The Larmor nutation

PRAMANA © Printed in India Vol. 48, No. 3,

__ journal of March 1997

physics pp. 775-786

The Larmor nutation

K N S R I N I V A S A R A O and A V G O P A L A RAO

Department of Studies in Physics, University of Mysore, Manasagangotri, Mysore 570 006, India

MS received 10 June 1996; revised 22 November 1996

Abstract. The classical non-relativistic problem of the motion of a charged particle in an external central force field and a weak uniform magnetic field is revisited to show that the motion of the kinetic angular momentum L = r x p of the particle, in the so-called Larmor approxi- mation, is not a simple precession but is actually a composite motion involving precession as well as a high frequency nutation. The precession-nutation motion of L is discussed in the Larmor approximation when the Larmor-frame-orbit of the charged particle is an ellipse (or a circle) for the case of the two central forces namely the Coulomb and the Hooke-law-force, which are the only two central forces known to permit closed orbits.

Keywords. Larmor precession and nutation; charged-particle orbits in electromagnetic fields;

atom in a magnetic field.

PACS Nos 03-20; 32"30; 31"90; 32-60; 41"90; 46"90

1. Introduction

It appears that only one exact solution is available for the classical non-relativistic problem of a charged particle moving under the combined influence of an attractive central electric field and a uniform magnetic field. This special solution describes the circular orbit of a charged particle in a plane perpendicular to the applied uniform magnetic field. M o r e genera', exact solutions for this p r o b l e m do not exist although a variety of a p p r o x i m a t e solutions m a y be obtained t h r o u g h L a r m o r ' s theorem.

L a r m o r ' s t h e o r e m as applied to this problem [ 1 - 5 ] implies that every b o u n d e d - o r b i t [6] of the charged particle moving in an inertial frame u n d e r the action of the central force field and a sufficiently weak magnetic field is approximately a b o u n d e d orbit in the same central field, but without the magnetic field, when viewed from an a p p r o p r i a t e rotating frame called the L a r m o r frame. If we assume that the m o t i o n of the charged particle in such a precessing orbit generates a 'pointlike magnetic dipole-moment whose magnitude is not affected by the motion it undergoes' [4J, then it follows that the magnetic dipole precesses uniformly with the L a r m o r frequency [7]. T h e assumption that the orbit is a pointlike magnetic m o m e n t is appropriate to p e r m a n e n t magnets or systems on an atomic or smaller scale and obviously is not valid for macroscopic charged particle orbits. Therefore, it is of some interest to study the m o t i o n of the magnetic m o m e n t It or the orbital angular m o m e n t u m L = It/V, where ~ is the gyromagnetic ratio, in the case of macroscopic orbits.

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K N Srinivasa Rao and A V Gopala Rao

Since the L a r m o r theorem is valid for macroscopic orbits also, we m a y expect that u n d e r the conditions of validity of the L a r m o r theorem, the m o t i o n of the vector L m a y be m o r e complex than a simple precession unlike in the case of microscopic orbits.

Secondly, since a precessing charged particle orbit is essentially a 'magnetic-top', we m a y expect the L a r m o r motion of L to involve nutation also in complete analogy with the p r o b l e m of a massive top in a gravitational field. In fact, L a n d a u and Lifshitz [8]

show that in the case of a point charge executing b o u n d e d periodic m o t i o n under the influence of a central electrostatic field and a weak magnetic field, a suitably time- averaged m o t i o n of L is the familiar L a r m o r precession. In this paper, by restricting ourselves to a study of the motion of L in the L a r m o r approximation, we show that the detailed m o t i o n of L consists of a precession as well as a high frequency nutation.

2. Equation o f m o t i o n for L and its solution in the L a r m o r a p p r o x i m a t i o n

The e q u a t i o n of m o t i o n of a negatively-charged particle I-9] of charge - q < 0 and mass m, m o v i n g in an inertial frame under the combined influence of an external central force fieldf(r) f and a uniform magnetic field B is given by

d(mv)/dt = f ( r ) f - qv x B/c, (2.1)

where r is the position vector of the particle, f is a unit vector along r, v = dr/dr is the velocity of the particle and c is the speed of light in vacuum. (Note: We work in Gaussian C G S units.) O n cross-multiplying this equation by r, we get

d L / d t = - 2mr × (v x ¢o), (2.2)

where L = r × my is the kinetic angular m o m e n t u m of the particle,

o~ - qB/2mc, (2.3)

and [o~l is the ' L a r m o r frequency'. Now, if we write the term - 2mr × (v x co) as the sum of two equal halves, express one of them using the Jacobi identity as

