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On a motion of an equilibrated gyroscope in the newtonian force field
ON A MOTION OF AN EQUILIBRATED GYROSCOPE IN THE NEWTONIAN FORCE FIELD (OB ODNOR
DVIZHENII UBAVNOVESBENNOBO V NIUTONOVSKOM POLE SIL)
PMM Vo1.27, Ill.
h'o.6, 1963, A.
QIROSROPA
pp.io99-lioi
ARKHANQEL' SKI1 (Moscow)
(Received
1..It
is
well
June
known [l, 21 that
solid
about
a point
force
field
are
fixed
in its
5,
the
1963)
equations
center
of
of
motion
gravity
of
a’ heavy
in the Newtonian
(1.1, with
the
particular
-f = A’&/’
solution
p,
y’
where A’, B’ and C’ are constants the dimensionless variables
p1
and taking
into
a-‘/p,
=
account
q1 =
(1.2)
C’a-‘lzr
y” =
B’&‘q,
zz
(a = 3glR)
to be determined.
a-‘/a
q,
we obtain
Introducing
rl =
a -‘/zr_
the
equations
1, =
a”l’t
(1.2) in
(1.1)
(1.3,
(1.4) dpx AQ(C-B)(l-B’C’) from which C-B A
41’1
we are
able
u+v--w=-
to
=
A’
0,
determine C-B A
C’Pl dl,
A’,
B’
(C’ -
+
and C’ as
A’ = v’zw
’
Since the determinant of the linear system is consistent, its solution
1684
B’) qlrl = 0
system
(1.5)
sgn u (1.5)
equals
zero
and the
A motion
v
U=l-a+wa,
depends
1 -
=
on one arbitrary
Let
us consider
the
u >G, which,
by (1.5)
in particular,
an
of
equilibrated
constant
occurs
the
Then the case tions
second
group
equations
of
Euler
possess A’pc
A’>B’>C’
condition
u > v >.
w
(1.7)
which
$
of
of
equations
motion
of
where A’,
B’
the
integrals
first
B’q12 $
C’r?
(1.8)
in
a solid
and C’
(1.4)
can be taken
about
a fixed
are moments of
A”p12
= D-l,
72 f Assuming
$
that
v-
2=
the
-
inertia.
in the
There
Bf2q12 + C’2r12 =
equa-
1
(1.9)
of integration notation
7’2 -f r”2 = 1
we can
> D,
express
/-
C’
DA’(A’__C’)cn
=I
~1 =
n (t1 -
to),
These
[31 the
formulas
for
pl,
and
q1
formulas,
ql=
D -
(A’ - D) (B’ DA’B’C’
together
with
C’
DB’(B’-_sSn
I/
C’) ’
(1.2)
r,
(.-1’ k2 = (B’ -
and (1.3),
&;;!C,)dnr
n ==
give
WP - C) C’)( A’ - D) the
(1 ,lo)
solution
of
problem. 2.
of
B’
as the
point
form D -
Pl =
satisfied,
C>B>A
where D is an arbitrary constant, and the second constant is set to be equal to unity (in the conventional Greenhill P2D2 = 1) because of the trivial integral of system (1.1)
r1 in the
is
when
w < 1,
classical
(14
case: w >o;
under
a d- b # 1)
10.
following v > 0,
1685
(a = A IC, b = B /C,
wb
b +
gyroscope
Let
time
us
the
assume that following
r10 is inequality
large
and that
at the
initial
instant
iS satisfied.
Further, t0 =0 .
assuming
that
y,,’
= 0,
we obtain
from formulas
(1.10)
that
1686
Iu.A.
Let
us introduce
the
small
Arkhangel
‘skii
parameter h = To” I rr,
The Quantities power
series
I -
u =
A’ =
h2 ( . . .),
[1L (1 -
b)]-’
n=
While
B’,
C’,
D,
~1, V,
(2.2)
10, n and k will
be expressed
as
in A
a +
Eulerian
A’,
+
‘g-$+
v =
l. (.
I-
A(.
b & hB ( . . .) ,w = Aa (1 -
B’ =
. .),
[h (1 -a)]-’
I(...),
ka = ka (1 -
. .),
investigating
+
the
motion
of
a) (1 c’=
~a”? (b To“0
a rigid
b)+h’(...)
lu,
4 f
(2.3)
D=h~,“-2+11s(...)
i4 (
body we shall
. . .
use
) the
angles
By (1.2), (1.3). (1.9) and (1.10) the first formula in (2.4) can be reduced to cos 8 = y. ” dn T, and by retaining only the first two terms in the expansion of dn T in series of powers of the small parameter k2, we obtain
by (2.3)
case= By (1.2), equation
(1.3),
in
(2.4)
(1 - -ro”‘?@- 4 hoea
To” I-aa [ (1.9).
