Collection of Problems in Classical Mechanics
5,2
5.2. Find the frequency of the small oscillations for the system depicted in Fig. 1 1 . The system rotates with an angular velocity l 2 in the field of gravity around a vertical axis. 5.3. A point charge q of mass tn moves along a circle of radius R in a vertical plane. Another charge q is fixed at t h e lowcst point OF the circle /Fi_g. 12). Find the equilibrium position and the frequency of the small oscillntions for the first point charge.
Problems 5.8. Find the eigen-oscillationsof the system in Fig. 14 in a unifnrni field of-qaviQ for the case when the particle can only move vertically-. 5.9. Find the stable small oscillations of a pendulum when its point of moves unifomly along a circle of radius R with frequency 9 (Fig. 15). The pendulum length is / [ I >> R).
A
1
1
rn
P
.
m
5.4. Describe the motion along a curve close to a circle for a point particle in the central field U(r) = - all" (0 -= n < 2). 5.5, Find the frequencies of the small oscillations of a spherical pendulum (a particle of mass m suspended from a string of length I), if the angle ordeffectiorr from the vertical oscillates about the value On. 5.6. Find the correction to the frequency of the srnaEL oscillations of a diatomic molecule due to its angular momentum M . 5.7. Determine the eisen-oscillations of the system shown in Fi9. 13 for the case when the particle movcs (a) horizontally, and (b) vertically . How does the frequency d e p n d on the tension in the springs in the
equilibrium position ?
5.10. Find the stable o~cillationsin the voltage across a capacitor and the current in a circuit with an e.m.f. CT(I) = UOcos CUI (Fig. 16).
Collection of Problems in Classical Mechanics
5.19
Problems
6.5
(c) Find the stablc current through rhe circuit of Fig. 16 in which there is an e.m.f. U ( I ) = V ( t / z - n ) for nt 5 r -C (nt l ) t . The internal resistance of the battery is zero.
6.3. Find the eigen-vibrations of a system described by the Lagrangian
5.20. An oscilIator with eigen-frequency w o and with a friction force acting up011 it given byhr = - 2 m 2 i has an additional force F(t) acting
What is the trajectory of a point with Cartesian coordinates x, y? 6.4. Find the normal coordinates of the systems with the following
upon it. (a) Find the average work done by F ( t )when the oscillator is vibrating in a stable mode for the case when
L
~agrangians:
(b) Repeat the calcuIations for the case when
( c ) Find the average over a long time interval of the work done by the
force F ( t ) = fl cos rijlr -If~ - cos W Z I ,
when the oscillator performs stable vibrations.
Id) Find the total work done by the force y( -w) = y8(w)
for the case where the oscillator was a t rest at
t =-
+
L = +(k2 +y=)--;(49 +?) +,xy ; (b) L = a ( t n l i 2 + m ~ 2 ) + j i j - + ( x Z + y a ) . (a)
21a and b).
- y(w) e""'dw,
dg2).
6.5. Find the eigen-oscillations of a system of coupled circuits (Figs.
F ( I )= fl cos wt tf2 cos 2wt.
+
= 8 x 2 4 -y'L)--+(cu;x=L
-
6. SMALLOSCILLATIONS OF S Y S ~ M WITH S SEVERAL DEGREES OF FREEDOM
6.1.Find the normal oscillations of the system of Fig. 19 for the case when the particles can move only vertically. 6.2. Three masses which are connected by springs move along a circle (Fig. 20). The point A is fixed. Find the eigen-vibrations of the system. Find the normal coordinates and express the Lagrangian in terms of those coordinates.
-2 2 d
M
CI*
5
O
I2
--Zu F* .E j $ +o s E
ID"
C
0
*
8 .-E g 3,
2 5 ~ 2h0. 2o
z
- g g"
LC-
0
C
,
N % w ? . J 5 c
EX
%ZEF+%Z v r w t 3 u s f e * r" "y
-? 5
g .L?
-s
2z w4 3
-
r2
62%
--
2 .s * €
0
?-'E;C" u-'. 0"lQ
.z
"3:,.
4 3 .
2
0
0
0 *'-@ t ; bO4
2;
Z L
4a2
F-
Gc 5U uS as
G Men
L0 a 2 0.B: E d ? ;"2 ET
J=
c
~
A 3
5:
CLI
E
g < r L z.5 '-*O g =. -3~e%a rgt gh
5%
-E G
2 .L_5
v:f=
-z 0
R h
s5 2rz-=3
-6
6) u u 2 2 3
L.
U
,,m
ry
z
33 -2 .u.5 2.2
.o C
b-.
; ..9
cl & '
0 5 ZG
m u 2 5 5 <
.+ 2 = . 5 0 2 +.
