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Physics LibreTexts

19.4: Appendix - Orthogonal Coordinate Systems

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The methods of vector analysis provide a convenient representation of physical laws. However, the manipulation of scalar and vector fields is greatly facilitated by use of components with respect to an orthogonal coordinate system such as the following.

Cartesian coordinates (x,y,z)

Cartesian coordinates (rectangular) provide the simplest orthogonal rectangular coordinate system. The unit vectors specifying the direction along the three orthogonal axes are taken to be (ˆi,ˆj,ˆk). In cartesian coordinates scalar and vector functions are written as

ϕ=ϕ(x,y,z)

r=xˆi+yˆj+zˆk

Calculation of the time derivatives of the position vector is especially simple using cartesian coordinates because the unit vectors (ˆi,ˆj,ˆk) are constant and independent in time. That is;

dˆidt=dˆjdt=dˆkdt=0

Since the time derivatives of the unit vectors are all zero then the velocity ˙r=drdt reduces to the partial time derivatives of x, y, and z. That is,

˙r=˙xˆi+˙yˆj+˙zˆk

Similarly the acceleration is given by

¨r=¨xˆi+¨yˆj+¨zˆk

Curvilinear coordinate systems

There are many examples in physics where the symmetry of the problem makes it more convenient to solve motion at a point P(x,y,z) using non-cartesian curvilinear coordinate systems. For example, problems having spherical symmetry are most conveniently handled using a spherical coordinate system (r,θ,ϕ) with the origin at the center of spherical symmetry. Such problems occur frequently in electrostatics and gravitation; e.g. solutions of the atom, or planetary systems. Note that a cartesian coordinate system still is required to define the origin plus the polar and azimuthal angles θ,ϕ. Using spherical coordinates for a spherically symmetry system allows the problem to be factored into a cyclic angular part, the solution which involves spherical harmonics that are common to all such spherically-symmetric problems, plus a one-dimensional radial part that contains the specifics of the particular spherically-symmetric potential. Similarly, for problems involving cylindrical symmetry, it is much more convenient to use a cylindrical coordinate system (ρ,ϕ,z). Again it is necessary to use a cartesian coordinate system to define the origin and angle ϕ. Motion in a plane can be handled using two dimensional polar coordinates.

Curvilinear coordinate systems introduce a complication in that the unit vectors are time dependent in contrast to cartesian coordinate system where the unit vectors (ˆi,ˆj,ˆk) are independent and constant in time. The introduction of this time dependence warrants further discussion.

Each of the three axes qi in curvilinear coordinate systems can be expressed in cartesian coordinates (x,y,z) as surfaces of constant qi given by the function

qi=fi(x,y,z)

where i=1, 2, or 3. An element of length dsi perpendicular to the surface qi is the distance between the surfaces qi and qi+dqi which can be expressed as

dsi=hidqi

where hi is a function of (q1,q2,q3). In cartesian coordinates h1, h2, and h3 are all unity. The unit-length vectors ˆq1, ˆq2, ˆq3, are perpendicular to the respective q1, q2, q3 surfaces, and are oriented to have increasing indices such that ˆq1׈q2=ˆq3. The correspondence of the curvilinear coordinates, unit vectors, and transform coefficients to cartesian, polar, cylindrical and spherical coordinates is given in Table 19.4.1.

Curvilinear q1 q2 q3 ˆq1 ˆq2 ˆq3 h1 h2 h3
Cartesian x y z ˆi ˆj ˆk 1 1 1
Polar r θ ˆr ˆθ 1 r
Cylindrical ρ φ z ˆρ ˆφ ˆz 1 ρ 1
Spherical r θ φ ˆr ˆθ ˆφ 1 r rsinθ
Table 19.4.1: Curvilinear coordinates

The differential distance and volume elements are given by

ds=ds1ˆq1+ds2ˆq2+ds3ˆq3=h1dq1ˆq1+h2dq2ˆq2+h3dq3ˆq3

dτ=ds1ds2ds3=h1h2h3(dq1dq2dq3)

These are evaluated below for polar, cylindrical, and spherical coordinates.