- - m r x (v x t o ) = m v x (to x r) + m e x (r x v), and add it to the other in equation (2.2), we obtain

d L / d t = to x L + dA/dt, (2.4)

where we have defined

A =- mr x (to x r). (2.5)

Although (2.2) and (2.4) are the same, the latter is m o r e convenient in discussing the m o t i o n of L. In fact (2.4) shows at once [10] that the m o t i o n of L would be a pure precession a r o u n d B only when the second term d A / d t on the right h a n d side of (2.4) is either zero or is negligible in comparison with the first term to × L. Since b o t h these terms are of the same order in B, it is clear that the m o t i o n of L can never be a simple precession however weak the magnetic field B m a y be.

T h e vector A above has a simple interpretation from the standpoint of the so called L a r m o r frame, i.e., a reference frame which rotates with the angular velocity to relative to the inertial frame in which the problem is being studied. If we denote quantities 776 Pramana - J. Phys., Vol. 48, No. 3, March 1997

The Larmor nutation

referred to the L a r m o r frame by an asterisk (*) a n d the time-rates-of-change with respect to the inertial a n d L a r m o r frames by d/dt a n d d * / d t respectively, then we h a v e [5]

d A / d t = d ' A / d r + to x A,

which is a general relation valid for all vector functions A(t). Also, we a s s u m e t h a t the origins o f the t w o frames coincide, so that we have r(t) = r*(t) a n d this relation t o g e t h e r with the fact t h a t to is a constant, leads to the following well-exploited relations for velocity a n d acceleration:

v = v* + to x r; v = dr/dt; v* = d * r / d t ; (2.6) a = a* + 2to x v* + to x (to × r); a = dr/dr; a* = d*v*/dt. (2.7) In the s a m e m a n n e r , we obtain the following relation connecting kinetic a n g u l a r m o m e n t a in the two frames of reference

L = L * + A ; L = r × m v ; L * = r × m v * (2.8)

This relation, s o m e h o w , seems to have received little attention in the literature. T h e vector L* in the a b o v e e q u a t i o n m a y be interpreted as the a n g u l a r m o m e n t u m relative to the r o t a t i n g f r a m e a n d the vector A as the a n g u l a r m o m e n t u m of t r a n s p o r t . T o a p p r e c i a t e the significance of A we m a y note that, for example, in the case of a rigid body, the v e c t o r L* = ~ r x m v* (where the s u m m a t i o n extends o v e r all the constitu- ent particles) vanishes in the b o d y rest frame and therefore the entire a n g u l a r m o m e n t u m a p p e a r s as the t r a n s p o r t a n g u l a r m o m e n t u m A in this frame.

O n using (2.6)-(2.8), (2.1) and (2.4), we obtain

ma* = md*2r/dt 2 = f ( r ) f + mto x (to x r) (2.9) and

d * L * / d t = to x A, (2.10)

which are exact e q u a t i o n s c o r r e s p o n d i n g to (2.1) a n d (2.2), but written in the r o t a t i n g L a r m o r frame. In the so-called L a r m o r a p p r o x i m a t i o n which c o r r e s p o n d s to situations in which the m a g n e t i c field B is sufficiently weak a n d orbit of the particle is b o u n d e d in the sense described earlier, we neglect the second t e r m on the right h a n d side of (2.9) in c o m p a r i s o n with the first and obtain

m d * 2 r / d t 2 ,,~f(r) f. (2.11)

T a k i n g the vector p r o d u c t of this equation with r then yields

d * L * / d t ~ 0, (2.12)

which is the e q u a t i o n of m o t i o n of L* in the L a r m o r a p p r o x i m a t i o n . Thus, in the L a r m o r a p p r o x i m a t i o n , L* is a constant as viewed from the r o t a t i n g frame, or, equivalently, it is a precessing vector satisfying

d L * / d t m to x L*, (2.13)

as viewed f r o m the inertial frame. It is gratifying to note that there is at least o n e a n g u l a r m o m e n t u m vector in the problem, namely L*, that executes a simple precessing m o t i o n Pramana - J. Phys., Vol. 48, No. 3, March 1997 777

K N Srinivasa Rao and A V Gopala Rao

in the L a r m o r approximation. However, it is L, and not L*, which is the angular m o m e n t u m of the particle in the inertial frame and in view of (2.8), the motion of L (in the inertial frame) is more complex than a simple precession because L is the sum of a precessing vector L* and the (moving) vector A.