-L -= m -2 + &
6
--
L4sv2CL~g
q
y q
6 g W ?5;
C
" E0, ea ~ ~ g . 5 .cI
/
I.'
Coltection of Problems in Classical Mechanics
6.32
Problems
7.7
(a) Show that if the eigen-oscillations or I he system are non-degenerate. the amplitudes of the oscillations of the diikrent particles are distributed either symmetrically or antisymmetrically with respect to the centre. (b) Show that if there is degeneracy, one can always choose the normal oscillations such that t hey are either symmetrical or antisymmetrical.
6.33. Show that if the points A and B (Fig. 33) move symmetricaIly o r antisymmetrically, several o f t he eigen-vibrations of tllc system or Fig. 33 will not be excited. 6.34. Use symmetrq. considerations to find the vectors of the normal vibrations o f the system of particles of Fig. 26. 631. Find the stable oscillations of the system of Fig. 23, if the point A moves according t o acos cot. Assume that there are frictional forces acting upon the particles p r o p ~ ~ n nto a ltheir velocities. 6.36. Describe the motion of the system o f Fig. 22 if a t t = 0 the particles are at rest at their equilibrium positions while t he point A mnlmc< according to a cos cor. Take the masses to he equal (ml= rnn = m). 6.37. Determine the tip-vibrations in 3 planc of n molecule which ha< the shape of an equilateral triangle. Assume than the potential encrgr; depend5 only on \he distances between the atoms and that all atoms arc the same. The a n s u l ~ rmomcntum is equal to zero u p to tcrms sf first order in the amplitude of the oscillations.
6.,%, Use symmetry argumentsto determine the degree of degeneracy OF various frequencies for the case of n "molecule" consisting of four identical atom$ which has the form of a recular tetrahedron at equilibrium.
7.3. Fjnd the eign-vibrations of N particles which are connected hy springs and which can move along a circle (Fig. 33c). All particles and the elastic constants of all springs are the same. Check that if the motion i s that of a wave travelling along the circle, the energy flux equals the product of the linear energy density and the group velocity. 7.4. Deternine the frequencies of the eigen-vibrations of a system of particles moving dong a straight line for the following cases (take the hint of problem 7.1 into account): (a) 2N particles, alternating with rnasces rn and M connected by springs of elastic constant x {Fig. 33d); Ib) 2 N particles of mass m connected by springs with dternating elastic constants x and K (Fig. 33e); (c) 2N-t- 1 particles of mass rn connected by springs with alternating elastic constants x and A' (Fig. 33f). 7.5. (a) Determine the eign-vibrations o f the system of Fig. 33a if the point A rnevcs according to a cos mi.
rb) Do the same for the system of Fig. 33b.
Chains of particles connected by springs are the simplest n-rndel used in the theory of solids (see, for example, Wannier, 1959, or Kittel, E96X).The eIectrical analogues of such linec: are r.f. lines employed in radio engineering.
7.1. Bterrnine the frequencies of the normal oscillations of a system of h; identical particles with masses rn connected hy identical sprinps with elastic conctants x and rnovin~alony a straight line (Fig. 33a). Hin!: E x p s r the n o m d oscillations in terms of standin," waves.
7.2. R e p a t this For the system of Fig. 33b with one free end.
7.6. D o the same for the system of Fig. 33d.
7.7. Determine the ieigqe-v&brations of a sysiem o f A panicles which move along a straight line for the following cases:
.. ..N- F ; the elastic conqtants of all the are ~ the J same (Fb.34). Discuss the cases when mnN s-.m and
(a) ml = m # m,v, i = I , 2, S
P
~
-
W N- m ;
(b) xi = x /Fig. 35). -d
St'dy
%?+I.
i = 1.2. . . .,N ; all the rnavsef are the same
the CasH when z,,,
=+ rr and
when q,;, cc z.
u
5 2 C
*
-
3
d
S
.5 L
0
'A d
E; C1
3E
F3
.-S, 'J
C
.3
& 0 5
7 C
d rl)
rd
K
m T:
.-CLE .5
..
3
5 7
h
-.=2 El) .":
a
3
N. 0 m5
r=:
Collection of ProbTems in Classical Mechanics
'J
9.8
9.8. An isotropic ellipsoid of revolution of mass rn moves in the gravitational field produced by a fixed point of mass A{. Use as generalised coordinates the spherical polar coordinates of the centre of mass and the Euler angles and determine the Lagrangian of this system. Assume the size of the ellipsoid to be small compared to the distance from the centre of the fieEd. Hitrt: The potential energy of the system is approximately equal to
where R = (XI, X2, X3j is the radius vector of the centre of the ellipsoid, DEpthe mass quadrupole moment tensor (see problem 9 . 3 , and F(R) = -y M I R is the potential of the gravitational field (compare Landau and Lifshitz, 1962,$ 41). 9.9. Write down the equations of motion for the components of the angular momentum along moving coordinate axes w h c h are chosen t o lie aIong the principal axes of the moment of inertia tensor. Integrate these equations for the case of the free motion of a symmetric top. 9.10. Use the Eules equations lo study the stability of rotations around the principal axes of the moment of inertia tensor of an asymmetric t o p 9.11. (a) A plane disk, symmetric around its axis, rolls over a smooth horizontal plane without friction. Find i t s motion in the form of quadrature~.