Two-dimensional polar coordinates (r,θ)

The complication and implications of time-dependent unit vectors are best illustrated by considering twodimensional polar coordinates which is the simplest curvilinear coordinate system. Polar coordinates are a special case of cylindrical coordinates, when z is held fixed, or a special case of spherical coordinate system, when ϕ is held fixed.

Consider the motion of a point P as it moves along a curve s(t) such that in the time interval dt it moves from P(1) to P(2) as shown in Figure 19.4.1. The two-dimensional polar coordinates have unit vectors ˆr,ˆθ, which are orthogonal and change from ˆr1,ˆθ1, to ˆr2,ˆθ2, in the time dt. Note that for these polar coordinates the angle unit vector ˆθ is taken to be tangential to the rotation since this is the direction of motion of a point on the circumference at radius r.

The net changes shown in figure of Table 19.4.2 are

dˆr=ˆr2ˆr1=dˆr=|ˆr|dθˆθ=dθˆθ

since the unit vector ˆr is a constant with |ˆr|=1. Note that the infinitessimal dˆr is perpendicular to the unit vector ˆr, that is, dˆr points in the tangential direction ˆθ.

Similarly, the infinitessimal

dˆθ=ˆθ2ˆθ1=dˆθ=dθˆr

which is perpendicular to the tangential ˆθ unit vector and therefore points in the direction ˆr. The minus sign causes dθˆr to be directed in the opposite direction to ˆr.

The net distance element ds is given by

ds=drˆr+rdˆr=drˆr+rdθˆθ

This agrees with the prediction obtained using Table 19.4.1.

The time derivatives of the unit vectors are given by equations ??? and ??? to be,

dˆrdt=dθdtˆθ

dˆθdt=dθdtˆr

Note that the time derivatives of unit vectors are perpendicular to the corresponding unit vector, and the unit vectors are coupled.

Consider that the velocity v is expressed as

v=drdt=ddt(rˆr)=drdtˆr+rdˆrdt=˙rˆr+r˙θˆθ

The velocity is resolved into a radial component ˙r and an angular, transverse, component r˙θ.

Similarly the acceleration is given by

a=dvdt=d˙rdtˆr+˙rdˆrdt+drdt˙θˆθ+rd˙θdtˆθ+r˙θdˆθdt=(¨rr˙θ2)ˆr+(¨rθ+2˙r˙θ)ˆθ

where the r˙θ2ˆr term is the effective centripetal acceleration while the 2˙r˙θˆθ term is called the Coriolis term. For the case when ˙r=¨r=0, then the first bracket in ??? is the centripetal acceleration while the second bracket is the tangential acceleration.

This discussion has shown that in contrast to the time independence of the cartesian unit basis vectors, the unit basis vectors for curvilinear coordinates are time dependent which leads to components of the velocity and acceleration involving coupled coordinates.

Coordinates r,θ
Distance element ds=drˆr+rdθˆθ
Area element da=rdrdθ
Unit vectors

ˆr=ˆicosθ+ˆjsinθ

ˆθ=ˆisinθ+ˆjcosθ

Time derivatives of unit vectors

dˆrdt=˙θˆθ

dˆθdt=˙θˆr

Velocity v=˙rˆr+r˙θˆθ
Kinetic energy m2(˙r2+r2˙θ2)
Acceleration a=(¨rr˙θ2)ˆr+(r¨θ+2˙r˙θ)ˆθ
Table 19.4.2: Differential relations plus a diagram of the unit vectors for 2-dimensional polar coordinates.
c1.PNG
Figure 19.4.1: Diagram for Table 19.4.2.

