The equations of motion in curvilinear coordinates, tensor notation and Coriolis force

The formulation of the equations of motion of a particle is simple in Cartesian coordinates using vector notation. However, depending on the problem, for example when describing the motion of a particle as seen from a non-inertial system of references (e.g. a rotating planet, like earth), there is advantage in using curvilinear coordinates and also tensor notation. When the particle's movement is observed from such a rotating referential, we also see "acceleration" that is not due to any force but to the fact that the referential itself is accelerated. The material below discusses and formulates these topics, and derives the expression for the Coriolis and centripetal force in cylindrical coordinates as seen from a rotating system of references.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.681 or newer.

Vector notation

Generally speaking the equations of motion of a particle are easy to formulate: the position vector is a function of time, the velocity is its first derivative and the acceleration is its second derivative. For instance, in Cartesian coordinates

Newton's 2^nd law, that in an inertial system of references when there is force there is acceleration and viceversa, is then given by

where  represents the total force acting on the particle. This vectorial equation represents three second order differential equations which, for given initial conditions, can be integrated to arrive at a closed form expression for  as a function of .

Tensor notation

In Cartesian coordinates, the tensorial form of the equations (4) is also straightforward. In a flat spacetime - Galilean system of references - the three space coordinates  form a 3D tensor, and so does its first derivate and the second one. Set the spacetime to be 3-dimensional and Euclidean and use lowercaselatin indices for the corresponding tensors

The position, velocity and acceleration vectors are expressed in tensor notation as done in (1), (2) and (3)

Setting a tensor  to represent the three Cartesian components of the force

Newton's 2^nd law (4), now expressed in tensorial notation, is given by

The three differential equations behind this tensorial form of (4) are as expected

Things are straightforward in Cartesian coordinates because the components of the line element  are exactly the components of the tensor

and so, the form factors (see related Mapleprimes post) are all equal to 1.

In the case of no external forces,  and the equations of motion, whose solution are the trajectory, can be formulated as the path of minimal length between two points, a geodesic. In the case under consideration, because the spacetime is flat (Galilean) those two points lie on a plane, we are talking about Euclidean geometry, that information is encoded in the metric (the 3x3 identity matrix (5)), and the geodesic is a straight line. The differential equations of this geodesic are thus the equations of motion (11) with  , and can be computed using Geodesics

Curvilinear coordinates

Vector notation

The form of these equations in the case of curvilinear coordinates, for example in cylindrical or spherical variables, is obtained performing a change of coordinates.

This change keeps the z axis unchanged, so the corresponding unit vector  remains unchanged.

Changing the basis and coordinates used to represent the position vector , it becomes

where since in (1) the coordinates () depend on t, in (15), not just  and  but also the unit vector depends on t. The velocity is computed as usual, differentiating

The second term is due to the dependency of  on the coordinate  together with the chain rule . The dependency of curvilinear unit vectors on the coordinates is automatically taken into account when differentiating due to the Setup setting geometricdifferentiation = true.

For the acceleration,

where the term involving  comes from differentiating  in (16) taking into account the dependency of  on the coordinate  This result can also be obtained by directly changing variables in the acceleration , in equation (3)

Newton's 2^nd law becomes

In the absence of external forces, equating to 0 the vector components (coefficients of the unit vectors) of the acceleration we get the system of differential equations in cylindrical coordinates whose solution is the trajectory of the particle expressed in the (

In this formulation (21) with , although , no acceleration in the  direction, is naturally expected, the same cannot be said about the other two equations for  and . Those two equations are discussed below under Coriolis and Centripetal forces. The key observation at this point, however, is that the right-hand sides of both unexpected equations involve , rotation around the z axis.

Tensor notation

The same equations (19) and (21) result when using tensor notation. For that purpose, one can transform the position, velocity and acceleration tensors (6), (7), (8), but since they are expressed as functions of a parameter (the time), it is simpler to transform only the underlying metric using TransformCoordinates. That has the advantage that all the geometrical subtleties of curvilinear coordinates, like scale factors and dependency of unit vectors on curvilinear coordinates, get automatically, very succinctly, encoded in the metric:

The computation of geodesics assumes that the coordinates () depend on a parameter. That parameter is passed as the first argument to Geodesics

These equations of motion (23) are the same as the equations (21) computed using standard vector notation, differentiating and taking into account the dependency of curvilinear unit vectors on the curvilinear coordinates in (16) and (17).  One of the interesting features of computing with tensors is, as said, that all those geometrical algebraic subtleties of curvilinear coordinates are automatically encoded in the metric (22).

