definitions we can see that
Also,
The quantities g_{ij},g^{ ij} are defined as
From the above equations we have
The identity map defined by can be shown to be
The scalar product of two vectors in curvilinear coordinates is
A secondorder tensor can be expressed as
The components are called the contravariant components, the mixed rightcovariant components, the mixed leftcovariant components, and the covariant components of the secondorder tensor.
The components of the secondorder tensor are related by
The action can be expressed in curvilinear coordinates as
The inner product of two secondorder tensors can be expressed in curvilinear coordinates as
Alternatively,
If is a secondorder tensor, then the determinant is defined by the relation
where are arbitrary vectors and
Let the position of a point in space be characterized by three coordinate variables (ξ^{1},ξ^{2},ξ^{3}). The coordinate curve ξ^{1} represents a surface on which ξ^{2},ξ^{3} are constant. Let be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write ^{[3]}
The fields are called the curvilinear coordinate functions of the curvilinear coordinate system .
The ξ^{i} coordinate curves are defined by the oneparameter family of functions given by
with ξ^{j},ξ^{k} fixed.
The tangent vector to the curve at the point (or to the coordinate curve ξ_{i} at the point ) is
Let be a scalar field in space. Then
The gradient of the field f is defined by
where is an arbitrary constant vector. If we define the components c^{i} of vector such that
then
If we set , then since , we have
which provides a means of extracting the contravariant component of a vector .
If is the covariant (or natural) basis at a point, and if is the contravariant (or reciprocal) basis at that point, then
A brief rationale for this choice of basis is given in the next section.
A similar process can be used to arrive at the gradient of a vector field . The gradient is given by
If we consider the gradient of the position vector field , then we can show that
The vector field is tangent to the ξ^{i} coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, . All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point .
Since is arbitrary, we can write
Note that the contravariant basis vector is perpendicular to the surface of constant ψ^{i} and is given by
The Christoffel symbols of the second kind is defined as
This implies that
Other relations that follow are
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
The vector field can be represented as
where are the covariant components of the field, are the physical components, and
is the normalized contravariant basis vector.
The divergence of a vector field ()is defined as
In terms of components with respect to a curvilinear basis
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
Now,
Noting that, due to the symmetry of ,
we have
Recall that if [g_{ij}] is the matrix whose components are g_{ij}, then the inverse of the matrix is [g_{ij}] ^{− 1} = [g^{ij}]. The inverse of the matrix is given by
where A^{ij} are the cofactor matrices of the components g_{ij}. From matrix algebra we have
Hence,
Plugging this relation into the expression for the divergence gives
A little manipulation leads to the more compact form
The Laplacian of a scalar field is defined as
Using the alternative expression for the divergence of a vector field gives us
Now
Therefore,
The gradient of a second order tensor field can similarly be expressed as
If we consider the expression for the tensor in terms of a contravariant basis, then
We may also write
The physical components of a secondorder tensor field can be obtained by using a normalized contravariant basis, i.e.,
where the hatted basis vectors have been normalized. This implies that
The divergence of a secondorder tensor field is defined using
where is an arbitrary constant vector.
In curvilinear coordinates,
Let () be the usual Cartesian basis vectors for the Euclidean space of interest and let
where is a secondorder transformation tensor that maps to . Then,
From this relation we can show that
Let be the Jacobian of the transformation. Then, from the definition of the determinant,
Since
we have
A number of interesting results can be derived using the above relations.
First, consider
Then
Similarly, we can show that
Therefore, using the fact that [g^{ij}] = [g_{ij}] ^{− 1},
Another interesting relation is derived below. Recall that
where A is a, yet undetermined, constant. Then
This observation leads to the relations
In index notation,
where is the usual permutation symbol.
We have not identified an explicit expression for the transformation tensor because an alternative form of the mapping between curvilinear and Cartesian bases is more useful. Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have
Similarly,
From these results we have
and
The cross product of two vectors is given by
where ε_{ijk} is the permutation symbol and is a Cartesian basis vector. Therefore,
and
Hence,
Returning back to the vector product and using the relations
gives us
In an orthonormal righthanded basis, the thirdorder alternating tensor is defined as
In a general curvilinear basis the same tensor may be expressed as
It can be shown that
Now,
Hence,
Similarly, we can show that
For cylindrical coordinates we have
and
where
Then the covariant and contravariant basis vectors are
where are the unit vectors in the r,θ,z directions.
