Application of Intrinsic Differentiation to Curvilinear Coordinate Systems

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Microfiche (MF) - 5d APPLICATION OF INTRINSIC DIFFERENTIATION TO ORBITAL PROBLEMS INVOLVING CURVILINEAR COORDINATE SYSTEMS by James C. Howard Ames Research Center Moflett Field, Gal$

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, 0. C. JUNE 1965 * f

NASA TN D-2861

APPLICATION OF INTRINSIC DIFFERENTIATION TO ORBITAL

PROBLEMS INVOLVING CURVILINEAR

COORDINATE SYSTEMS

By James C. Howard

Ames Research Center Moffett Field, Calif.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $2.00 9

APPLICATION OF INTRINSIC DIFFE€EJ!ULATIONTO OR6ITA.L

PROEEEMS INVOLVING CURVILINEAR

COORDINATE SYSTEMS

By James C. Howard Ames Research Center

SUMMARY

Absolute or intrinsic differentiation has been shown to be an effective tool for solving a certain class of vector problems. In many cases, the solution of problems involving the rates of change of vectors may be obtained more directly by this method than by conventional methods. A single formula with sufficient generality to handle a wide range of problems renders application a purely mechanical process, requiring little or no ingenuity on the part of the analyst. In order to demonstrate its utility, the method has been used to process certain fundamental vectors associated with the orbits of planetary bodies or space vehicles. The processing of these vectors leads to Lagrange's planetary equations. It is shown that the method has distinct advantages when used for obtaining the rate of change of the argument of perifocus of an orb it ing b cdy .

INTRODUCTION

It is well known to students of tensor analysis that problems involving the rates of change of vectors of any variance are conveniently solved by the method of covariant and intrinsic differentiation. This method of dealing with vectors, and the more general entities called tensors, has found little or no application to the problems of ordinary vector analysis probably because all such problems are amenable to solution by conventional methods. However, when covariant and intrinsic differentiation are divorced from the complexi- ties inherent in tensor analysis, it is found that certain problems can be solved more directly and with much less ingenuity on the part of the analyst than is required by conventionalmethods. This is a consequence of the geo- metrical simplification inherent in the method when curvilinear coordinates are used. For example, intrinsic differentiation my be used to determine the influence of perturbing forces on the time rates of change of certain fundamental vectors associated with the orbits of planetary bodies or space vehicles. When this is done, a more direct determination of the rates of change of some of the orbital elements is possible. In particular, the intrinsic

derivative of a vector which lies in the orbital plane, and points in the . direction of the perifocus, leads more directly to a rate of change of the argument of perifocus which includes the influence of changes in the longitude of the ascending node. Furthermore, since the orientation of a plane is uniquely determined by the normal to its surface, the orientation of an orbital plane in space is determined by the angular momentum vector. This fact may be used to advantage in finding the time rates of change of the elements defining the orientation of an orbit plane in the presence of perturbing forces. The intrinsic derivative my be used to obtain the rate of change of the angular momentum vector and hence the rates of change of orbital plane inclination and nodal longitude. These two examples have been chosen to illustrate the utility of the method. Moreover, the method may be applied with equal facility to any situation in which vector changes are involved.

NOMENCLATURE

vector

contravariant vector components in the x coordinate system

covariant vector components in the x coordinate system

the semimajor axis

unit vector which lies in the orbital plane and points in the direction of the perifocus - system of base vectors reciprocal to ai(x)

system of base vectors in the x coordinate system

contravariant vector components in the y coordinate system

covariant vector components in the y coordinate system

system of base vectors reciprocal to bj(y)

system of base vectors in the y coordinate system

eccentricity

vector lying in the orbital plane and pointing in the direction of the perif ocus