9.14
Problems
Answer in detail thc following questions: Under what conditions does the angle of inclination of the disk to the remaill constant ? ~f the djsk rolls in such a way that its axis has a fixed (horizontal) direction in space, at what angular velocity will the rotation around this ads k stable? Ifthe disk rotates around its vertical diameter, at what angular velocity willthe motion be stable? (b) A disk rolls without slipping over a horizontal surfacc. Find the ,quatiom of motion, and answer for this case the same questions as under (a). (c) Repeat this for a disk which rolls without slip omr a horizontal plane, without rotation around a vertical axis? (d) A disk rotates, without slipping, around its diameter which is at right angles to an inclined plane, which makes a small angle x with the horizontal, on which the disk is placed. Find the displacement of the disk over a long time interval. 9.12. (a) Find in terms of quadratures the law of motion of an inhornogeneous sphere which is slipping without friction on a horizontal plane. The mass distribuzion is symmetric with respect to the axis passing through the geometric centre and the centre of mass of the sphere. Study the effect of small dry friction forces for the case when the motion of the sphere, if the dry friction forces are neglected, would be such that the angle ktween the vertical and the axis of symmetry remained cunstant. IbS Find the equations of motion for the sphere of (a) if it rolls without slipping over a horizontal plane. (c) Find the equations of motion of this sphere for rolling without slipping down an inclined surface. 9.13. A particle is dropped from a height 11 with zero initial velocity. Find its displacement from the vertical in the directions of the West and the South. 9-14. A partjcle moves in a central field potential U(r). Find the equation for its trajectory and describe its motion in a coordinate system which is rotating uniformly with an angular velocity 53 pasallel to its angular momentum M.
t This me$ns that the cohesion ofthe disk to the plane at the point ofcontact is SO firm that the area of cootact neither slips not rotates. Thc energy loss due to rolling friction can bc neglected.
Answers and Solutions Collection of Problems in Classical Mechanics
2.33
and thus
and averaging this equation leads to (B) = [BA 01,
/
where
eM
0=
dpzI = m2tan Q 8
do =
a
-
{n' -3n(n. n'))
with (b)
tan Hence we have
-~
1 do - b 8= 2 dr-a -
~
-
a
when
0 -z 8 c 0,";
0,
when
0,
( 0,
(7)
z COS
)a2sec' +B d b ,
I f b(a
-= 8.
-b) d2wlsinZa 0 sin 0,
when
0 c 0 c 8, = 2 arctan
when
0, < 0.
(c) da = A-Z/("-l)(n tan QB)("+')/("-'l d2w/2(n-l) sin 8 cos2+8.
that is, 0 is the precessional velocity of the orbit.
3.l.(a) It can be seen from Fig. 78 that the angle of deflection of a particle is twice the angle of the slope of the tangent to the surface of revolution at the point of collision. We have therefore
[
I
In deriving (5) we have used the fact that (rip) will be along the major axis of the ellipse, that is, parallel to B. From (5) wesee that the vector B rotates with constant magnitude with an angular velocity 0. Equation (3) can be written in 'he form
(M)= IMAS-21,
'
-
da=
n' = -,!/~IIJ~I.
te
The possible deflections of the particle lie within the range of angles from zero, as e b, to 8, = arctan (bla), as e 0. We have thus
mc8(l -r2)X = MliMl,
do - a2d2w 4 COS' c0sZ+8
i, 1
I
3.2. It is the paraboloid of revolution e2 = a.z/E. The reader should check whether the trajectories of the particles scattered by the potential U = -a/r and those of the particles scattered by the paraboloid approximate one another as r --. 3.3.
do =
I'
a2n2
+e
~ C O S
lo,
(ncos@-I)(n-ws +O) i t s ]d ~ w , (I +nz-~n C O S + O ) ~ . when 0 c 0 c 0 , = 2 arccos n, when 8, c 0 < n, . :
where n = d m ) .W h y is there a dSerence between this scattering cross-seaion and the one for scattering by a potential well (see Landau and Lifshitz, 1960, 8 19, problem 2)?
3.4. (a)
I0,
'
when
E c a2/2p.