In passing, we make an interesting observation on the canonical angular m o m e n t u m L c = r x (p -- qA/c) of the charged particle in this problem. In a suitable gauge, the vector potential for a uniform magnetic field B m a y be written as A = - r × 11/2. With this A, we obtain Lc = L - A. Comparing this with (2.8) then shows that Lc = L*. In view of this relation, (2.13) m a y also be interpreted as describing the L a r m o r precession ofL¢. Larmor's theorem in this form, for L¢, may also be obtained directly without using the rotating frame [11].

We now proceed to determine the motion of L in the inertial frame. First, we mention very briefly about the only exact solution available for equation (2.1). This solution describes the uniform circular motion of a particle of charge -- q a n d mass m in the C o u l o m b force field of a positive charge q' at rest at the origin of coordinates of the inertial frame. With a denoting the radius of the circle and Icol denoting the constant angular frequency, the solution mentioned is given by r = (a cos co't, a sin co't, 0), where co' = co __+ x/co02 + ¢92; co = qlBl/2mc; coo = x/qq'/ma3 > 0 and the z-axis of the inertial frame has been chosen such that B = (0,0,B). A simple calculation then yields L = (0, O, ma2oy) = constant. (For a further discussion of this solution see the following references: Purcell [12], Reitz et al [13] or Matveev [14].) Leaving aside this exact solution which a n y w a y has a constant L and is therefore not of interest to us, we consider approximate solutions of equation (2.1) obtained t h r o u g h the L a r m o r the- orem. (In fact, our interest in these solutions is restricted to a study of the motion of the angular m o m e n t a rather than the orbits themselves.) We have a simple prescription for obtaining the orbital angular m o m e n t u m vector associated with a n y such solution of (2.1) obtained through the L a r m o r theorem: First, we pick up an arbitrary bounded- orbit solution r = r o (t) of the problem (2.11). (It is important to note t h a t r o (t) so chosen is in fact an exact central force orbit and not an approximate one as required by (2.11).) Then, using this vector ro(t), we calculate vector field A(t) defined in (2.5). Next, we calculate the conserved orbital angular m o m e n t u m vector L* = r o x (m dr0/d t) of this orbit and use it to obtain the vector field L = L* + A, which is the required orbital angular m o m e n t u m in the inertial frame (in the L a r m o r approximation).

We now use this prescription and calculate the L associated with an arbitrary, exact solution ro(t ) of(2.11). Let to(t) expressed in the basis (i,j, k) of the inertial frame be given by

ro(t ) = x(t)i + y(t)j + z(t)k. (2.14)

Further, let the z-axis of the inertial frame be chosen such that B is parallel or antiparallel to k. Then

= cok, (2.15)

where co > 0 when B is parallel to k and negative otherwise. Next, using (2.14), (2.15) and the definition (2.5), we obtain

A = m r o x ( - x r o ) = - m c o { x z i + y z j - ( x 2 + y2)k}. (2.16)

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T h e L a r m o r n u t a t i o n

Since we k n o w that L* is a precessing vector as seen from the inertial frame (i,j, k), a n d hence satisfies (2.13) (now exactly, as to(t) is an exact solution of(2.11)), we m a y take it to be the general solution of (2.13) given below:

L* = Lo(sin0sin~0i - sin0cos ~0j + cos0k). (2.17)

Here Lo a n d 0 are two constants of integration and [151

~0 = oJt. (2.18)

The angles 0 = constant and tp - n / 2 = o~t - n / 2 are evidently the polar and azimuthal angles of L* with respect to the inertial frame (i, j, k) and these angles clearly show t h a t the vector L* precesses around the z-axis with angular frequency

Itol.