-ia , -ja

angulnr momentum vector

orbital plane inclination

mass of central body

mass of space vehicle or planet

perturbing force vector position vector

coordinate transformation

time

velocity vector

components of the position vector in the x coordinate system

components of the position vector in the y coordinate system

Christoffel symbol of the first kind

Christoffel symbol of the second kind

constant coefficients

true anomaly

eonecker delta

dynamical constant, G(M + m)

longitude of ascending node line

argument of perifocus as measured from ascending node line

w+R

Superscripts

indices of c ontravariance

Subscripts

indices of covariance

ANALYTICAL CONSIDERATIONS

Transformation Laws for Scalar Components and Base Vectors

Scalar components.- When referred to a general curvilinear coordinate system, a vector may be expressed in the following form

3 If in some expression a certain index occurs twice, this means that the expression is to be summed with respect to that index for all admissible values of the index, that is, n AiEi =zAiEi i=i - where Ai are the tensor components of the vector A, and Zi, a system of base vectors. In accordance with established notation, tensor components will be denoted by superscripts and the corresponding base vectors by subscripts. In the literature, these vectors are referred to as contravariant vectors, to distinguish them from other vectors which are denoted by subscripts. For the problems considered in the present report, the distinction between these vectors disappears. However, it is necessary to keep the distinction in mind, because if general coordinate transformations are contemplated the transformation law for the components of a contravariant vector denoted by superscripts, differs from that for a vector denoted by subscripts. The latter vectors are referred to as covariant vectors. For a coordinate transformation T from a coordinate system x to a coordinate system y given by

. ., x") the law of transformation for the components of a contravariant vector Ai is given by (see appendix A and ref. 1) :

aj B'(y) = Ai(x) (3) axi

where Ai(x) are the contravariant components in the x coordinate system and BJ(y) are the components when referred to the y coordinate system. For the same transformation of coordinates, other vectors, such as the gradient of a scalar point function, obey a different transformation law. These are the covariant vectors denoted by subscripts. The appropriate transformation law for these vector components is (see appendix A)

where Ai(x) are the covariant components in the x coordinate frame and Bj(y) are the covariant components when referred to the y coordinate frame. As the following argument shows, the distinction between these two transf orma- tton laws vanishes when the transformation -T is orthogonal Cartesian. Let x1 be the components of a position vector r when referred.to the x coordinate system which is orthogonal Cartesian. Likewise, let yJ be components of the same vector when referred to another orthogonal Cartesian system. The transformation of coordinates T is given by

4 yi = a.xij (5) J - where the u$ are constants. The position vector r is invariant with respect to coordinate transformations. Hence the square of the vector is also invariant. Theref ore

i i jxk = 8&jxk xjxj =yiyi= ajap theref ore ii aj% = 8; where 8$ is the Kronecker delta, that is, (see ref. 2)

j = j 1 for k 6k = 0 for j # k

Equation (6) is.the orthogonality condition which may be used to solvft equation (5) for xJ. If both sides of equation (5) are multiplied by a’ k’

theref ore

Theref ore, j ii x = ajy

From equation (5), it is seen that -- ax j and from equation (7)

It follows from equations (8) and (9) that

5 .

As a consequence of equation (lo), the distinction between contravariant and covariant vectors disappears, when coordinate transformations are confined to orthogonal Cartesians systems. This also explains why there is no preoccupa- tion with these vectors in the study of ordinary vector analysis. - Base vectors.- Subscripts assigned to a system of base vectors ai indicate that they are covariant in character, and obey the convariant transformation law. See equation (4) and appendix A. Therefore, if Zi(X) are a system of base vectors in the x coordinate system, and bj(y) are the corresponding base vectors in the y coordinate system, then

- In this connection it should be noted that to every-system of base vector ai, there exists a reciprocal system of base vectors B1 with the following property -J - j- IJ a . ai=Ei=ai. a (12) where 1 for j=i

A superscript is assigned to the reciprocal base vectors to indicate their contravariant character, and to emphasize the fact that they obey the contra-- variant transformation law. (See eq. (3) and appendix A.) Hence,-jf ai(x) are the reciprocal base vectors in the x coordinate system and bJ(y) are the corresponding vectors in the y coordinate system, then

In a curvilinear coordinate system the base vectors are, in general, not unit vectors, but are functions of the coordinates; that is, - - ai = ai(x 12x,. . ., xn> (14)

The base vectors may be obtained 9s follows: let dF be the differential of a position vector F and let dxl be the correspogding differentials of the position components, Then by substituting dF for A, and dxi for A' in equation (l), we have

dy = ki5-i (16)

6 - From equation (15) the base vectors ai are given by

- In an orthogonal Cartesian frame of reference, the base vectors ai consti- tute a triad of mutually orthogonal unit vectors, that is, vectors of unit length. However, in problem formulation, it is usually convenient to use a more general curvilinear coordinate system. When this is done, the magnitudes of the base vectors generally differ from unity.