(1.10)
cos
_
and (2.3)
B v = kam,
B=
’ “a 7,
m=1-_,
(2.6)
hD-1 - 1 1 - 7,“s = -
we obtain
6=
&
...
$
I
the
(2.5) second
1 + hP(1 -b)
= T@“--’ +
.
+
h (...,
(2.6)
[41 dt
= -
r,L
B sn E Ja = 2 cn E dn E
J1=iTBji_$$$$#]*
(2.7)
us set
8 (7 + E) = Retaining (2.7)
t)
0
we can reduce
$-&=6[J1r$ Jal*gB:z,-$, Let
2r
to
v SIP2
Integrating
(I
only
peix,e the
first
(7 -
(Ine0 (ttz -+ El~j = -
5) = pe-ix
terms
in the
expansion
in powers
2ix)
of
h in
we have
sn 5 0’ __-(8 cn E dn 5 0 (5) J,
=
1
zLa
-dkzm+..., (2 -
a -
x= b) +
. . .,
-f
I/m
(m +
Jz -= +
1) sin 27 + (,&)‘I
...
(2.8)
A motion
By (2.3). takes
the
(2.6)
of
an equilibrated
and (2.9)
the
formula
1667
gyroscope
for
the precession
angle
(2.7)
form (2.10)
Ilr-_0,=~~‘iah(2-a-b)t-_b’b~~~~sin2~,t+... By (1.10). (2.4)
the
(2.5)
following
and (2.10)
we obtain
expression
for
q__5_ In formulas cos
8,
vyO* (system which
(2.5).
= y,”
(1.1) is
ro -
is
[b -
(2.10)
and r.
related
y$
the
a $
from the angle
~~74
and (2.11)
-
two formulas
arbitrary
constants
autonomous)
we can add a fourth
arbitrary
strength
through
of
(2.11)
(2.11).
appear:
t + h for
to
h on the
in
rotation:
3b)]} t + . . .
a -
three
last
natural
By substituting
is large).
(r.
of
t
constant the
qo,
following
formula:
To =
Let
us note
that
the
angles
8,
las
in the
case
the
small
the
functions
sn
Let
T,
cn
O,, = -
$. == -k ar’/‘A (2 Consider
tant
8,,
from the middle
of
not
then
D will
also
be independent
T
and (2.10)
sphere.
a rigid
will
(t + h) +
. . .,
Euler
on h.
of
body,
centered
rectangle
on the
made up of angles
meridian
y.
f s sin by the
6,
point,
angles
and of f s.
of A.
using
(2.5).
(2.10)
form
810=0,+
fixed
Con-
A and the Jacobi
in the
two parallels
formu-
we introduce
in powers
be put
determine
corresponding
depend
be expanded of
which
s sin 2r, (t + h) + . . ., (s = h2 (b -
b) t -
by the
cannot
the motion
ssin6,cos2ro
a unit
case
ro,
(2.5)
a-
from the
in the
k2 will
and dn
T
a spherical
middle. parallel
substantially if
(2.12) and (2.11)
(2.10)
Indeed,
h = y,“/
quantity
formulas
6 -
surface
Euler.
us now consider
and (2.11);
9 -
of
r,h + . . .
(2.5),
y and 9 differ
parameter
sequently,
3.
formulas
5$
ssin6,+ a)/4cos 0,) (3.1)
and form on its distant
from the
two meridians Then,
the
dis-
trajectory
on our unit sphere rotating with the con(61, I+JI) traced by the z -axis stant angular velocity i, = l/2 s l’21 (2 - a - b) about the fixed axis II'
is the
ellipse (3.2)
which
is tangent to the spherical rectangle at the midpoint of its the z-axis performs in the first When tracing this ellipse, sides. approximation a periodic motion with the period T = w/ro, and at the instants
of
time
1688
1u.A.
J-t
t1*-
(nlf
3/2) Zr,
Q,
~~
Arkhangel’skii
2-c(n1 +
~-
,
f,*
the z-axis will pass through the parallel with the outer meridians section
of
the middle
The natural very
little
Let
(yo =
Then,
meridian
rotation
from
of
formulas
from the
integrals
yo’
=
at all
values
velocity
r.
of about
The case
fixed
y. ” = 0 reduces
no interest,
or
(if
investigated
under
= 0)
C’
ye”
large
the middle
that
= q.
body rotates
(2.11)
angular
= 1 and ye”
follows
axis
of
differs
with
p.