To understand how are the geodesic equations computed in one go in (23), one can perform the calculation in steps:

For step 1, we have

Set this metric where

Step 2, the geodesic equations in tensor notation with the coordinates depending on the time t are computed passing the optional argument tensornotation

Step 3: compute the components of this tensorial equation

These are the same equations (23).

Having the tensorial equation (26) is also useful to formulate the equations of motion in tensorial form in the presence of force. For that purpose, redefine the contravariant tensor  to represent the force in the cylindrical basis

Newton's 2^nd law (19)

now using tensorial notation, becomes

where we recall (see related Mapleprimes post) that to obtain the vector components entering  from these tensor components we need to multiply the latter by the scale factors (), the component of  in the direction of  is given by .

Coriolis force and centripetal force

After changing variables the position vector of the particle got expressed in (15) as

A distinction needs to be made here, according to whether the unit vector  depends or not on the time , the former being the general case. When  is a constant, the value of the coordinate  - the angle between  and the x axis - does not change, there is no rotation around the z axis. On the other hand, when , the orientation of  and so the coordinate  changes with time, so either the force acting on the particle has a component in the  direction that produces rotation around the z axis, or the system of references - itself - is rotating around the z axis.

Likewise, the expression (15)  can represent the position vector measured in the original Galilean (inertial) system of references, where a force is producing rotation around the z axis, or it can represent the position of the particle measured in a rotating, non-inertial system references. Hence the transformation (14) can also be interpreted in two different ways, as representing a choice of different functions (generalized coordinates) to represent the position of the particle in the original inertial system of references, or it can represent a transformation from an inertial to another rotating, non-inertial, system of references.

This equivalence between the trajectory of a particle subject to an external force, as observed in an inertial system of references, and the same trajectory observed from a non-inertial accelerated system of references where there is no external force, actually at the root of the formulation of general relativity, is also well known in classical mechanics. The (apparent) forces observed in the rotating non-inertial system of references, due only to its acceleration, are called Coriolis and centripetal forces.

To see that the equations

that appeared in (27) when in the inertial system of references , are related to the Coriolis and centripetal forces in the non-inertial referencial, following [1] introduce a vector representing the rotation of that referencial around the z axis (when, in the inertial system of references, the non-inertial system rotates clockwise, in the non-inertial system  increases in value in the anti-clockwise direction)

According to [1], (39.7), the acceleration in the inertial system is expressed in terms of the quantities in the non-inertial rotating system by the sum of the following three vectorial terms.

First, the components of the acceleration measured in the non-inertial system are given by the second derivatives of the coordinates () multiplied by the scale factors, which in this case are given by () (see this post in Mapleprimes)

Second, the Coriolis force divided by the mass, by definition given by

Third the centripetal force divided by the mass, defined by

Adding these three terms,

So that

and where the right-hand side of (36) is, actually, the result computed lines above in (18)

From (37), in the absence of external forces  and so the acceleration  measured in the rotating system is given by the sum of the Coriolis and centripetal accelerations

In other words: in the absence of external forces, the acceleration of a particle observed in a rotating (non-inertial) system of references is not zero.

Expressing this equation (40) in terms of () we get

resulting in the three equations

which are the equations returned by Geodesics in (23)

References

[1] L.D. Landau, E.M. Lifchitz, Mechanics, Course of Theoretical Physics, Volume 1, third edition, Elsevier.

Vectors in Spherical Coordinates using Tensor Notation

Edgardo S. Cheb-Terrab¹ and Pascal Szriftgiser²

(2) Laboratoire PhLAM, UMR CNRS 8523, Université de Lille, F-59655, France

(1) Maplesoft

The following is a topic that appears frequently in formulations: given a 3D vector in spherical (or any curvilinear) coordinates, how do you represent and relate, in simple terms, the vector and the corresponding vectorial operations Gradient, Divergence, Curl and Laplacian using tensor notation?

The core of the answer is in the relation between the - say physical - vector components and the more abstract tensor covariant and contravariant components. Focusing the case of a transformation from Cartesian to spherical coordinates, the presentation below starts establishing that relationship between 3D vector and tensor components in Sec.I. In Sec.II, we verify the transformation formulas for covariant and contravariant components on the computer using TransformCoordinates. In Sec.III, those tensor transformation formulas are used to derive the vectorial form of the Gradient in spherical coordinates. In Sec.IV, we switch to using full tensor notation, a curvilinear metric and covariant derivatives to derive the 3D vector analysis traditional formulas in spherical coordinates for the Divergence, Curl, Gradient and Laplacian. On the way, some useful technics, like changing variables in 3D vectorial expressions, differential operators, using Jacobians, and shortcut notations are shown.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.640 or newer.

Start setting the spacetime to be 3-dimensional, Euclidean, and use Cartesian coordinates