How does the cross-section change, when a changes sign?
u
k..z
aJ
52 2
.-C ,z
9'L ."g E-
0
"lo; a-rc
fl>&
L
o "b, iZ
.-2 5I1 . c5
-
3 E k e
5 2; aJ &
.U12 0
@I
E k G
-m - 2 E y
8 2 ~ 2 a
LJ=
E 03zl c E.E "E.E 2 I
0
3
"VE
-Z e -3 m s ,a's c, '2 2
0
1 IFZU
srz
I
Collection of Problems i n Classical Mechanics
Answers and Solutions
3.1 f
3.10
roots when
-= H,,
8
=-
2e
E-
'
Using equation ( I ) and the relation
we can write the expression for the cross-section f l f l = = " ( l d e : + ~ d p= l ' )-2n ---(X~~.T~--X~C/.Y~), 2 in the form do =
I
I
x:
-
I
FIG.84
When '<< O m - it turns out that XI c< I and q << 1, so that do = dZw/29p. When em-8<<8~n we can solve ( 1 ) b y expanding xe-r' in senenneilr the maximum. We then get XI "- =
.
I r2/OIfTo
and hence
d2u
do =
V'=
v ( r-2xz nF.
-2sz
(1)
I
--
2 z / i ~ 1/9,,> ~;, Figure 84 shows dfl/d2was a function of 0. The sinpIarity at 6 = Q,,,is problem 3.8). Discuss whether the presence of the in the cross-s~tionat 0 = 8, is connected with the approximations made in the solution of the problem, '
1!
~1
I
d2
~ t a hvelocity Y before the collision will have a 3.11 (a) A v.- 2dn V ) after the collision, where n is a unit vector = :locity = to the surface of the ellipsoid. Using the relations' hich is (o,o. I 1, n = N - (~[ x l a 2 ] ,[yl'W. trlc23)we get
d h the axis along we can wite ' = polar v(sin6 cos q, sin 0 sin q ~cos , R), and from (1) we then have 2z2 22 t a n v = a ~ ,cosO= s i n l o = N2c2 (2) bx N2i?
(-)2($+i+)*
For fie cross-section we have
where Xl. -
When
O=O,,=--
xV 3 ~ f l
2
- 2EB
we haye
V d* ~ do= 4;?2E03'
equation for the ellipsoid. To evaluate the Iacobian it is convenient to introduce an auxiliary variable U such that x = a 2 ~ ~ ~ ~y = v ,bWsinq.
* We know from differential geometry that na v ( x t / d +g/bet:?/r4- l), while N is determined by the relation ne = 1. 119
-..
Y
Ccl
0 U
.-
T l m
'0 C
rJ
-F
z
err -L
-F -.4
C
0 3
&
C
Y
2 YII L
wr c
.+
c-*'
kh
2% I .bG
F:f2 2!
CU
w
m
II
5 I*
.
1' %
p 2 u II
,-
+
.1"
-v 7.-, $ ' "
w
e 1 5 L 1,
ti
I";
0:* 2-a 4
2 Q
c m 2 m c s c *
-$ 'J
-2
a,
;p
ao;
-5G zc
u
qz 3 3 5 . $ 2 ,=E7
=
a m
4 8 3-
Y > d
l1 = O K , '$Em2
8 2 -4~ 22.2 ~ 5 7
h
-- & = U
V
=
u w : ,'"
h
-1
0.2-
0.2
-;c 2 0
2 ~
Y..r: y
2az ..= rl 5 ~5 ;'
R w-
T-. 30 E wc
e
E$4="
II
I
2 3 E
SuQJ2
Y
V
a'& rD !a
- -g
a.E
I1 -2
I
II
-5 -9 -2
--
r:-
W!
..
. ;
:g ": bp
z5.5
5 5o 2g
-? b8,5 n
a
E
3
I 5 - a E
+
rd
C
L
s m O ma*;'
3 k' En
r'
b r L ,
m E: Y e )
5
:~ 2 c.
0
0-
2 lLT? 5 ,. c,.4 5 - 7 3 2
m m
2
3"
a % 5
- rr a ~$2; 2E
ip"=%-
0
V1
5.P U
PA 2
s ac+-M 2T.
=
Tp
g. z
1
II
3 % k7
7
"-:- 2 . : G
, \ 2+-'- z s
F-
w, *-
F <
2E U E ;11
- .07 3 (P
2
Ed
Z N _ o
"
C-
?;a
- 9
-y _ q E
c
g W F C 2 2 g
E-u 5 Z
K
2
%
3'"sp. *- g E' " 3 3,
~
I
-
d
3
.
-;,' *
-14 rb
c4A U
I N
el-
.?
IG
-> -tl
--.. vr
IN I1 *,
.-
4
g
$
E " 0
.A
.5
5-g
eW g 5% L
. O E j
$ a ,
g.2 .-IZ .% e 8 5.; L
2 - a ~ O l u r J
2dw
5g ;;:
n
L.
