DYNAMICS IN SPHERICAL COORDINATES
J.M.
Redwood 2007
Introduction
This worksheet is intended as a brief introduction to dynamics in spherical coordinates.
Definition
and Sketch
Consider
a point P
on the surface of a sphere such that its spherical coordinates form a right
handed triple
in 3 dimensional space, as illustrated in the sketch below. These coordinates
are usually referred to as the radius, polar angle (or colatitude) and azimuth
(or longitude) respectively, and follow the socalled American convention.
The unit vectors are indicated in the sketch by ,
normal to the surface of the sphere at P;
,
tangent to the meridian at P;
and ,
tangent to the latitude circle at P.
For a fuller definition see Maple's help pages at sphericalcoordinatesDefinition.
The
transformation equations from spherical to Cartesian coordinates are:
The
transformation equations from Cartesian to spherical coordinates are:
or
Velocity
Denoting
vectors by bold face type, let r be the vector joining the centre of the sphere
to P and
be its unit vector. Let v
and a
be the velocity and acceleration respectively of P.
In what follows, it is to be understood that r,
v
and a,
together with their magnitudes in the directions of the unit vectors, are
functions of time. In order to present equations that are easier to
read the usual functional notation (t)
will be omitted.
The velocity
of P
is given by
But
∴
equation (a)
Expanding
the second term on the rhs of the above,
equation (b)
Substituting
in the above from the following table of partial derivatives of the unit vectors,
and
with respect to r,
ϕ
and ϑ
Equation
(b) becomes
Substituting
this in equation (a)
It
will be noted that this equation contains no derivatives of the unit vectors.
The two rightmost terms are tangential velocities and are much as might
be expected from experience with polar coordinates.
Acceleration
The
acceleration, a,
of the point P
is obtained by differentiating v
w.r.t. time, expanding the terms containing derivatives of unit vectors in
the same way as above, substituting from the table, and collecting terms.
The result is:
Again,
this equation contains no derivatives of the unit vectors: indeed none are
present in any of the higher derivatives of r.
The three double differentials w.r.t. time are accelerations of the
coordinates. The terms in
and are
accelerations toward the centre and the polar axis (OZ)
respectively. The terms with a coefficient of 2 are Coriolis accelerations.
The Coriolis term in the
coordinate is as might be expected from experience with polar coordinates.
The first Coriolis term in thecoordinate
is caused by the velocity of P
about OZ
i.e. ,
while the second in that coordinate may be thought of as being caused by the
velocity of P
about OZ
in the equatorial plane (OXY)
and the component of
, i.e. ,
along the projection of OP
in that plane.
Force and
Moment
If
m
is the mass of a particle at the point P,
the force, F,
needed to accelerate it is,
The
moment (or torque) needed to produce the acceleration is obtained from
This reduces
to
The components
of the moment may be thought of as righthanded (clockwise) rotation vectors
with their axes along the angular unit vectors.
Worksheet
The
velocity and acceleration shown above are now written as Maple input so that
they can be readily copied and pasted into a worksheet. The unit vectors
,
are inconvenient in a worksheet and will now be written as
and
respectively, which is the format used in Maple's Vector Calculus package.
>

restart;
with(VectorCalculus):

>

SetCoordinates(spherical[r,phi,theta]):

>

OP
:= <r(t)phi(t)theta(t)>:

>

v
:= <diff(r(t),t)r*diff(phi(t),t)r*sin(phi)*diff(theta(t),t)>:

>

a
:= <diff(r(t),t$2)r*(diff(phi(t),t))^2  r*sin(phi)^2*(diff(theta(t),t))^2
2*diff(r(t),t)*diff(phi(t),t) + r*diff(phi(t),t$2) r*sin(phi)*cos(phi)*(diff(theta(t),t))^2
2*sin(phi)*diff(r(t),t)*diff(theta(t),t) + r*sin(phi)*diff(theta(t),t$2)
+
2*r*cos(phi)* diff(phi(t),t)* diff(theta(t),t)>:

>

M
:= <0m*r*(2*sin(phi)*diff(r(t),t)*diff(theta(t),t) + r*sin(phi)*diff(theta(t),t$2)
+
2*r*cos(phi)*diff(theta(t),t)*diff(phi(t),t) )m*r*(2*diff(r(t),t)*diff(phi(t),t)
+
r*diff(phi(t),t$2)  r*sin(phi)*cos(phi)*(diff(theta(t),t)))^2>: 
The
command PDETools[declare] is now used to indicate the derivatives of r,
ϕ
and ϑ
w.r.t. time will be shown with primes in order t make the equations easier
to read.
>