L = Lxi + Lyj + Lzk, (2.19)

where

L x = L o sin0sin~p - m o ~ x z ; L y = - - L o sin0 cos~p -- m t o y z ;

Lz = Lo cos 0 + m o g ( x 2 + y2). (2.20)

These equations yield L once the central force orbit ro(t ) is specified as in (2.14). This solves the problem in principle. However, in actual calculations it is more convenient to specify the planar orbit ro(t ) in terms of two variables rather than in terms of the three variables x, y, z as done above. We do this by introducing the rotating L a r m o r frame (I, J, K) with corresponding coordinates, X, Y, Z, as follows 1-16]. The plane of ro(t ) is taken to be the X - Y plane so that L* is always along the Z-axis. Further, we choose the X-axis to lie along the line'of nodes formed by the X - Y plane with the plane of the orbit ro(t ) as shown in figure 1. Then, constructing the rotation matrix connecting the

kq

/

\

Figure 1. Euler angles specifying the orientation of the Larmor frame relative to the laboratory (inertial) frame.

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K N S r i n i v a s a R a o a n d A V G o p a l a R a o

two frames in terms of the Euler angles, we arrive at the c o o r d i n a t e t r a n s f o r m a t i o n relating the two frames:

x = X cos ~o - Y cos 0 sin ~o, y = X sin q~ + Y cos 0 cos q~, z = Y sin 0. (2.21) Substituting these in e q u a t i o n (2.20), we finally obtain the required formulae

L x = L o sin 0 sin q~ - m e ) X Y sin 0 cos ~0 + me) y2 sin 0 cos 0 sin q~, Ly = - L o sin 0 cos q~ - moo X Y sin 0 sin ~o -- m co y2 sin 0 cos 0 cos ~o,

L z = L o cos 0 + m e ) X 2 + m e ) y 2 cos 2 0. (2.22)

T h e c o m p o n e n t s of L relative to the rotating frame (I, J, K) are similarly found to be given by

L = me)sin 0 ( - X Y I + X 2 J ) + (L o + m e ) r ~ cos 0)K, (2.23) where r02 -- (X 2 + y2).

Before concluding this section, we restate the L a r m o r a p p r o x i m a t i o n in terms of a small dimensionless p a r a m e t e r . We m a y recall that for a b o u n d e d orbit r = r(t), r - Irl lies in the interval 0 < rmi n ~ r ~< rm, X < oo. Now, let the m a g n i t u d e of the central force f ( r ) ~ have its smallest value on the orbit, say, w h e n r = r 1. Then, evidently, If(r)E]/> [f(rl)[. We also note that [mto x (to x r)[ ~< me)2rmax. Thus, the L a r m o r ap- p r o x i m a t i o n condition [f(r)t >> [into x (to x r)l m a y be expressed as

If(r1) [ >> mrmaxe) 2. (2.24)

If we n o w define a characteristic a n g u l a r frequency f~ > 0 associated with the b o u n d e d orbit of the charged particle by

m~-~2rmax = If(rOI , (2.25)

the condition (2.24) reads

f~ >> Icol. (2.26)

Thus, the L a r m o r a p p r o x i m a t i o n describes a situation in which the dimensionless p a r a m e t e r le)l/~ is 'small'. Incidentally, the m o d u l u s of to a p p e a r s in (2.26) a b o v e because co, we m a y recall, could be positive as well as negative d e p e n d i n g on whether B is parallel or antiparallel to k.

3. Tracing the motion of L in the case of elliptic Larmor frame orbits

In this section, we study the m o t i o n of the unit vector 1,17] [, along L in the case of such inertial frame orbits r(t) for which the corresponding L a r m o r f r a m e orbit (LFO) r 0 (t) is an ellipse. We k n o w that (Bertrand's theorem: see G o l d s t e i n I-4], p. 93, or L a n d a u a n d Lifshitz [1], p. 32) the attractive inverse square law force a n d the H o o k e law force (also called the space oscillator force) are the only two central forces which permit closed, a n d hence elliptic orbits. Hence our discussion would c o v e r the cases of b o t h these forces.