.." .U .-B fJ
(=
La T-r
.S
=I a
n o g Z U m (J
u
-5-" 6,
-
7 2 2
zz gej 03'
e o E
OJ
-a& 3
.E
Collection of Problems in Classical Mechanics
6.37
if we include terms which are quadratic in the u,, [he angular nlomcn. turn
(MI = m x [ u =A k l = ml424s-"rscF7l
112)
cnn be non-zero. if there i s a phase difference between the oscillations q, and q ~ . It is interesting to analyse what changes are introduced i l we consider now the possibility that the potential enerFy may depcnd on the angcJ b e h m n the bands. It is clear that svch a dependence will not aRcCt the fwquency o f t he q, oso'llation. The frequencies of the q 2 and ql orillation will be changed. but a twofold ckpeneracy will remain. Indeed. together with a q-oscillation we can also have another oscillation obtained from the original q-oscillation by a rotation over 12W. fir frequency m u a bc r l ~ esame as that of the original oscillalion. On the other hand, the 4-coordinate will diRer (only the yl oscillation remains the same under a rotation over 120"). We find thus two independent oscillations with the s a w frequency. The normal coordinates mus! in this case satisfy only one condition: they must be orthogonal to q,; in particular, q, and qs remain nor-
mal coordinates. 6.38. Let the oscillation for which the molecule retains its $ h a p (Fir. 1 0 2 4 have a frequency m l .
Answers and Solutions
7.q
'
The frequency CO, of the oscillation which retains its Form under rotations around OD over k 1 3 (Fig. 102b) is, in general, different from wa. One can obtain another displacement of the atoms by taking a reflection in the plane BCO; we obtain oscillations which differ from the second only in that atoms. A and D change roles. The frequency of that oscillgtionm3 = oz,Similarly, in the reflection in the plane AOC the roles of atoms R and D are changed, while the frequency remains the same,
,,,
= roz. This fourth osciIlat ion cannot be reduced to a su~rposition the previous ones as, in contrast to those, it is not ~rtI'm?etricwith respxt to the pjane AOD. The oscillation which is symmetric with respect to the planes AOR and DOC (Fig. 102c) has a frequency (us which i s direrent Sron~oland 1,. A rotation over an angle 2 ~ , ' 3around OD which results i n a cyclic permutarion of A, B, ant! C,leads to an oscillation which is symmetric *ith respect t o the planes COA and DOE, and its frequency w, = ws. The molecule has thus three eigen-frequencies wliich are, respectively, non-degenerate and twofold and threefold degenerate. In conclusion we note that the molecules considered in problems 6.37 and 6.38 are, clearly, not to be found i n nature. However, a similar approach can be used also for real molecules.
w4
,f
7. O~C~~LATIOKS aF L ~ E A CHAINS R 7.1. The Lnprnngian of the system is
where the x,are the displacements of the nth mass from its equilibrium position. We also introduce the coordinate of the equilibrium position of the hth mass, y, = !la, where u is the equilibrium length of one spring. The Lagrangan equations of motion
are equivalent to the set
with the additional condition Xg
= X,v*1
= 0.
5 x 'I%
2 VJ
C L)
. c T
i 2
.
w
+ (0
WII
2
z
N
'NO
C,,
0 0
3 g 27 a,
--2
3
E .S 2
w &
C)
% 2 z E
= :; =l
5 2 C
g$z
0-"2 8 .E g t z 3-2:
.-g
5
2 ;j
0
f l a w 3-
g.5
s
0
e 2 : m <2
W
-2 '3; - * C
F
h
E gs-.
Answers and Solutions
Collection of P~oblernsin Classical Mechanics
The set (2) has a non-trivial solution, provided (T
-
-.-,
) -/I,$"=
~ ~ p - 2 1 c , * ~ ~ ) ( z *, l p z t ( m ~ ~ r
0.
whence !'I, 2 =
*/+2/-*il_l .
here
~5
;) = Ite(ae"'"lT) = cos wtr cos C O ~ T
0;;
-+
(!I;
ZT~J
sin C O ~ T?in t + ~ .