Vector Derivatives and the Christoffel Symbols - The scalar product of any two base vectors ai and Ej may be defined as follows :

Likewise, the scalar product of the reciprocal base vectors Xi and Zj may be defined as

-ia. zj = gij = $ . ai (19)

The symmetry of gij and gij follows from the nature of the scalar product. Certain combinations of the partial derivatives of the scalar products with respect to the system coordinates are useful in obtaining the derivative of a vector, or in formulating the equations of motion in a general curvilinear coordinate system. The definitions that follow are ascribed to Christoffel and are called Christoffel symbols (see ref. 3). There are two of these symbols, the first of which is defined as

The Christoffel symbol of the second kind is

The utility of the Christoffel symbols is immediately apparent when an attempt is made to find the partial derivative of a base vecLor, or its reciprocal, with respect to any system coordinate. Any vector A may be expressed in the form of equation (1). Furthermore, since the base vectors are in-general functions of the coordinates, it follows that the derivative of A with

7 respect to any coordinate must involve the Christoffel symbols. From equation (l), the partial derivative of the vector A with respect to the coordinate xk is given by

Likewise,

and

Since

it follows that

From equations (23) through (26) azi - - . ak = [ij, k] axj - From equation (12), the rate of change of the base vector ai with respect to xj assumes the form & - -k 2axJ = [ij, k]a (28)

Equation (28) gives the required rate of change of the base vector, with respect to a system coordinate, in terms of the Christoffel symbol of the first kind and the reciprocal base vectors. A more convenient form is obtained if both sides of equation (28) are multiplied scalarly by the reciprocal base vector Z2 to yield aq, - . x2= [ij, k]Zk -2a ax j From equation (lg), it is seen that -k -2 kl a *a =g

8 . theref ore

In terms of the defining formula (211, equation (30) may be rewritten as follows :

Theref ore,

By substitution- of equation (32) in equation (22) the partial derivative of a vector A with respect to the system coordinate xk is

The indices i and 2 in the second term on the right side of equation (33) are dum$ indices, and may therefore be replaced- by any other convenient indices. In order to have a common base vector ai, equation (33) may be rewritten as follows

Furthermore, since

and

--=-aAi dxk dAi axk at at the intrinsic derivative, or the derivative with respect to time, may be obtained from equation (34) in the following form:

1As already indicated, a repeated index implies summation with respect to that index. Since the sunmation index can be changed at will, it is usually referred to as a dummy index. Of course, the range of admissible values of the index must be preserved.

9 In an orthogonal Cartesian reference frame

Therefore, since all these scalar products are constants, it follows that the Christoffel syxhols vanish. In this case, the covariant I er vative, (34) reduces to the sum of the partial derivatives of the components along a set of fixed axes

Likewise, the intrinsic derivative of a vector reduces to the ordinary time rates of change of the components along a set of fixed axes.

For a general space of three dimensions, equation (35) assumes the form - dt = [($ + fl) zl + ($ + fe) z2 + (g+ f3) z3]

f2 = [Al({&}$ + {‘}&12 dt + 0%)+ A2(p}g21 dt + {2}g22 dt + {f3}g)

+A3(f}e+{ 2 }-+(‘}&)I dx2 31 dt 32 dt 33 dt

3 dxl 3 dx2 3 dxl 3 dx2

+ + + f3 = [A1&} dt (12) dt + {;3} $) +.A2({21} at {2*} dt {23} S)

The formidable looking equations (36) through (39) for the intrinsic derivative of a vector in a general space of three dimensions contain 27 Christoffel symbols. Because of the symmetry of the Christoffel symbols, V

