(7) ~ + ~ cos O. 5~ + (/~-I~)~ ~.,-- o. ON THE UNSYMMETRICAL TOP.

ON THE UNSYMMETRICAL TOP.

B Y

PETER FIELD

of ANN ARBOR, MICH., U. S. A.

I. This n o t e is a c o n t i n u a t i o n of t h e p a p e r p u b l i s h e d in Vol. 56, pp.

3 5 5 - - 3 6 2 , of t h i s J o u r n a l a n d t h e n o t a t i o n is t h e s a m e as was used a t t h a t time. T h e m o m e n t s of i n e r t i a f o r t h e p r i n c i p a l axes of t h e body a t t h e fixed p o i n t are I1, I s , a n d I 3 a n d t h e c o m p o n e n t s of t h e a n g u l a r velocity a l o n g t h e s e axes a r e wl, ~%, o)a. T h e c e n t r o i d is a t (o, o, h), E u l e r ' s angles a r e d e n o t e d by (0, ~0, ~0), a n d t h e w e i g h t of the b o d y by w.

z. I f t h e u s u a l n o t a t i o n f o r a d e r i v a t i v e w i t h r e s p e c t to t i m e is used, E u l e r ' s e q u a t i o n s f o r t h i s ease are

! 1 ~1 + (I3 - - I s ) ~% o)~ - ~ h w sin O cos ~T Is ~ + (I1 - - I ~ ) c% ta I - - - - h w sin 0 sin 5 ~ + ( / ~ - I ~ ) ~ ~.,-- o.

T h e values of r c%, aJa are

~o I = ~J cos 9~ + ~b sin 0 sin co., = - - 0 sin ~o + ~ sin 0 cos q~

~o~= ~ + ~ cos O.

I n a d d i t i o n t h e r e a r e t h e t w o well k n o w n i n t e g r a l s

11 ~0 ~, ~- I ~ co ~ + L3 ~o ~ - - T = - - z w h c o s O 40--33617. A c t a mathematica. 62, Imprim6 le 18 avril 1934.

(~)

(2) (3)

(4) (S) (6)

(7)

314 P e t e r Field.

I~col sin 0 sin q) + I2oJ ~ sin 0 cos 9 + I3 ~~ cos 0 = k

(8)

k a n d T b e i n g c o n s t a n t s o f i n t e g r a t i o n .

3. T h e case w h i c h it is d e s i r e d t o s t u d y is t h a t in w h i c h I~ a p p r o a c h e s I 1 a n d I 3 b e c o m e s s m a l l i n s u c h a w a y t h a t I1 ---T~ /3 a p p r o a c h e s t h e v a l u e p.

I f Ix, v,, /x .... a n d I v , , , a r e t h e m o m e n t s o f i n e r t i a o f t h e b o d y f o r its t h r e e p r i n c i p a l planes, t h e v a l u e o f p in t e r m s of t h e m is

I x l zt - - l y l zl

H e n c e if we c h o o s e t h e axes so t h a t Ix,=, is g r e a t e r t h a n I v .... it m a y be a s s u m e d t h a t p is a p o s i t i v e c o n s t a n t w h o s e v a l u e lies b e t w e e n zero a n d one. S u c h a s i t u a t i o n c a n arise in case t h e b o d y w i t h a fixed p o i n t h a s t h e s h a p e of u rod.

4. W i t h t h e s e r e s t r i c t i o n s o n t h e v a l u e s o f t h e m o m e n t s of i n e r t i a , equa- t i o n s I, 2, 3, 7, a n d 8 b e c o m e

h u,

G1 - - to.~ to e ~ ~ - sin 0 cos q9 1 1

(I')

h w sin 0 sin 99 (2')

~5~ + % to1 = 11

~53 = p a'l o~_~ (3')

T - - 2 w h eosO

+ (73

~o I sin 0 sin 99 + ~o~ sin 0 cos q9 = ~ 9 k (8')

T h e v a l u e s o f to 1 a n d to~ g i v e n in (4) a n d (5) s u b s t i t u t e d in (7') a n d (8') g i v e

T h e r e f o r e

a n d

0.0 + g,2 sin 2 0 ~ T - - 2 w h cos 0 k

11 a n d y) s i n ~ 0 = ~ .

k

~0 - ix sin" 0 (9)

I~, s i n ' 0 t~* = / 1 sin -~ 0 ( T - 2 w h cos O) - - k ~. (I o)

On the Unsymmetrical Top. 315 These equations d e t e r m i n e ~p and O in t e r m s of t h e time and show how ~p a n d 0 v a r y in t h e course of t h e motion. T h e y n e e d n o t be discussed as t h e y are the well k n o w n equations of m o t i o n f o r a spherical p e n d u l u m .

5. I f co~, c%, c%, and ~b are replaced by t h e i r values in t e r m s of ~ a n d 0, e q u a t i o n (3') becomes

where

= go + /1 sin 2 0 J = I~ s i n 2 0

0 ~ s i n 2 ~ + ~ c o s 2 pkO

- - P I ~ sin 2 O

- - 2 - - + 0 "~ sin2(q) 4 ~) (II)

- - k e ~ arc t a n - - : . . . .

/1 O sin 0

6. T h e i n t e g r a t i o n of e q u a t i o n (I I) is a simple m a t t e r if 0 ~ o. So f a r as t h e v a r i a t i o n in 0 and ~p is concerned, this c o r r e s p o n d s to t h e case of a conical pendulum. E q u a t i o n (If) t h e n becomes

p k 2 p k 2

- - 2 I~ s i n ~ 0 sin 2 q~ - - I~ sin 2 0

F r o m this it follows t h a t

~ 2 _ _ P k2 sin 2 ~ 0 + ~ I~ s i n 2 0

~o b e i n g t h e value of ~o when ~o = o.

F i n a l l y

f]/r

sin ~ cos ~o.

dq~

p k 2 . 2 1~ s i n ~/ q~

7. T h e case h ~ o . Geometrically, this is a p a r t i c u l a r case of P o i n s o t motion. A n a l y t i c a l l y , it is a p a r t i c u l a r case of t h e solution given by Euler.

I t seems of i n t e r e s t to n o t e the simplifications i n t r o d u c e d b y t h e special values of t h e m o m e n t s of inertia. I n this case it is no r e s t r i c t i o n to t a k e the a n g u l a r m o m e n t u m v e c t o r a l o n g t h e vertical OZ. This m a k e s 0 = -z a n d f r o m (Io)

2 '

= V I I T . From (9)

316

Equation (I

I)

Peter Field.

]~ x- x-

~ - - I 1 s i n ~ 0 - - 5 ' 0 : ~ t + ~0~

__ p X "~ p T

2 113 sin s 0 sin 2 q~ = 2~l sin 2 9~.

This equation differs from the one treated in paragraph six only in that sin * 0 has been replaced by unity.