The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

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Mathematical Operators

The information contained in this appendix can be found in a very large number of published references. We have drawn the following primarily from the presentation in Appendix 1.B of Binney and Tremaine (1987); hence, if clarification is needed of any of the following, Binney and Tremaine should be consulted first.

Gradient

The general (curvilinear coordinate) form of the gradient operator that is valid in any orthogonal coordinate system is:

i=1,2,3 ei (1/hi) xi

[Equation VI.M.10]

On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.

Mathematica Application

The accompanying application permits you to determine the gradient of virtually any analytically expressible scalar function G(x), in any one of fourteen different orthogonal coordinate systems, utilizing the symbolic manipulation capabilities of Mathematica. While this very general, interactive tool is potentially very powerful, it is strongly recommended that you not try to use it until you have had some experience using the related, but less general, applications that have been taylored for Cartesian, cylindrical, or spherical coordinates.


Divergence

Consider any vector function F(x) with orthogonal components (F1, F2, F3); the divergence of this function is usually written as F. The general (curvilinear coordinate) form of the divergence operator that is valid in any orthogonal coordinate system is:

F [1/(h1h2h3)]{ x1(h2h3 F1) + x2(h3h1 F2) + x3(h1h2 F3) }

[Equation VI.M.11]

On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.

Mathematica Application

The accompanying application permits you to determine the divergence of virtually any analytically expressible vector function F(x), in any one of fourteen different orthogonal coordinate systems, utilizing the symbolic manipulation capabilities of Mathematica. While this very general, interactive tool is potentially very powerful, it is strongly recommended that you not try to use it until you have had some experience using the related, but less general, applications that have been taylored for Cartesian, cylindrical, or spherical coordinates.


Laplacian

Consider any scalar function G(x); the Laplacian of this function is:

2G [G].

[Equation VI.M.12]

The general (curvilinear coordinate) form of the Laplacian operator that is valid in any orthogonal coordinate system is:

2G = [1/(h1h2h3)]{ x1[(h2h3/h1) x1G] + x2[(h3h1/h2) x2G] + x3[(h1h2/h3) x3G] }

[Equation VI.M.13]

On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, or spherical coordinates.

Mathematica Application

The accompanying application permits you to determine the Laplacian of virtually any analytically expressible scalar function G(x), in any one of fourteen different orthogonal coordinate systems, utilizing the symbolic manipulation capabilities of Mathematica. While this very general, interactive tool is potentially very powerful, it is strongly recommended that you not try to use it until you have had some experience using the related, but less general, applications that have been taylored for Cartesian, cylindrical, or spherical coordinates.


Lagrangian Time-Derivative

Throughout this H_Book, we will use the operator symbol "D" to represent the Lagrangian (or total ) time-derivative. The Lagrangian time-derivative is related to the Eulerian (or partial ) time-derivative through the following expression:

D d/dt = t + v

[Equation VI.M.14]

When dealing with any fluid dynamical system, it is extremely important that you understand the physical relationship between the the total and partial time derivatives that is implied by this mathematical expression. If you don't understand the difference between partial and total time derivatives in this context, read the accompanying discussion of Lagrangian vs. Eulerian representations.


Time-Derivative of Unit Vectors

The Lagrangian (or total ) time-derivative of any unit vector ei in any orthogonal curvilinear coordinate system can be written in the following form:

Dei d(ei)/dt = (x1ei)(dx1/dt) + (x2ei)(dx2/dt) + (x3ei)(dx3/dt).

[Equation VI.M.15]

As detailed, for example, in Part I (page 26; equation 1.3.6) of Morse and Feshbach (1953), given the definitions of xi and hi for a specified coordinate system, the various coefficients in this expression can be evaluated according to the following tabulated expressions:

Evaluating: (xjei)
x1
x2
x3
e1
- e2 (1/h2)x2h1
+
- e3 (1/h3)x3h1
e2 (1/h1)x1h2
e3 (1/h1)x1h3
e2
e1 (1/h2)x2h1
- e3 (1/h3)x3h2
+
- e1 (1/h1)x1h2
e3 (1/h2)x2h3
e3
e1 (1/h3)x3h1
e2 (1/h3)x3h2
- e1 (1/h1)x1h3
+
- e2 (1/h2)x2h3

[Equation VI.M.16]

Hence, we deduce that for any curvilinear coordinate system,

De1 = e2A + e3B,
De2 = - e1A + e3C,
De3 = - e1B - e2C,

[Equation VI.M.17]

where:
A [Dx2] (1/h1) x1h2 - [Dx1] (1/h2) x2h1
B [Dx3] (1/h1) x1h3 - [Dx1] (1/h3) x3h1
C [Dx3] (1/h2) x2h3 - [Dx2] (1/h3) x3h2

As an example, given the definition of the position vector [VI.M.7] x in any curvilinear coordinate system, the above relations must be utilized when determining how to express the vector velocity (v dx/dt) or the vector acceleration (a d2x/dt2) in that coordinate system.


Given the definition of the position vector [VI.M.7] x in any curvilinear coordinate system, the definition of the direction cosines [VI.M.1], and the relationships [VI.M.5 & VI.M.6] between the various direction cosines that is demanded by the condition of orthogonality, show that quite generally the velocity vector v is given by the expression,

v = e1 (h1Dx1) + e2 (h2Dx2) + e3 (h3Dx3).

[Equation VI.M.18]


On a few accompanying pages, we have detailed the derived expressions for v and a in Cartesian, cylindrical, and spherical coordinates.


Convective Operator

When "D" operates on any scalar function G(x), determining how to handle the v "convective operator" on the right-hand-side is straightforward. Specifically, for any orthogonal coordinate system,

(v) G = i=1,2,3 ( vi/hi ) xi G.

[Equation VI.M.19]

However, it is less obvious what form the convective operator should take when "D" operates on a vector function. In general, when v operates on the vector F(x), the result is a vector whose jth component takes the following form:

[ (v)F ]j = (v) Fj + i=1,2,3 {[ Fi/(hihj) ] [ vj xihj - vi xjhi ]} .

[Equation VI.M.20]

On a few accompanying pages, we have detailed the specific form that this operator takes in Cartesian, cylindrical, and spherical coordinates.

Familiar Coordinate Systems

Cartesian
Cylindrical
Spherical

Relatively unfamiliar coordinate systems

Kuzmin
T1
Stäckel [not yet documented]


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