{tj} = {h}

10 and the number of independent Christoffel symbols reduces to 18. Furthermore, for the three-dimensional spaces most commonly used, equation (36) reduces to a manageable form. If a base vector of unit length be denoted by gi, then in a cylindrical coordinate system, - a1 = 31 ,

As a consequence of equations (41), there are only two nonzero Christoffel symbols in a cylindrical coordinate system, embedded in a space of three dimensions. These are

{:2} {:2} = -xl 1 ana

Hence, a vector referred to this coordinate system has a time rate of change as follows:

In three-dimensional spherical coordinates,

- A a1 = a1 f gll = 1 - a2 = x1g2 , g22 = (x1)' (44) - A 2 a3 = x1 sin x2a3 , g33 = (xl sin x') In this case there are six nonzero Christoffel symbols. These are

{A}= -xl {:3} = -sin x2 cos x2

(45) (122) = {Zl} = 3 {;3} = {:> = 5

= -x1 sin2 x2 {A} = {;2} = cot x2 When these values of the Christoffel symbols are substituted in equation (35), the time rate of change of a vector, referred to a three-dimensional spherical coordinate system, assumes the following form:

+[~+{~2}(A1$+A2~)+A3{2})dx3]&2dt 33 dt

Z3 + [s+ {3}(A1 + A3 e)dt + {23}(.. edt + A3 g)]dt (46)

APPLICATIONS

Intrinsic Derivatives of Certain Fundamental Vectors Associated With the Orbits of Planetary Bodies and Space Vehicles

Certain fundamental vectors associated with an orbit in space, and which are subject to change in the presence of perturbing forces, provide a useful area of application for intrinsic differentiation. One example of such a vector is the angular momentum vector h which lies in the direction of the normal to the orbital plane. This vector may be used to determine the orientation of an orbital plane in space. Furthermore, the rate of change of this vector in the presence of perturbing torques may be used to determine the rates of change of orbital plane inclination and nodal longitude. -The intrinsic derivative of a vector e, which lies in the orbital plane and is directed to the perifocus, may be used to determine the rates of change of the argument of perifocus and the eccentricity. A third vector may be used to complete the orthogonal triad.

- x2 The angular momentum vector. - Equation (35) may be used to obtain the rates of change of the angular momentum vector, and hence the rates of change of the orbital ele- X' ments defining the orientation of A an orbital plane in space. In a2 spherical coordinates, equation (35) assumes the form shown in equation (46). The coordinate uses are chosen as follows (see sketch (a)): Sketch (a)

12 1. -The g1 axis is taken to be coincident with the angular momentum vector h.

2. The & axis is in the direction of increasing polar angle, that is, in the direction of increasing orbital plane inclination.

3. The third axis & is in the direction of the ascending node line and completes the mutually orthogonal triad of axes.

In this coordinate system the base vectors are given by equations (44). Let the vector appearing in equation (46) have components as follows

A1=h, A2 = A3 = 0 ( 47) When substitutions are made from equations (44), (4>), and (47) in equation (46), it is found that

I therefore

- These are the component rates of change of the vector h, referred to the x coordinate system with base vectors 55. However, it mybe more convenient to use some other reference frame. Assuming that the perturbing forces are

referred to the radial and transverse direc- A A tions in the plane of the orbit, and in the b3 bi direction of the no-lto the orbital plane, the vector dh/dt should be transformed to a reference frame having these directions. If the coordinatfts in this ref- 6 A erence frame be denoted by 9, where i=1, I 2, 3, a coordinate transformation TI must be determined before the components of &/at can be referred to the y coordinate frame. Let bj(y) be a system of base vectors in the y coordinate system (see sketch (b)) . In terms of the natation already established, the transformation of base vectors assumes the following form

61 = -sin(y + w)& + cos(y + w)g3

A b2 = -cos(y + w)& - sin(y + w)&

A b3 = $1 a3 Sketch (b) If Ai(x) are the components of ah/dt in the x coordinate system and BJ(y) are the corresponding components in the y coordinate system, the transformation law, equation (3) gives