Note that the components of the metric tensor are such that
which shows that the basis is orthogonal.
The nonzero components of the Christoffel symbol of the second kind are
The normalized contravariant basis vectors in cylindrical polar coordinates are
and the physical components of a vector are
The gradient of a scalar field, , in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form
Similarly, the gradient of a vector field, , in cylindrical coordinates can be shown to be
Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be
The physical components of a secondorder tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are
Using the above definitions we can show that the gradient of a secondorder tensor field in cylindrical polar coordinates can be expressed as
The divergence of a secondorder tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,
Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,
where are covariant basis vectors, are contravariant basis vectors. Also, let () be a background, fixed, Cartesian basis.
Let be the position vector of the point with respect to the origin of the coordinate system. The notation can be simplified by noting that . At each point we can construct a small line element . The square of the length of the line element is the scalar product and is called the metric of the space. Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,
Using the chain rule, we can then express in terms of threedimensional orthogonal curvilinear coordinates (ξ^{1},ξ^{2},ξ^{3}) as
Therefore the metric is given by
The symmetric quantity
is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.
Note also that
where are the Lamé coefficients.
If we define the scale factors, , using
we get a relation between the fundamental tensor and the Lamé coefficients.
If we consider polar coordinates for R^{2}, note that
(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.
The orthogonal basis vectors are g_{r} = (cos θ, sin θ), g_{θ} = (−r sin θ, r cos θ). The normalized basis vectors are e_{r} = (cos θ, sin θ), e_{θ} = (−sin θ, cos θ) and the scale factors are h_{r} = 1 and h_{θ}= r. The fundamental tensor is g_{11} =1, g_{22} =r^{2}, g_{12} = g_{21} =0.
If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for ndimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.
Normally in the calculation of line integrals we are interested in calculating
where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term
by the chain rule. And from the definition of the Lamé coefficients,
and thus
Now, since when , we have
and we can proceed normally.
Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
Again, in curvilinear coordinates, we have
and we make use of the definition of curvilinear coordinates again to yield
Therefore,
where is the permutation symbol.
In determinant form, the cross product in terms of curvilinear coordinates will be:
In orthogonal curvilinear coordinates of 3 dimensions, where
one can express the gradient of a scalar or vector field as
For an orthogonal basis
The divergence of a vector field can then be written as
Also,
Therefore,
We can get an expression for the Laplacian in a similar manner by noting that
Then we have
The expressions for the gradient, divergence, and Laplacian can be directly extended to ndimensions.
The curl of a vector field is given by
where Ω is the product of all h_{i} and ε_{ijk} is the LeviCivita symbol.
An inertial coordinate system is defined as a system of space and time coordinates x_{1},x_{2},x_{3},t in terms of which the equations of motion of a particle free of external forces are simply d^{2}x_{j}/dt^{2} = 0.^{[4]} In this context, a coordinate system can fail to be “inertial” either due to nonstraight time axis or nonstraight space axes (or both). In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any noninertial coordinate system (in this sense), extra terms appear, called Christoffel symbols. Strictly speaking, these terms represent components of the absolute acceleration (in classical mechanics), but we may also choose to continue to regard d^{2}x_{j}/dt^{2} as the acceleration (as if the coordinates were inertial) and treat the extra terms as if they were forces, in which case they are called fictitious forces.^{[5]} The component of any such fictitious force normal to the path of the particle and in the plane of the path’s curvature is then called centrifugal force.^{[6]}
This more general context makes clear the correspondence between the concepts of centrifugal force in rotating coordinate systems and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.^{[7]}^{[8]}^{[9]}) For a simple example, consider a particle of mass m moving in a circle of radius r with angular speed w relative to a system of polar coordinates rotating with angular speed W. The radial equation of motion is mr” = F_{r} + mr(w+W)^{2}. Thus the centrifugal force is mr times the square of the absolute rotational speed A = w + W of the particle. If we choose a coordinate system rotating at the speed of the particle, then W = A and w = 0, in which case the centrifugal force is mrA^{2}, whereas if we choose a stationary coordinate system we have W = 0 and w = A, in which case the centrifugal force is again mrA^{2}. The reason for this equality of results is that in both cases the basis vectors at the particle’s location are changing in time in exactly the same way. Hence these are really just two different ways of describing exactly the same thing, one description being in terms of rotating coordinates and the other being in terms of stationary curvilinear coordinates, both of which are noninertial according to the more abstract meaning of that term.
When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.