After n periods the oscil tat ion
We sllorv in Fig. 114 the repions of instability against parametric
resonance. 8.9.
has changed to Any oscillation is n superposition of oscillations such as x,, ?: in:parnicular, a real oscillation (which is the only one which has a direct physical meaning) ~ ( t= ) : Ael''lr+d*e-icrlt, 0 < t < z,
(compnre problem I m S; 30 of Landau and Lifsfiitz, 1960). 8.10, (a)
is the sum x r ( i ) + x , ( i ) with
If':b-= 1 , J : t , , I I = 1 and the oscillations +r,.,(t) (and at the same tinlc x(r)) rcrnain bounded. If, however, y =- I , we have p i w 1 , and the amplitude of the oscillations increases without bound. This is the casc of the onset of parametric resonancc. One verifies easily that if the frequency difference i~ small, If01 -wz{ = col, this condition is satisfied if the frequencies lie ctost to nx IT :
-
We draw attention to the fact that the relation UeK~f r-li~characferistic for intcrrnolecular forces.If we substitute into (1) the values r = e2 (5 X 10-'Oest~)z,m lQ-27p. a cm, w 10lqHZ,which are typica! for atoms, we get U,, IW'" erg cmR/r"whh is close to the correct value for van der l a d s interact ions, as far as order of magnitude is concerned. This result may serve aq sn indication of the physical nature of this interaction, A complete calculation of the van der W a a l ~forces ix only possible using quantum mechanics. 8.11. T h e motion along the z-axis is nearly uniform, z =. I:?. In the x-1'plane there acts upon t hc particle a fact oscillating force = 2A.r sin kar. f,, = -2Ay sin kt'l.
- - -
-
30 zpl' 5
-
-8s 2 .s
8
72 *.u
d
-- 2 gz 2
2 '-2 2 mclor
W
r
o
u 2 5
s .rmg
&
"4
0 %
2'Lh
5 2
w,a
3
U P - ,
5
2,
.-g #Y ;;ia 0 -0 5 ,g en
c 9 0
G
$43 0
2-
5 3 5
$32
-2.:
G E E
053-
g 0
g ;g b
~
-2 -g Z E e
GmE P5 -Y E 2 E
- =r 2 u
,-
G
a
1 I
5 E q c
5 %
C C .- ~
5
0
m
h $
.-
k k a
O 2
-. 4 T o .-" 5 2
- -
Z Q J2
2-c-
*P
Cu $ $ O +-
c e 8 .2 O b -
u,
LI 'C
11
'Ll
.o K,
z z 2
q c
=
oa * -
G n u I1 E G
as -." V
FB
" & 0.5 Y 7
2
~
24
9
I?
'P
iri 1 C
II
-
a, ti4
pl
t L
11 0,
h
w
V
>is w
CI
1
I
2 11
Q h
+.
KI:, Q
I cJ I
a. II
h, L"
II L
-2
67
0
-2
-Q 5
.-C
el>?-
m
.-
kc
"> z2
II
L
-3
0
e 2s
.2 5
z'2 Y
'
O
*5
> N CI
k
el' N
2
+
g
d
-
at,
2
L
<
4-
GI
m
2 E
+ .-5
I1
t-4
I!
8" +I 3 4
-
b
6, .Ez
=c j c a 'j
4 cr
5
$5 s u
E
2
ca
IZ w
h
-?
w
G4.r
L
2
,i
2
--
>
%
0
i O " % nE QT $n r .
-. c
w Q r
2
--b
?
?7: L
0
"5
-
Q-
C 1 1 rn m 2
2 5 ~ E: w 9 'rn -3
s o w
g 2
25
7 '3 c c
gg
3 -3
2
e
E.2
% c o w
'-
x
En
Y d 2 ;1 .s
3
y v l r j
636 & c 2 '3 ,. 8 > , c 5 -$ 2 A -2 z 2.2
I .
*:
z4gs oe $B 5 5 4 . 2 2
G
9 5
1 I
g
2< IJ
& = U D
3
C
3Fi"
,z>*
9 U
T:
rpu:
ou"-E
..TE =; .-6 2
. SO v , n -t
- c L -
93
GUJ
z : -ie r;.S%
0
I'
$:I1
I
0 n 'S 0
m Y
-
2"
gu x
0 s V:
N3
11 L
Collection of Problems in Classical Mechanics
12.6
I
-l;iyn) corresponds to counlcsclockwix (clmkwise) motion, and ,
Eo"
01=-arcs1n~- , CI
E$ f i 3 = ~ + a r ~ ~i I-l 0
If p = 0 (Ee2 = 01, we have $
p =- -
4 2
tan(J-n-t{-[sr-O]) P n -- -, tan 9 n
12.8
The equations For the orbi~.