Bj(y) 5= eAi(x)

However,

See appendix A. Therefore, equation (3) may be written in the following alternative form

h Bj(y) = (bj $i)Ai(X)

From equation (48)

Substituting from equations (51) in equation (50), and using equations (49) we obtain the transformed coqonents of the vector &/at as follows

dx2 2 dx3 B1(y) = -h sin(y + W) - + h cos(y + w) sin x dt

B3(y) = -ah dt However, - -=FXPdE dt - where r is the position vector of the planet-or space vehicle in the y coordinate system with base vectors bj, and P is the perturbing force vector. Theref ore, \ - A AA - r X = rbl X (bjbj) P

14 From equations (52) and (33), it follows that

dx2 h cos(y + u) sin x2 - h sin(y + u) - --0 dx3dt at - h sin(? + w) sin x2 at + h cos(y + w) -dx2at = r(:3 P) dh & - -at = r(b2 P) These equations give

- --dx3 - r sin(y + w) (g3 . p) (55) at h sin x2

Equation (54) gives the rate of change of inclination, and equation (55) the rate of change of nodal longitude. n

The orbital vector- F.- Con- sider the vector e, which lies in the orbital plane and is directed to the perifocus. (See sketch (c) and ref. 4.) In a cylindrical reference frame with coordinates xi, the base vectors -denoted by ai have the following values: - a1 = $1 - a2 = XI& (56) - a3 = $3 In this case, it should! be noted that x2 is the nodal longitude. In the case previously considered, x2 was the orbital plane inclim- tion. In this reference frame, Sketch (c) which is chosen because the influence of nodal longitude- and orbital plane inclination appear in the formla- tion, the vector e has components as follows (see appendix B and sketch (c)):

-e = e (cos + sin u cos i a;? + sin sin is3 XI that is,

- 2- 3- e = AIZi = A%= + A a2 + A a3 where A1 = e cos w

~2 = (e sin w cos i)/xl (57)

A3 = e sin w sin i

In cylindrical coordinates, equation (35) assumes the form shown in equation (43). Substituting from equations (42) and (37) in equation (43) gives

CE = { [k (e cos w) - x1-dx2. e sin w cos i] g1 dt dt X1

+ [%(e sinx; cos i dx2 cos w + e - sin w cos dt X1

d (e sin w sin i)& +-dt Theref ore,

= cos w - e sin WC; - e -dx2 sin w cos i) CEdt [(G dt

sin w cos i + e; cos w cos i - e -di sin w sin i + e - dt dt

sin w sin i + eL cos w sin i + e -di sin w cos i) $31 dt

These are the component rates of change of the vector F referred to the coordinate system with base vectors Zi. As in the case of the angular momentum vector, it may be more convenient to use some other reference frame. Assuming again that the perturbing forces are referred to the radial and transverse directions in the plane of the orbit, and in the direction of the normal to the orbital plane, the vector &/at must be transformed to a reference frame having these directions. A coordinate transformation T2 must be determined before the components of dZ/dt can be referred to this coordinate frame. Let the base vectors- in this reference frame be again denoted by bj, then the base vectors bj and Zi are related as follows:

A bl = cos(y + w)&l + sin(y + w) cos ig2 + sin(y + w) sin i&

A A A h b2 = -sin(y + w)al + cos(y + w) cos ia2 + cos(y + w) sin ia3

A b3 = -sin i& + cos ia3A

Corresponding to the transformation of coordinates T2, the transformation law for the components of the vector dF/dt is given by equation (30). In this case, the components Ai(x) appearing in the right side of equation (50) may be obtained from equation (58). They are

16 dx2 sin w cos i

A2 = (6 sin w cos i + eij cos w cos i - e -di sin w sin i + e -dx2 cos w) (60) at at }