Cylindrical Coordinates (ρ,ϕ,z)

The three-dimensional cylindrical coordinates (ρ,ϕ,z) are obtained by adding the motion along the symmetry axis ˆz to the case for polar coordinates. The unit basis vectors are shown in Table 19.4.3 where the angular unit vector ˆϕ is taken to be tangential corresponding to the direction a point on the circumference would move. The distance and volume elements, the cartesian coordinate components of the cylindrical unit basis vectors, and the unit vector time derivatives are shown in Table 19.4.3. The time dependence of the unit vectors is used to derive the acceleration. As for the two-dimensional polar coordinates, the ˆρ and ˆθ direction components of the acceleration for cylindrical coordinates are coupled functions of ρ, ˙ρ, ¨ρ, ˙ϕ, and ¨ϕ.

Coordinates ρ,ϕ,θ
Distance element ds=dρˆρ+ρdϕˆϕ+dzˆz
Volume element dv=ρdρdϕdz
Unit vectors

ˆρ=ˆicosϕ+ˆjsinϕ

ˆϕ=ˆisinϕ+ˆjcosϕ

ˆz=ˆk

Time derivatives of unit vectors

dˆρdt=˙ϕˆϕ

dˆϕdt=˙ϕˆρ

dˆzdt=0

Velocity v=˙ρˆρ+ρ˙ϕˆϕ+˙zˆz
Kinetic energy m2(˙ρ2+ρ2˙ϕ2+˙z2)
Acceleration a=(¨ρρ˙ϕ2)ˆρ+(ρ¨ϕ+2˙ρ˙ϕ)ˆϕ+¨zˆz
Table 19.4.3: Differential relations plus a diagram of the unit vectors for cylindrical coordinates.
c2.PNG
Figure 19.4.2: Diagram for Table 19.4.3.

Spherical Coordinates (r,θ,ϕ)

The three dimensional spherical coordinates, can be treated the same way as for cylindrical coordinates. The unit basis vectors are shown in Table 19.4.4 where the angular unit vectors ˆθ and ˆϕ are taken to be tangential corresponding to the direction a point on the circumference moves for a positive rotation angle.

Coordinates r,θ,ϕ
Distance element ds=drˆr+rdθˆθ+rsinθdϕˆϕ
Volume element dv=r2sinθdrdθdϕ
Unit vectors

ˆr=ˆisinθcosϕ+ˆjsinθcosϕ+ˆkcosθ

ˆθ=ˆicosθcosϕ+ˆjcosθsinϕˆksinθ

ˆϕ=ˆisinϕ+ˆjcosϕ

Time derivatives of unit vectors

dˆrdt=ˆθ˙θ+ˆϕ˙ϕsinθ

dˆθdt=ˆr˙θ+ˆϕ˙ϕcosθ

dˆϕdt=ˆr˙ϕsinθˆθ˙ϕcosθ

Velocity v=˙rˆr+r˙θˆθ+r˙ϕsinθˆϕ
Kinetic energy m2(˙r2+r2˙θ2+r2sin2θ˙ϕ2)
Acceleration

a=(¨rr˙θ2r˙ϕ2sin2θ)ˆr+(r¨θ+2˙r˙θr˙ϕ2sinθcosθ)ˆθ+(r¨ϕsinθ+2˙r˙ϕsinθ+2r˙θ˙ϕcosθ)ˆϕ

Table 19.4.4: Differential relations plus a diagram of the unit vectors for spherical coordinates.
c3.PNG
Figure 19.4.3: Diagram for Table 19.4.4.

The distance and volume elements, the cartesian coordinate components of the spherical unit basis vectors, and the unit vector time derivatives are shown in the table given in Figure 19.4.3. The time dependence of the unit vectors is used to derive the acceleration. As for the case of cylindrical coordinates, the ˆr, ˆθ, and ˆϕ components of the acceleration involve coupling of the coordinates and their time derivatives.