can i n gencrnl not be integrated to procluce clernentary f~~nctions. H orvever, one can easily describc t h e motion qualitatively if one notes hat equation ( I ) rvhich gives a relation between the angles (J and a is, apart from the notation, the same as the equation for thc motion of a spherical pendulum (see problem 1 of 24 of Landau and Lifshitz, 1960). The particle thus moves in such a w a y [hat the point where ils radius vector intersects the surface of a sphere of radius I describes the same curbe as does a spherical pendulum of length I, energy Pi2rnF, ant1 angular momentum pp i n the field of gravity g = -a/mi? This curve is enclosed bctweeii two "parallel" circle%on the sphere corresponding to 0 = Or and 0 = 02. If P < 2Ep2/a 1, one can easily integrate equations (1) and (2):
-I0 = n-l/(e- ;-a2) CDS [ 2 2 ; Cma/IpI) arsinh (rdr)]+ e t - {-zZ,
The particle moves aIong the orbil of Fig. 123b, and we have r = - d%/i r ( I -= 0 ; the particle falls into the centre at t = 0). 12.7. The complete integral of the Hamilton-Jacobi equation is (see Landau and Lifshitz, 1960, 4 48)
2r. 1/E
-
F =EfF???/a. (3) 2~ It is clear from (3) that a partide whtn falling into the centre moves in the region between tiyo conical sul+faces Ol z (1 5 (I? rotating :]round the z-axis, while one complete rotation around the z-axiscorresponds to two cornpIete oscillations in the angle 0. I n this approximation the orhit is dosed for a spherical pc~ldulurn(it is an ellipse). 12.8. (a) If the particle does not fall into the cenlre, the equation for a finitc orb11 is
O=n-
--z2 - ; O < a .
+=
Answers and Solutions
4 2 + ~E* f ( 2 -~
COS
4 = I + e cos [.fC/t)]. r
I1 1
where
Thc gencralised momenta are the same as in problem t 2.4a. The partjclc can fall into the ccrltre, if P = 2rn(Ee2-a cos a ) -; O (which i s clearly satisfied if a' -i 2Ep"la c;.: 1). Tn that case r decreases monotonically from to 0 (compare equation (7) of the preceding problem). and the angle y increases monotonically from 0 to -. On the initial section of the orbit the angle 0 decreases from a value Of = r - a to a valuc where vanishec (if x2 -= 2Ep2Jac 1. it turn? out that 01= x -4(2Ep/a)) aftcr which j t again increases to the value a,, and so on. ah
-,
while the constnnts b and satisfy thc inequalities E -. Q and IF 0 - Z /I-= 2tna, the orbit fills the r e ~ o ABCDEF n (Fy. 124).
that is, it approaches O I I ~ point in this arbitrarily closely.
8
0.
"
G'
+
-k.i
-7- C Yi
C
I
!= .-
M,C
"
S', ~ I wlm
L
4
& M I
?
h+l
I
nrn
t-
,-.-- .-= --
5
2 : 2 2
J r A
0
..'
5
-
N
II
2 CI
tI
-2 -5
:. g -E0 6
o h :
Ern* TL b .=
2
9
s
> g & 2 ; n '3
'A
u-a 3
k-
+2 1
2
yzoz+
"2;
.'
-l::
2n 6*S " "2U To
-
4
3 z l l 0
" 4 3 - 23
.= -2 '2 2
--=c-mFo
--
c
2 U ; . = r f z
u C 3 ~ = c S U o
e
0
--
m s ' r 'Z
m
Z-Ee p L i 3
- -?Ss:a 2- 2 ' ~ 2
0
*~
d f
4 0 v kr. 0 & ~ 4ud cd . ~
Y
~
O h $ z s2
E
r
n
'-PC
W
U
$
'
z
a m
II.suC SoC 1o3c1 r : - .= E 4PXSle2 c 3
N A .--
9
os*2 ".-t .3 2 5 s 2 & e 03,
-E .au-c E . ~e 2LI:AsL g.2 = 5
QJ
ml:
WE-
li
L
Q
0 .9 z bGa m zo za . T
s.z
+
8 . 2 p > E 7 #1 a
5g2,$5s2
X
uLL-z S F !
E P L S o =
3- 2 .2 us:
'C]
0.2
2
2%
--2
2 ~
r0
2 2 u -2 r .-
--
0
- m
b L
2 E.2
I
- *n
s
& Z
0
0 "A
02
5
21,s m
-5
o
n,
z ,
5
2.2 0 za E E.Z m u 5g 2
= YG; Y
n u
-
E.8
J E W 5
4 %$.% F i
bl--0
s
0
m
r
.E 8
Collection of Problems in Classical Mechanics
13.7 1
We can use here cquntions (22.9) and (22.10) from Landau and L i f s ~ l i ~ ~ , 1960, for the quantity .i-+ifox-. Integrating by parts we get ~ (= i)
)ol:.
Jfir
I
I?!;-
r(r)sin miiii - [x(o) I?)
13.18
Answers and Solutions
We can find the function L ( R ) w i t h r ~ u tevaluating the integral (1). The substilution r = R.Y gives =
X
Lil(r
cos te,r r / r
IJER2, M),
rvhence ER2 = constant. We gel I hus Tor the angle of incidence s i n s = -fmin - - = . - 114- = constant. R ~ ' ~ IR J E
-
Tf the farce changes slowly, I ( ( ) uill thus oxillate near 4 0 ) . It' F ( t )
-
13.15. (a) Eccy2'G-"I;
constant as t -, the total change in the adiabatic invariant, I(-) -I(O), can be very small (see problem 5.1 8).