A3 = (6 sin w sin i + e& COS w sin i + e di sin w cos at When substitutions are made from equations (39) and (60) in equation (5O), it is found that the vector dz/dt has the following comyonents in the directions of the base vector Ej Bl(y> = (6 cos 7 + ea sin y + e -dx2at sin y cos i)l Bz(y) = (-Gsin y + e; cos y + e -a2dt cos 7 cos i (61) B3(y) = (e aat sin w - e - sin i cos J In order to express the rate of change of the vector 'E as a function of the perturbing force vector P, it is necessary to examine the equation of motion of a particle in a noncentral, gravitational force field. It is shown in equation (Bl9) that

By equating components from equations (61) and (62), and remefiering that in this case dx2/dt = dQ/dt, we obtain the following equations:

- 2h A dR 6 cos 7 + e& sin y = - (b2 P) - e sin 7 cos i P (63) - e& cos y - 6 sin y = - (y sin g2+ CL tl) P - e at cos 7 cos i (64)

A - e- sin w e -dR sin i cos w = - sin 7(b3 P) at - at h (65) The rate of change of the eccentricity e. and the rate of the change of the argument of perifocus may be obtained by solving equations (63) and (64). When this is done, it is found that - - cos yfbl P) + -& sin y(2 + e cos 7)(;2 P) - -at cos i] (66) It is seen that the influence of changes in the longitude of the ascending node is given by the third term on the right side of equation (66), that is,

-a; = -cos i ah However, if the argument of perifocus as measured from the inertially fixed X1 axis is denoted by 5, then Z=b+n (68) In this case, the total contribution from the rate of change of the nodal longitude is

From equation (55) h= r sin(y + w) - 63 PI h sin i theref ore

Substituting from equation (70) in equation (68) gives

A + -r sin y(2 + e cos y)b2 + sin(y + U)tan k = { (& COS y) eh

The derivation of this result should be compared with the approach used in reference 4, where spherical geometry had to be used to determine the influence of inclination and nodal longitude. In that case, the problem had to be solved in two parts, whereas in the present case the problem is solved in one step without appeal to system geometry, subsequent to the choice of curvilinear coordinates. Likewise, on solving equations (63) and (64) for G, it is found that

cos y(2 + e cos y) + e

If required, the rate of change of the semimajor axis may be obtained with the aid of equation (72) and the following relationship

h2 = pa(1 - e2)

18 . therefore ah 2h -at = p(1 - e2)$ - 2p.aei. theref ore

theref ore

Theref ore,

A A = e [e sin ybl + (1+ e cos 7)b2 P (73) h *I -

Ames Research Center National Aeronautics and Space Administration Moffett Field, Calif ., Feb. 4, 1965 APPErJDIX A

TRANSFORMATION FORMULAS FOR THE BASE VECTORS THEIR RECIPROCALS

The transformation formulas and, hence, the covariant or contravariant character of the base vectors and their reciprocals may be obtained as follows: Let the differential of a position vector be denoted by dF. Then if ?Zi(X) are the base vectors in the x coordinate system, and Ej(y) the base vectors in the y coordinate system, the differential aF may be expressed in the following alternative forms

theref ore

Likewise,

theref ore

- It is seen from equations (A2) and (A3) that the base vectors ai and bj obey the covariant transformation law; consequently, the use of subscripts is justified.

Reciprocal Base Vectors

To each system of base vectors Bi there exists a reciprocal system of vectors EJ with the following property

where 8i is the Kronecker delta, that is,

. 1 for j=i 6: = o for j # i

Scalar multiplication of each side of equation (A2) by bj(y) gives on using (A4)(see ref. 3)

I Similarly, from equations (A3) and (Ab) it is seen that

Equation (Al) referred to the reciprocal system of base vectors assumes the form -. fi = Zi(x)aXi = bJ(y)dyj (A71 theref ore

theref ore

and

theref ore

From equations (A7) and (A8)

theref ore

Likewise, from equations (A7) and (AS)

From equations (A10) and (All), it is seen that the reciprocal base vectors Z1, obey the contravariant law of transformation; theref ore, the superscript notation is justified.