It is important to note that the angular unit vectors ˆθ and ˆϕ are taken to be tangential to the circles of rotation. However, for discussion of angular velocity of angular momentum it is more convenient to use the axes of rotation defined by ˆr׈θ and ˆr׈ϕ for specifying the vector properties which is perpendicular to the unit vectors ˆθ and ˆϕ. Be careful not to confuse the unit vectors ˆθ and ˆϕ with those used for the angular velocities ˙θ and ˙ϕ.

Frenet-Serret coordinates

The cartesian, polar, cylindrical, or spherical curvilinear coordinate systems, all are orthogonal coordinate systems that are fixed in space. There are situations where it is more convenient to use the Frenet-Serret coordinates which comprise an orthogonal coordinate system that is fixed to the particle that is moving along a continuous, differentiable, trajectory in three-dimensional Euclidean space. Let s(t) represent a monotonically increasing arc-length along the trajectory of the particle motion as a function of time t. The Frenet-Serret coordinates, shown in Figure 19.4.4, are the three instantaneous orthogonal unit vectors ˆt, ˆn, and ˆb where the tangent unit vector ˆt is the instantaneous tangent to the curve, the normal unit vector ˆn is in the plane of curvature of the trajectory pointing towards the center of the instantaneous radius of curvature and is perpendicular to the tangent unit vector ˆt, while the binormal unit vector is ˆb=ˆt׈n which is the perpendicular to the plane of curvature and is mutually perpendicular to the other two Frenet-Serrat unit vectors. The Frenet-Serret unit vectors are defined by the relations

dˆtds=κˆn

dˆbds=τˆn

dˆnds=κˆt+τˆb

The curvature κ=1ρ where ρ is the radius of curvature and τ is the torsion that can be either positive or negative. For increasing s, a non-zero curvature κ implies that the triad of unit vectors rotate in a right-handed sense about ˆb. If the torsion τ is positive (negative) the triad of unit vectors rotates in right (left) handed sense about ˆt.

Distance element ds(t)=ˆt|dr(t)dt|dt=ˆtv(t)dt
Unit vectors

ˆt(t)=v(t)|v(t)|

ˆn(t)=dˆt/dt|dˆt/dt|

ˆb(t)=ˆt׈n

Time derivatives of unit vectors ddt(ˆtˆnˆb)=|v|(0κ0κ0τ0τ0)(ˆtˆnˆb)
Velocity v(t)=dr(t)dt
Acceleration a(t)=dvdtˆt+κv2ˆn
Table 19.4.5: The differential relations plus a diagram of the corresponding unit vectors for the Frenet-Serret coordinate system.
c4.PNG
Figure 19.4.4: Diagram for Table 19.4.5.

The above equations also can be rewritten in the form using a new unit rotation vector ω where

ω=τˆt+κˆb

Then equations ?????? are transformed to

dˆtds=ω׈t

dˆnds=ω׈n

dˆbds=ω׈b

In general the Frenet-Serret unit vectors are time dependent. If the curvature κ=0 then the curve is a straight line and ˆn and ˆb are not well defined. If the torsion is zero then the trajectory lies in a plane. Note that a helix has constant curvature and constant torsion.

The rate of change of a general vector field E along the trajectory can be written as

dEds=(dEtdsˆt+dEndsˆn+dEbdsˆb)+ω×E

The Frenet-Serret coordinates are used in the life sciences to describe the motion of a moving organism in a viscous medium. The Frenet-Serret coordinates also have applications to General Relativity.

Problems

1. The goal of this problem is to help you understand the origin of the equations that relate two different coordinate systems. Refer to diagrams for cylindrical and spherical coordinates as your teaching assistant explains how to arrive at expressions for x1, x2, and x3 in terms of ρ, ϕ, and z and how to derive expressions for the velocity and acceleration vectors in cylindrical coordinates. Now try to relate spherical and rectangular coordinate systems. Your group should derive expressions relating the coordinates of the two systems, expressions relating the unit vectors and their time derivatives of the two systems, and finally, expressions for the velocity and acceleration in spherical coordinates.


This page titled 19.4: Appendix - Orthogonal Coordinate Systems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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