T o study the m o t i o n of L, we determine the a z i m u t h ¢PL a n d the p o l a r angle 0L of the vector L as functions of time. The angle tpL is given by tan tpL = L r / L x. O n 780 Pramana - J. Phys., Vol. 48, No. 3, March 1997

The Larmor nutation

using (2.22) and performing some elementary trigonometric rearrangement, we may invert this relation as

7¢ 7~

q~L : q~--~ + t/ = 03t--~ + t/, (3.1)

where we have defined t / = t/(t) by

tan t / = -- ( m 0 3 / L o ) X Y ( 1 + ~xY 2 COS 0)- 1. (3.2) On the other hand, the p o l a r angle 0 L of L may be calculated from the relation

cos 0 L = L JILl. (3.3)

Now, if the L F O {X(t), Y(t)} is specified, we may calculate the angles 0 L and ~PL as functions of time by using equations (3. i)-(3.3). Then, the m o t i o n of [, m a y be studied by considering the vector

L± = sin 0 L cos ~PLi + sin 0L sin ~PLJ, (3.4)

which is the projection o f [, o n t o the x - y plane of the inertial frame. T h e tip of this vector would trace a circle if the motion of L is a pure precession. Any non-circular tracing clearly indicates the presence of nutation. This is as far as we can go with a general central force field. Thus, we consider the two specific cases of interest.

3.1 Coulomb law elliptic orbits

We begin by recalling some essential properties of an elliptic orbit ([1], pp. 32-39) of a negatively-charged particle of charge - q < 0 and mass m moving in the C o u l o m b field of a n o t h e r positively-charged particle of charge q' > 0" at rest at the c o o r d i n a t e orgin (which also happens to be one of the two foci of the elliptic orbit). The p a r a m e t r i c equations describing such an elliptic orbit are

X = a p ; r = b a ; p - ( c o s ~ - e ) ; a - = s i n k ; t = ( T o / 2 ~ z ) ( ~ - - e s i n ~ ) , (3.5) where X -= ap(t) and Y = ba(t) are the Cartesian coordinates of the particle tracing the ellipse. T h e constants a, b and e are respectively the semi-major axis, semi-minor axis and the eccentricity of the ellipse. The eccentric angle ~ = ~(t) increases by 2n for one complete passage of the charged particle round the ellipse. T h e origin of time has been chosen such that ~ = 0 at t = 0 and ~ = 2~ at t = T o, where T o is the period of the elliptic motion. T h e angular frequency 03 0 associated with T o, is given by 030 = 2x/To = Lo/mab = x / - - q q ' / m a 3, where the positive constant L o is the magnitude of the conserved angular m o m e n t u m associated with the elliptic orbit. Lastly, we note that the variables p and a in (3.5) are dimensionless and have the ranges - (1 + e) ~< p ~< (1 - e) a n d - l ~ < a ~ < l .

Now, using (3.5), (3.3) and (2.22), we obtain COS 0 L =

k + ~{(a/b)p 2 + (bk2/a)a 2}

x/1 + 2ke{(a/b)p 2 + (b/a)a 2} + e 2 {(a2/b2)p 4 + (1 + k 2)p2a2 + (bZk2/a2)a4}

(3.6) Pramana - J . Phys., Voi. 48, No. 3, March 1997 781

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where we have d e n o t e d cos 0 by k and have i n t r o d u c e d the dimensionless para- meter e = co/coo- If we now observe that e and 0 are constants and the variables p and cr are periodic under g---, ~ + 27r which c o r r e s p o n d s to t---, t + T o (see remarks following (3.5)), it follows that cos 0 L given by (3.6) is a periodic function of time having the period T o. This means that, in the L a r m o r a p p r o x i m a t i o n , L nutates with a time-period which is precisely the orbital period T o of the charged particle in its elliptic L F O .

Next, since the function ~/(t) defined through (3.2) is also periodic under t ---, t + T 0, we note that whenever T =27r/[col = N T o where N is an integer, the azimuth ~0 L of L changes b y 27r in a time T during which 0L, which has a time period T o = T / N , would go t h r o u g h exactly N cycles. In other words, when T = N T o, the tip of the unit vector L would trace a closed curve. When co and co o are n o t c o m m e n s u r a t e , the tip of L does not trace a closed curve. In any case, the m o t i o n o f L in the L a r m o r a p p r o x i m a t i o n is a composite m o t i o n involving both precession a n d nutation.

