The Motion of a Point Upon the Surface of an EllipsoidAuthor(s): Thomas CraigSource: American Journal of Mathematics, Vol. 1, No. 4 (1878), pp. 359-364Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369380 .

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THE MOTION OF A POINT UPON THE SURFACE OF AN

ELLIPSOID.

BY THOMAS CRAIG, Fellow of the Johns Hopkins University.

THE ideas contained in tlhe following brief paper were suggested to me in reading the twenty-eighth Chapter of Jacobi's Vorlesungen fiber Dynamik. A similar investigation- may have been given before, but I have never seen anything on the subject.

Let the point be acted upon -by a force directed constantly towards the center of the, ellipsoid and varying as the distance from the center. Let 3, a constant, denote the force at uniit's distance from the center, and let a denote the force in the direction of the normal to-the surface. We have then for the equations of motion of the point

d2x x dt2 a2' +

'(1) dtY-=aY B

d2z z dt-2 62a 2 +zX

the ellipsoid being given by the equation x 2 y2 z2

22 62 Mty+ e( b Z

Multiplying equations (l) by x - 2 respectively, and adding, we obtain a2 d2 + a27 + c2 It2 =a[t 2 2x& d2~z [-2 +9 +z2]

Calling V/p the reciprocal of the length of the perpendicular from the center on the tangent plane at the point x, y,~ z, we may wsrite this equation

) 2dt2+ydt2 z d2z

2 + b2- a2 +c2dt2 OP+dd 359

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360 CRAIG, The MJiotion of a Point u.pon the Surface of an Elli.psoid.

Differentiating the equation of the ellipsoid twice with respect to t, we have x dx+ y dy z -dz a dt 62dt C2 dt x d . dy z d2z a dt.2 b dt c2 dt2

+ 1 (d#) + 1 (dy) +; (Idz =0;

designate by P the last three terms of this last equation, then we have from (a) (2) - P = ap +13. Again, equations (1) give

r1 dx d2xc 1 dyd2y 1 ddzl 2 rS dwyct dz _

2 __ _d_ a x & + __ _ __-

_a b dt dt+2

+ dtd 4 dt ?4dt

+ xs dxc + y~ dy z dzl + a dt b2 dt C 2 dt

this is simply (3) dP

a_ p

dt dt Eliminating a from (2) and (3), we have

1 dp + I dP 0 -P dt P +,j d1;t

the integral of which is (4) p (P + _A const.

. . . ~~~~dx dxy dz . Multiplying equations (1) by d. 't and d- respectively, and adding, gives

dx d2x dy d2y dz d2z [ dx dy dz dt2 + dt dt + dt dt2 -13 dt + Y dt dt

Integrating this we get

(5) 2 + ( =2 + (== + +

z2) + B,

where B - const. of integration. For 13 0 or when no force acts towards the center, we have the simple case,

s = V/B. t + const. or the arc varies directly as the time. Any point on the ellipsoid can be given as the intersection of this surface with the two confocals

, 2 y2 z2

a2 + ? + b2 +x C+ +X

a +., b2 - C2 + A

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.CRAIG, Thte Motion of a Point upon the Surfacce of an Ellipsoid. 361

Then we have 24 and 2X as the elliptic coordinates of the point, 4 = const. being the equation of one set of lines of curvature and 2 =const. the equa- tion of the other -set, the intersection of any two of which determines the position of a point upon the surface. We have now (Salmnon's Geom. of Three Dimen., page 124),

x2 + y2 + z2 - a2 + b2 + C2 _ (X + X2);

also, (6) dS2= ec+ dy+Z2=Xl - 2 I ,A I 72d2]

4 (d2s, d+d+(?)(4I)c2+ 24) (a2+X2)(62+X2)(c2+X2)

and for the differentials d;v, dy, dz the known values

dx = -Na(2c [X Xa2d/%2\/4 V2 (a2 -b2) (a_2__) a2+ __ _a + X2

(7) 1 6 = [d A b2 + X2 + da2 <b2 + I (62) a2) 62 + X1

dz= _ 1 b _0 Fdb.-&c+2bd2.\/c 21

2 (c2 a2) (c2 _ 62) Lc2?+4 + c2 +

and finally for the perpendicular from the center to. the tangent plane

(8) 24 ? Z 2 = 244 a b~ c abc From the expressions for dx, dy, and dz we can readily obtain.

dX2 dy2 dz2 2 dX2 d22 1 (9) a2 62 + 2 -2 L 2)+)(C2+ 2 + 2- >'

the first inember of which equation, divided by dt2, is- the quantity P, therefore

(10) P __-_2 [(dt) ( )1

where for convenience we write q: = (a2 + 24) (b2 + 4) (C2 + 24)

-= (a2 + 22) (b2 + 22) (C2 + 22)

Substitutinig for P its value as obtained from equation (4), this last becomes

(11) I-2 [-1 (d ) 2 1 (dX2)] 2 4 2.