(1.9)
intersection
(3.3)
as shown by formula
show that
and the
the
of
+ 1, + 2 ,...)
of
rotation
(1.2)
time,
(n1= 0,
and through the points the outer parallels (t2*).
with
now the cases
0)
2q1,
points (fl *)
the body
the uniform
us consider
1) “r”
~-
= 0. = 0
inter-
velocity
When y,” # 0.
(A’
rO. = 1 # 0).
B’
p = q = 0 (y = 0,
y’
with
angular
the
constant
= 0)
xl.
to
the
case
r.
to
the
case
p = q = y”=
more general
= 0 (if
conditions
C’
[d.
in
f 0) which 0,
is
which has
of
been
BIBLIOGRAPHY 1.
Beletskii, tela
novskogo motion force 2.
zakreplennoi polia
of
Stekloff,
Iu. S.,
zheniami
5.
Nauk
point Vol.
SW?,
135,
nrekhanika
the
113,
No.
de Clebsch pp.
et
niuto-
equations
in central
indbfini
Vol.
rendus,
tverdogo
2,
(Theoretical
1957.
sur
sur le
526-528,
of
Newtonian
le
mouvement
problbme
de
1902. Vo1.2.
Mechanics),
1960.
Sikorskii, tions
a fixed
dans un liquide
Conptes
dvizhenia tsentral’nogo
of
Remarque sur un probl&me
Teoreticheskaia
Fizmatgiz, 4.
Akad.
uravnenii pod deistvem
integrability
body about
solide
, P.,
tochki
(On the
Dokl.
V. A., corps
M. de Brun. Appel’
sil
a rigid
field).
d’un
3.
Ob integriruemosti
V. V., okolo
Elementy
k aekhanike
with
Application
teorii
(Elements to
ellipticheskikh of
Mechanics).
the
Theory
funktsii of
the
s prilo-
Elliptic
Func-
ONTI, 1936.
privedennogo v bystroe vrashchenie Arkhangel’ skii. Iu. A., 0 dvizhenii tiazhelogo tverdogo tela vokru’g nepodvizhnoi tochki (On the motion of a rapidly rotating heavy solid about a fixed point). PM Vol. 27.
No.
5.
1963.
Translated
by
T.L.
DVIZHENII UBAVNOVESBENNOBO V NIUTONOVSKOM POLE SIL)
PMM Vo1.27, Ill.
h'o.6, 1963, A.
QIROSROPA
pp.io99-lioi
ARKHANQEL' SKI1 (Moscow)
(Received
1..It
is
well
June
known [l, 21 that
solid
about
a point
force
field
are
fixed
in its
5,
the
1963)
equations
center
of
of
motion
gravity
of
a’ heavy
in the Newtonian
(1.1, with
the
particular
-f = A’&/’
solution
p,
y’
where A’, B’ and C’ are constants the dimensionless variables
p1
and taking
into
a-‘/p,
=
account
q1 =
(1.2)
C’a-‘lzr
y” =
B’&‘q,
zz
(a = 3glR)
to be determined.
a-‘/a
q,
we obtain
Introducing
rl =
a -‘/zr_
the
equations
1, =
a”l’t
(1.2) in
(1.1)
(1.3,
(1.4) dpx AQ(C-B)(l-B’C’) from which C-B A
41’1
we are
able
u+v--w=-
to
=
A’
0,
determine C-B A
C’Pl dl,
A’,
B’
(C’ -
+
and C’ as
A’ = v’zw
’
Since the determinant of the linear system is consistent, its solution
1684
B’) qlrl = 0
system
(1.5)
sgn u (1.5)
equals
zero
and the
A motion
v
U=l-a+wa,
depends
1 -
=
on one arbitrary
Let
us consider
the
u >G, which,
by (1.5)
in particular,
an
of
equilibrated
constant
occurs
the
Then the case tions
second
group
equations
of
Euler
possess A’pc
A’>B’>C’
condition
u > v >.
w
(1.7)
which
$
of
of
equations
motion
of
where A’,
B’
the
integrals
first
B’q12 $
C’r?