Regarding the n u t a t i o n frequency, we m a y note that it is given by co o for elliptic paths of non-zero eccentricities. Interestingly, when e becomes zero, i.e., when the L F O is a circle, the n u t a t i o n frequency changes to 2coo. This happens because, when e = 0, cos OL given by (3.6) becomes increasingly periodic; its period changes from 27r to 7r for (or equivalently, T O to To~2 for t), as only even powers o f p and a a p p e a r in it. We also note that since the m a x i m u m separation rma x between q' and - q on the ellipse (3.5) is a(1 + e), the characteristic frequency Q of (2.25) becomes Q = coo(1 + e) 3/2. Thus the L a r m o r a p p r o x i m a t i o n condition (2.26) now requires coo >> (1 + e)3/2[co I. As a conse- quence, the n u t a t i o n frequencies (coo or 2coo) are very large c o m p a r e d to the L a r m o r

frequency Icol.

An estimate of the magnitude of cos OL, may be o b t a i n e d as follows. We note that if we retain only terms of the order O(e) in (3.6), and substitute back for e as co/coo, w e obtain cos 0L = cos 0 + (coa/coob) sin 20p 2 + O(e2). Incidentally, this formula shows that

1.2 i ' i =

0 . 6

0 . 0

- 0 . 6

- 1 . 2 ' I ,~ i

- 1 . 2 - 0 . 6 0 . 0 0 . 6

(a)

1 . 2 F i g u r e 2 a .

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The Larmor nutation

, , , (b)

-0.6

-0.7

-0.8

-0,9

• I . |

-0.125 0.000 0.125

• I

-0.69

-0.70

-0.71 I ~ ~

-0.72

-0.73

-0.02 -0.01

F i g u r e s 2 b , c .

(C)

0.00 0.01 0.02

cos OL > COS 0 when co > 0 and cos 0L < cos 0 when co < 0. Also, observing that

(p2)max

= (1 + e) 2 and (p2)mi, = 0, we obtain

(COS0L)ma x -- (COS0)~ n = + (toa/toob)(1 + e) 2 sin20 + O(to2), (3.7) where the + sign corresponds to co > 0 and the - sign to 09 < 0. It is important to note that the leading term on the right hand side of (3.7) is linear in to and hence in the magnetic field strength IB[. Equation (3.7), however, does not yield an estimate of the nutation amplitude = (0L)m,x - (0)mi,. It may be obtained numerically by calculating cos 0L as a function of the angle ~ and inverting.

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In figure 2, we have s h o w n s o m e m o d e l precession-nutation curves for the C o u l o m b case. These curves have been d r a w n b y setting L o = 1, co = _+ 1 a n d 0 = re/4. W h e n 0 = ~/4, 0 L also lies in (0, x/2) so that sin 0L is positive. Therefore it m a y be calculated as x/1 - cos 2 0 L t h r o u g h equation (3.6). T h e angle q~L has been calculated f r o m (3.1) and (3.2).

Further, we have chosen three convenient values for co o n a m e l y 10, 100 and 1000 which yield d o s e d nutation curves. In each of the figures 2(a), 2(b) a n d 2(c), we have shown three different curves. T h e heavy-solid curve corresponds to to = 1, the light-solid curve corre- sponds to a~ = - 1 whereas the (dashed) circle of radius sin 0 = 1/x/2 does not represent a n y real m o t i o n of L, b u t has been d r a w n only for the sake of reference. (Of course, it c o r r e s p o n d s to the circle of (cos 0L)mi n = COS 0 = 1/X/2 w h e n co > 0 a n d to the circle of (cos 0L)ma x ---- COS 0 = 1/X/~ when to < (3). T h e c o m p u t e d values of the n u t a t i o n amplitude are as follows. F o r to = 1, it is 36-4, 12-4 and 1.6 degrees w h e n too = 10, 100, and 1000 respectivdy. Similarly, for t o - - - 1 , it is given b y 122.8, 21.4 a n d 1.6 degrees when too = 10, 100 and 1000 respectively. These values clearly s h o w t h a t the nutation amplitude decreases as [el decreases. In fact, the precession slows d o w n a n d the n u t a t i o n dies out as I~l decreases a n d eventually when [el touches zero, i.e., w h e n B = 0, L gets arrested at OL = O, and tp L = - - x / 2 (see (3.1), (3.6) a n d the r e m a r k s following (2.18)).