where D = abcA. Equation (5) becomes in elliptic coordinates

(12) 2-4 2 [2, (dl) -2 (-2)] = C- 3 (X2 + 22),

where for brevity we write C = j3 (a2 ? b2 + c2) + B. Eliminate t from 91

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362 CRAIG, The Motion of a Point upon the Su&face of an Ellisoid.

equations (11) and (12), and the result will be the differential equation, in elliptic coordinates, of the path of the point upon the ellipsoid: this is

r~~~ X AlS2 2d2 CdX2 d 2

\x x-p} V -- -[C-~ (Xl + %2)]L--- or

(13) VZdt _ S/X2d__ _ _2_

q) (D CVXP(D-C +L32) 8~ ~~~~~~Z (D (@- C2 1 -8t X2 + 922)

in which the variables are separated. If again we make / 0, that is, if after the first inipulse, no force acts upon the point but that in the direction

of the normal, this equation becomes, on making D 02

/ (a2 + i) (b2 + 1) (C 2? + X v (a2 +X2)(b2 + 2) (C2 + 2) (02 + X2)

this is the differential equation of a geodesic upon the ellipsoid. The integrals of these expressions will be elliptic for 0 0 O. If we change

the variables X, and 22 into new ones defined by the equations

X2=1

our differential equation becomes

(14) d< (V(a24 + 1)(b24 + 1)(c2 j + 1)(_D2-_C+)

'V(a2v + 1)(b2n + 1) (c2nj + 1) (Dj2 . Cv7+ 3

Write-D kandD = m, and call-1,-02v-1-l2 the roots of tbe

quadratic equations 02 + kE + m = 0, n' + k7 + m = 0;

and further-2 - a', b- 1b, e =l, and our equation can be written

(15) f V(/ + a')(0 + b')(a +C) + 1 + a2)

,__ _ __ = const. V/ (' + a') (n + b') (K + d) (n + )7) (n + 72)

These are not in general reducible to elliptic integrals. If now we find 2 as a function of Al, say x2 =f (S,) f and qi F (Al) -F, then substituting in

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CRAIG, The Motion of a Point upon the St face qf an Ellipsoid. 363

either of the equations (11) and (12) we will be able to find the tiIme t as a function of X1l.

Making this substitution in equation (12), we readily find

[ q f ( dt ) 2] f ) t- S 4PP[c-43 (x f dX1 .

When we can determine the values of f and F from equation (15), we will be able from this last to express the elliptic coordinates Al and t2 as func- tions of the time, a problem whiclh, for the. case of a simple pendulumrr or point constrained to move upon the surface of a sphere, has been solved by M. Hermite in Orelle's Journal, vol. 85.

We have seen that in general the path traced out by the moving point is not a geodesic, but was such a line where there was no central attracting force, or where ,B 0. It is interesting, however, to observe that, for another value of f, the point will move along a geodesic. Suppose that we have two quan- tities, x and a, the former either constant or a function of the time, the latter constant or a function of the arc s. And now make

r + sN IdI dx i<] Lt+ ?dt dt t=1dt

Oz r + c dd] s- dz = d- [ Jr dt] dt X dt

Our equations of motion now assume the forms d 2x axc dxc dt2 a2 + 13 d

d2ycay dy + A dt2 b2 + dt d 2z _ az dz dt2 2 dt

We have as before 1dxc d2xc 1 dy d2y 1 dz d2 x dilltv dyl z dzl

2 dt d dt d d _- d ? t b4dt e d

dt

a1 -(d-t) b (dt) C2 (t) or simply

ddPt a ddp dp -i 2dtP.

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364 CRAIG The Motion of a Point upon the Surface qf an Ellipsoid.

And again, x d2x +y (I y +z d 2z a [$2 + 2 +

aS d t2 bdt2 c2;d t2 4

Fdx & dy zdzl + W3 t- + b2 dt + <2dt] or simply

_P a.p. Eliminating a from these equations

1 dp +1 dP2 pdt Pdt

fronm which we have dp +

dP 2Iidt 2 dt + a dS d

Now writing 2 rdt = T

2 fds =

we have as the integral of this equation pP- GeT+E. p = C'+

We can also readily obtain from the equations of motion the following ds5 C)zAe%(T + Y)

dt These equations give

1 fd s \ -P Ctt2 -dt C"

which becomes in elliptic coordinates a_ rXl-2 IA d l22 d 2 2 1 A12 1-S2 r XIl 1KA

ac'4 L dt dt abc 4 L dt \ \dt

Putting abC, _ 02, and multiplying through by dt2, we have after simple reductions

Vsxidxi. =4 / X2d'%2 0, _ ~~ VXdA _

N/(D (X1 + 02) 5 (X2 + 02) or

Vx1d2vj ,Vx2dl2 AV (a2+ XI) (b 2+ XI) (c2 + XI) (O + XI) (a2 + %2) (b2+ X2) (G2 +X2) (02 +%2)

This is the differential equation of the path of the moving point which is, as we see, a geodesic upon the ellipsoid.

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