PDETools[declare](prime=t): 

(6.1) 

(6.2) 

(6.3) 

(6.4) 
Derivation
A
derivation of the above table of partial derivatives is given below. It
is not a proof; it is but one of several ways of demonstrating the validity
of the table.
In
the sketch, the points O,
Q,
P
and Z
are in the meridian plane perpendicular to the equatorial plane OXQY
. Ox,
Oy
and Oz
represent the Cartesian coordinates of the point P.
It will be seen from the sketch that Ox
and Oy
can be resolved to form Oq,
which itself resolves into components in the directions of the unit vectors
and .
Similarly, Oz
can be resolved along the same directions and its components added to those
of Ox
and Oy,
thus forming the vectors r
and ϕ.
The components of Ox
and Oy
perpendicular to Oq
form the vector .
The
unit vectors
and
along OX
and OY
may be resolved and equated to the components of the unit vectors
and
in the OXQY
plane, as shown in equation (
7.1 ) below.
>

restart:
with(VectorCalculus): SetCoordinates(spherical[r,phi,theta]):

>

cos(theta)*e[x]
+ sin(theta)*e[y] = sin(phi)*e[r] + cos(phi)*e[phi];


(7.1) 
Let
xyz
represent the unit vectors in Cartesian coordinates.
>

xyz:=
SetCoordinates(<1,1,1>,cartesian[x,y,z]);


(7.2) 
The
trigonometric functions used above to resolve x,
y
and z
in the directions r,
ϕ
and
are now written as a matrix of functions that transform Cartesian coordinates
to spherical coordinates. Let it be called c2s.
>

c2s
:= << sin(phi)*cos(theta)  sin(phi)*sin(theta)  cos(phi)>,
< cos(phi)*cos(theta)  cos(phi)*sin(theta)
 sin(phi)>,
< sin(theta) 
cos(theta)  0 >>:

The
orthogonality of c2s
is now tested and its determinant is obtained.
>

LinearAlgebra:IsOrthogonal(c2s);
simplify(LinearAlgebra:Determinant(c2s)); 

(7.3) 
Since
the transformation matrix, c2s,
is orthogonal, the spherical coordinates are orthogonal; and since they were
defined as such, this acts as a check on the validity of the transformation
matrix. The determinant of c2s
has a value of +1, and so the transformation to spherical coordinates requires
only a rotation of the axes, and thus the spherical coordinates are right
handed. Again, they were defined as such, and this acts as a further
check on the validity of the transformation matrix. (In passing, it
is worth noting that the transpose of c2s
is the transformation matrix from spherical to Cartesian coordinates.)
The
unit vectors in spherical coordinates are given by,

(7.4) 
Let
rtp
represent these unit vectors.
An
inert matrix of the partial derivatives of the unit vectors in spherical coordinates
is now constructed, thus:
>

`Derivatives
of spherical unit vectors` := <<seq(Diff(e[i],r),i in [r,phi,theta])>

<seq(Diff(e[i],phi),i
in [r,phi,theta])>
<seq(Diff(e[i],theta),i
in [r,phi,theta])>>:

This
inert matrix is equated to the Jacobian of the unit vectors in spherical coordinates
i.e. to their partial derivatives.
>

J
:= Jacobian(convert(rtp,Vector),[r,phi,theta]):

>

`Derivatives
of spherical unit vectors` = J; 

(7.5) 
The rhs
of (7.5)
can be compared with the rhs of the matrix of unit vectors (7.4)
and substitutions made from the latter. Noting that the magnitudes of
the Cartesian vectors in (7.1)
are unity, a similar substitution can be made from (7.1)
into (7.5)
.
>

J[1,2]
:= e[phi]; J[1,3] := sin(phi)*e[theta]; 
>

J[2,2]
:= e[r]; J[2,3] := cos(phi)*e[theta]; 
>

J[3,3]
:= sin(phi)*e[r] cos(phi)*e[phi];


(7.6) 
Hence,
>

`Derivatives
of spherical unit vectors` = J; 

(7.7) 
Writing
,
and
as ,
and
respectively, equation (7.7)
can
be written as a table thus:
Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for noncommercial, nonprofit use only. If you wish to use this application in forprofit activities, please make a donation to Cancer Research.