Vector Transformations

Equations- (A10) and (All) may be used to obtain the transformation law for a vector A, where - A = Aizi = AjXj (A121

21 - If = Ai(X)Ei(x) when the vector A is referred to the x coordinate system, and if -A = Bj(y)bj(y) when referred to the y coordinate system, invariance of A requires that

Bj(y)bj(y) = Ai(x)Zi(x) (A13 1

From equations (A2) and (Al3), the appropriate transformation law is obtained as follows: Bj(y) = a,jaxl Ai(x)

Equation (A14) is the contravariant transformation law for the components of the vector A. When A is referred to the x coordinate system with base vectors Ei(x), which obey the covariant transformation law, the components Ai(x) obey the contravariant tra_nsformation law, and, hence, the use of superscripts is justified. If A is referred to the reciprocal base system ai, then from equation (A12) - A = AiZi

On a transformation of coordinates from the x coordinate system to the y coordinate system, the invariance of A requires that

From equations (A10) and (Al5), the appropriate transformtion law is obtained as follows

- It is seen that when a vector A is referred to a coordinate system by use of reciprocal base vectors, which obey the contravariant law, the corresponding components of A obey the covariant law, and the use of subscripts is therefore justified.

22 EQUATION OF MOTION OF A PARTICIX IN A NONCENTRAL GRAVITATIONAL FORCE FIELD

The equation of motion of a particle of unit mass moving in an inverse- square law, central force field is

Vector xrgltiplication of each side of equation (Bl) by the angular momentum vector h gives

dt" rL

--- *A h = r X V = (9y)bS

On substitution from equation (B3) in equation (B2) it is seen that

theref ore

The integral of equation (B4) is given by

- where e is a constant vector of integration. The vector F may be expressed in terms of its scalar magnitude and a vector of unit length as follows : - A e = ea (B6) where & is a vector of unit length. See sketch (e). Substituting equation (B6) in equation (B5) gives 'Xh = + eaA CL il 037) - Equation (B7) may be solved to obtain the position vector r. Scalar multiplication of each side of equation (B7) by 7 gives the following equation for F:

23 Theref ore

and

r= WI-1 1 + e cos y

It is seen that e, the magnitude of the constant vector of integration appearing in equation (B6), is the orbital eccentricity. Vector multiplication of each side of equation (B7) by h gives

Therefore,

With the notation of sketch (c), the unit vector $ may be expressed in the following form:

A A A a = (cos y)bl - (sin y)b2 '

If the assumption of an inverse-square law central force field is not satis- 1 fied, the equation of motion must be modified accordingly. In the presence of a perturbing force P, the equation of motion becomes - - --dV -p-- - yr dt r3 - Furthermore, in the presence of the perturbing force vector P, the assumption of constancy no longer applies to the vector 'E. Hence,

Pg={gXh+VX (FXP) -&[(Fx') XTI} Theref ore,

and

p-=dz Pxh+Vx(FxF) dt The first term on the right side of equation (Bl3) may be written in the following alternative form: - AA AA - P X E = h(blb2 - bzbl) P 0314) Likewise,

When substitutions are made from equations (Bg) and (Bl?), the second term on the right side of equation (Bl3) assumes the form

- A vx(E;xP)=E $1;~ + + (b2 h

A Substituting for a from equation (B10) in (~16)gives

From equations (B14) and (B17) it follows that

Theref ore, AA AA hAA E = bA1C2 - [F sin Y(b2b2 + b3b3) + - b2bl]} ? at P REFERENCES

1. Sokolnikoff, I. S.: Tensor Analysis: Theory and Applications. John Wiley and Sons, Inc., London, 1951, p. 62.

2. Spain, B. : Tensor Calculus. Third ed., Oliver and Boyd, Edinburgh, 1960, p. 4. 3. McConnell, A. J. : Application of Tensor Analysis. Dover Pub ., Inc., N. Y., 1957, p. 141. 4. Howard, James C. : Theoretical Study of Orbital Element Control With Potential Applications to Manned Space Missions. NASA TN D-2237, 1964.

5. Wills, A. P.: Vector Analysis,With an Introduction to Tensor Analysis. Dover Pub., Inc., N. Y., 1958, p. 221.

I 26 NASA- Langley, 196 5 A-20 28