13-17. Equating the values of the adiabatic invariant before and after tlie switching on of the field. 13.13. (a)
where a, h, and e are the lengths of the edges of the parallelepiped. and Ik = constant. The dependence of the encrgy on a. 6. and r is determined not only by their initial values, but also by r he in~tialdistribution of energy over the different degrees of freedom. (b) Tlle position of the parallelepiped iq determined by three angles (for ingtance, the Euler angles). These angles will occur as parameters in the Hnrniltanian and will change slowly when the parallelepiped is rotated. In a system of coordinates with axes parallel to it< edges, we can separate the variables in he I-Tarnilton-Jacobi equation. The form af the Hnmiltonian js in that system of coordinates independent of the above-mentioned parameters. The invariance of the appropriate adiabatic invariants thus leads to the invariance of the absolute maynitudes 06 the component5 of the velocity along each of the edses. The angles of incidence oft he particlc on each of the sides are also invariant. 13.14. In spherical coordinates the variables separate. The a n ~ v l a mor mentum M i5 \trictIy conserved. (Moreover. M, is an adiabatic invariant correcponding t o the angle T.) The adiabatic invariant for the radial motion i5
.[
R
I,=
I 2%
rn,un
2/2rn~-~fi!~r-~di-.
(1)
(compare problem 13.3). 13.18.
E = llQ1 4- iZQ2
(in the notation of problem 6.4a). The orbit fills the rectanyle
The condition Ihnt the theory o f adiabatic invariants is applicable is 1 c,
2, << ,
i = I , 2.
-
Outside rhe region of degeneracy these conditions reduce to the same ones for mlll(fl.In the region of degeneracy lo:-wzl a.and the second condition is more restrictive and gives ti) di (the region of degeneracy is traversed during a time which is considerably longer than the period of the beats).
zt n -2 :rn C
I
5 8~
-
y t z 0 , w 2 +,
pz", o z >
5
*z= 6 0
~l e 0 cd U
U
Collection of ProbFerns in CIassical Mechanics
13.19
13.19. When the coupling ~ v isy not present, [he svstem splits into two independent osciltators with coordinates x and ?:. The CoWespondinp adiabatic invariants are I, = EJwl and I, = E,,'w. where I:', and 17, arc the energies of these oscillators. Wlien t hc couplitig is taken into account the sptern consists of two independent oscillators with coordinates QL and 0:. If the frequency changes ~ufficienrlyslowly. the quantities 11 = ErllL'I and I , = E,l_(2, arc invariant. Outside t hc region of de~enerncythe normal oscillations are strongly localised, and when ro, ro, rt turns out that = S. & = y, while when
-
Answers and Solutions
13.a
a,, ws, we have QI = .v, Q p = -x. Thus, when w z -= my, I, = 11, = 12,while when w l Q I ~ we , have I, = Jz, I,, = Il (see Fig. 133). . We shall illustrate this by the following example. Two pendulums, the l e n t h of one of which can be changed slowly, are coupled by a spring with a small stiffness (Fig. 134). When the Iengtbs /and L of the pendulum are appreciably different, the normal oscillatio~sare practically the same as the oscillations ofone or the, other penduEum. Let the pendulum AB initiaJly oscillate with amplitude q o . and I he pndulum CD with a very small !hen L is deereased the amplitude of the oscilEations of the CD remains smaII until its length becomes almost equal to I. When L = E. its amplitude increases (and when I = L both pendulums will osciljate with the same amplitude, T ~ l l / ini ,antiphase). When L is decreased further, practically all the energy transfers to the pendulum CD 3
and its amplitude becomes = rp~(ll~)', as for a separate pendulum. If we traverse the degeneracy region relatively fasl, a e rbl, such a transfer of energy between the oscillators wiI1 not take place, Jf, moreover, &, e o$, w 1 << &IWI, I, and I, will be invariant. 13.20. From the equations of motion,
we see easily that the coupling ktween the oscillators leads to a large energy transfer when 20,= ws. Let x=a(r)cos(wlt+lpl), y = b(t)cos(o~~r+y~).
Jf a
b, the term fix2 = *pa2 + +pa2 cos (Zwr
+2 ~ )
in (2) will pIay the role of an applied Force. Ieading to a resonance increase in ~ 7 .If, however, a &c b, the term
+
2pxy = 2phx cos (wz y)
leads to a parametric building-up of the oscillations in x. The region of resonance interaction (see probIem 8.7) is
In p n m l . a strong resonance interaction between the oscillarors occurs when m l = mmz, where 18 and m are integers, However, the width of the