(1.8)
in
a solid
and C’
(1.4)
can be taken
about
a fixed
are moments of
A”p12
= D-l,
72 f Assuming
$
that
v-
2=
the
-
inertia.
in the
There
Bf2q12 + C’2r12 =
equa-
1
(1.9)
of integration notation
7’2 -f r”2 = 1
we can
> D,
express
/-
C’
DA’(A’__C’)cn
=I
~1 =
n (t1 -
to),
These
[31 the
formulas
for
pl,
and
q1
formulas,
ql=
D -
(A’ - D) (B’ DA’B’C’
together
with
C’
DB’(B’-_sSn
I/
C’) ’
(1.2)
r,
(.-1’ k2 = (B’ -
and (1.3),
&;;!C,)dnr
n ==
give
WP - C) C’)( A’ - D) the
(1 ,lo)
solution
of
problem. 2.
of
B’
as the
point
form D -
Pl =
satisfied,
C>B>A
where D is an arbitrary constant, and the second constant is set to be equal to unity (in the conventional Greenhill P2D2 = 1) because of the trivial integral of system (1.1)
r1 in the
is
when
w < 1,
classical
(14
case: w >o;
under
a d- b # 1)
10.
following v > 0,
1685
(a = A IC, b = B /C,
wb
b +
gyroscope
Let
time
us
the
assume that following
r10 is inequality
large
and that
at the
initial
instant
iS satisfied.
Further, t0 =0 .
assuming
that
y,,’
= 0,
we obtain
from formulas
(1.10)
that
1686
Iu.A.
Let
us introduce
the
small
Arkhangel
‘skii
parameter h = To” I rr,
The Quantities power
series
I -
u =
A’ =
h2 ( . . .),
[1L (1 -
b)]-’
n=
While
B’,
C’,
D,
~1, V,
(2.2)
10, n and k will
be expressed
as
in A
a +
Eulerian
A’,
+
‘g-$+
v =
l. (.
I-
A(.
b & hB ( . . .) ,w = Aa (1 -
B’ =
. .),
[h (1 -a)]-’
I(...),
ka = ka (1 -
. .),
investigating
+
the
motion
of
a) (1 c’=
~a”? (b To“0
a rigid
b)+h’(...)
lu,
4 f
(2.3)
D=h~,“-2+11s(...)
i4 (
body we shall
. . .
use
) the
angles
By (1.2), (1.3). (1.9) and (1.10) the first formula in (2.4) can be reduced to cos 8 = y. ” dn T, and by retaining only the first two terms in the expansion of dn T in series of powers of the small parameter k2, we obtain
by (2.3)
case= By (1.2), equation
(1.3),
in
(2.4)
(1 - -ro”‘?@- 4 hoea
To” I-aa [ (1.9).
(1.10)
cos
_
and (2.3)
B v = kam,
B=
’ “a 7,
m=1-_,
(2.6)
hD-1 - 1 1 - 7,“s = -
we obtain
6=
&
...
$
I
the
(2.5) second
1 + hP(1 -b)
= T@“--’ +
.
+
h (...,
(2.6)
[41 dt
= -
r,L
B sn E Ja = 2 cn E dn E
J1=iTBji_$$$$#]*
(2.7)
us set
8 (7 + E) = Retaining (2.7)
t)
0
we can reduce
$-&=6[J1r$ Jal*gB:z,-$, Let
2r
to
v SIP2
Integrating
(I
only
peix,e the
first
(7 -
(Ine0 (ttz -+ El~j = -
5) = pe-ix
terms
in the
expansion
in powers
2ix)
of
h in
we have
sn 5 0’ __-(8 cn E dn 5 0 (5) J,
=
1
zLa
-dkzm+..., (2 -
a -
x= b) +
. . .,
-f
I/m
(m +
Jz -= +
1) sin 27 + (,&)‘I
...
(2.8)
A motion
By (2.3). takes
the
(2.6)
of
an equilibrated
and (2.9)
the
formula
1667
gyroscope
for
the precession
angle
(2.7)
form (2.10)
Ilr-_0,=~~‘iah(2-a-b)t-_b’b~~~~sin2~,t+... By (1.10). (2.4)
the
(2.5)
following
and (2.10)
we obtain
expression
for
q__5_ In formulas cos
8,
vyO* (system which
(2.5).
= y,”
(1.1) is
ro -
is
[b -
(2.10)
and r.
related
y$
the
a $
from the angle
~~74
and (2.11)
-
two formulas
arbitrary
constants
autonomous)
we can add a fourth
arbitrary
strength
through
of
(2.11)
(2.11).
appear:
t + h for
to
h on the
in
rotation:
3b)]} t + . . .
a -
three
last
natural
By substituting
is large).
(r.
of
t
constant the
qo,
following
formula:
To =
Let
us note
that
the
angles
8,
las
in the
case
the
small
the
functions
sn
Let
T,
cn
O,, = -
$. == -k ar’/‘A (2 Consider
tant
8,,
from the middle
of
not
then
D will
also
be independent
T
and (2.10)
sphere.
a rigid
will
(t + h) +
. . .,
Euler
on h.
of
body,
centered
rectangle
on the
made up of angles
meridian
y.
f s sin by the
6,
point,
angles
and of f s.
of A.
using
(2.5).
(2.10)
form
810=0,+
fixed
Con-
A and the Jacobi
in the
two parallels
formu-
we introduce
in powers
be put
determine
corresponding
depend
be expanded of
which
s sin 2r, (t + h) + . . ., (s = h2 (b -
b) t -
by the
cannot
the motion
ssin6,cos2ro
a unit
case
ro,
(2.5)
a-
from the
in the
k2 will
and dn
T
a spherical
middle. parallel
substantially if
(2.12) and (2.11)
(2.10)
Indeed,
h = y,“/
quantity
formulas
6 -
surface
Euler.
us now consider
and (2.11);
9 -
of
r,h + . . .
(2.5),
y and 9 differ
parameter
sequently,
3.
formulas
5$
ssin6,+ a)/4cos 0,) (3.1)
and form on its distant
from the
two meridians Then,
the
dis-
trajectory
on our unit sphere rotating with the con(61, I+JI) traced by the z -axis stant angular velocity i, = l/2 s l’21 (2 - a - b) about the fixed axis II'
is the
ellipse (3.2)
which
is tangent to the spherical rectangle at the midpoint of its the z-axis performs in the first When tracing this ellipse, sides. approximation a periodic motion with the period T = w/ro, and at the instants
of
time
1688
1u.A.
J-t
t1*-
(nlf
3/2) Zr,
Q,
~~
Arkhangel’skii
2-c(n1 +
~-
,
f,*
the z-axis will pass through the parallel with the outer meridians section
of
the middle
The natural very
little
Let
(yo =
Then,
meridian
rotation
from
of
formulas
from the
integrals
yo’
=
at all
values
velocity
r.
of about
The case
fixed
y. ” = 0 reduces
no interest,
or
(if
investigated
under
= 0)
C’
ye”
large
the middle
that
= q.
body rotates
(2.11)
angular
= 1 and ye”
follows
axis
of
differs
with
p.
(1.9)
intersection
(3.3)
as shown by formula
show that
and the
the
of
+ 1, + 2 ,...)
of
rotation
(1.2)
time,
(n1= 0,
and through the points the outer parallels (t2*).
with
now the cases
0)
2q1,
points (fl *)
the body
the uniform
us consider
1) “r”
~-
= 0. = 0
inter-
velocity
When y,” # 0.
(A’
rO. = 1 # 0).
B’
p = q = 0 (y = 0,
y’
with
angular
the
constant
= 0)
xl.
to
the
case
r.
to
the
case
p = q = y”=
more general
= 0 (if
conditions
C’
[d.
in
f 0) which 0,
is
which has
of
been
BIBLIOGRAPHY 1.
Beletskii, tela
novskogo motion force 2.
zakreplennoi polia
of
Stekloff,
Iu. S.,
zheniami
5.
Nauk
point Vol.
SW?,
135,
nrekhanika
the
113,
No.
de Clebsch pp.
et
niuto-
equations
in central
indbfini
Vol.
rendus,
tverdogo
2,
(Theoretical
1957.
sur
sur le
526-528,
of
Newtonian
le
mouvement
problbme
de
1902. Vo1.2.
Mechanics),
1960.
Sikorskii, tions
a fixed
dans un liquide
Conptes
dvizhenia tsentral’nogo
of
Remarque sur un probl&me
Teoreticheskaia
Fizmatgiz, 4.
Akad.
uravnenii pod deistvem
integrability
body about
solide
, P.,
tochki
(On the
Dokl.
V. A., corps
M. de Brun. Appel’
sil
a rigid
field).
d’un
3.
Ob integriruemosti
V. V., okolo
Elementy
k aekhanike
with
Application
teorii
(Elements to
ellipticheskikh of
Mechanics).
the
Theory
funktsii of
the
s prilo-
Elliptic
Func-
ONTI, 1936.
privedennogo v bystroe vrashchenie Arkhangel’ skii. Iu. A., 0 dvizhenii tiazhelogo tverdogo tela vokru’g nepodvizhnoi tochki (On the motion of a rapidly rotating heavy solid about a fixed point). PM Vol. 27.
No.
5.
1963.
Translated
by
T.L.