3.2 Hooke law elliptic orbits

W r i t i n g the H o o k e law central force as f ( r ) ~ = - mto0Er, we o b t a i n the following p a r a m e t r i c e q u a t i o n s to the elliptic o r b i t (with the c e n t r e o f a t t r a c t i o n r = 0 at the centre o f the ellipse) ( [1], p. 70)

X - - a c o s ~ ; r = b c o s ( ~ + 6 ) ; ~ = t o o t, (3.8) w h e r e we h a v e set ~ = 0 at t = 0. T h e o t h e r c o n s t a n t o f i n t e g r a t i o n 6, w h e n c h o s e n a p p r o p r i a t e l y , gives the v a r i o u s familiar Lissajous figures. W e h a v e specifically

1 . 2 . i i . i .

0 . 6

0 . 0

- 0 . 6

- I . 2 i I i I , I ,

- 1 . 2 - 0 . 6 0 . 0 0 . 6

Figure 3a

(a)

1.2

784 Pramana - J. Phys., Vol. 48, No. 3, March 1997

The Larmor nutation

- 0 . 6 4

- 0 . 6 6

- 0 . 6 8

- 0 . 7 0

-0.72

-0.74

-0.76

' I

/I

, I

-0.125

l

I •

!

I I

0.000 O. 125

(b)

-0.702

-0.704

-0.706

-0.708

-0.710

-0.712

I

-0.02 Figures 3b, c.

I I l (c)

I i I , I i

-0.01 0.00 0.01 0.02

Figures 2 and 3. Precession-nutation curves in the C o u l o m b (figure 2) and H o o k e cases (figure 3): These are curves traced by the tip of the angular m o m e n t u m vector projected on to the plane perpendicular B (i.e., the x - y plane) of the inertial frame.

L has been scaled such that its magnitude when B = 0, i.e. L o, is unity. The curves show nutation of considerable amplitude besides precession.

c o n s i d e r e d t h e c a s e 6 = - n/2 w h i c h y i e l d s X = a c o s tot a n d Y = b sin cot r e p r e s e n t i n g a c e n t r a l ellipse. F i g u r e s 3(a), t o 3(b) s h o w s o m e m o d e l p r e c e s s i o n n u t a t i o n c u r v e s d r a w n w i t h a = 10 u n i t s a n d b = 1 unit. F u r t h e r to o --- 10, 100 a n d 1000 r a d i a n s / s e c o n d respectively in the figures 3(a), 3(b) a n d 3(c). As before, in each figure, the h e a v y - s o l i d c u r v e c o r r e s p o n d s t o co = 1, the light-solid curve c o r r e s p o n d s to co = - 1 w h e r e a s the (dashed) circle o f r a d i u s sin 0 = l / x / 2 has been d r a w n for the s a k e of reference. Lastly, we n o t e t h a t Pramana - J. Phys., Voi. 48, No. 3, March 1997 785

K N Srinivasa Rao and A V Gopala R a o

the c o m p u t e d values of the nutation amplitude are as follows. F o r o9 = 1, it is 22.5, 3-8 and 0.4 degrees when o90 = 10,100 and 1000 respectively. Similarly, for o9 = - 1, it is given by 67.5, 4.4 and 0.4 degrees when o90 = 10, 100 and 1000 respectively.

A c k n o w l e d g e m e n t s

T h e a u t h o r s wish to t h a n k the referee for his c o m m e n t s which h a v e helped in i m p r o v i n g the presentation of the p a p e r considerably. T h e y also t h a n k D r M A Sridhar for generating the n u t a t i o n curves on a computer.

References

[1] L D Landau and E M Lifshitz, Mechanics (Pergamon Press, Oxford, 1976) pp. 32-39, 70 [2] R P Feynman, R B Leighton and M Sands, The Feynman lectures on physics (Reading,

Massachusetts, Addison-Wesley, 1964) pp. 34-6-34-7

[3] C Kittel, Introduction to solid state physics, seventh edition (John Wiley, Singapore, 1995) p. 418

[4] H Goldstein, Classical Mechanics (Addison-Wesley, London, 1980) pp. 93, 107-109, 176-177, 233

[5] K R Symon, Mechanics (Reading, Massachusetts, Addison-Wesley, 1960) pp. 271-278, 283-285

[6"] We call an orbit r = r(t) in which It[ remains finite and non-zero, i.e., 0 < rmi" ~< [r] ~< r ....

a bounded-orbit

[7] See equation (2.3) for definition

[8] L D Landau and E M Lifshitz, The classical theory of fields, third edition (Pergamon Press, Oxford, 1971)pp. 105-107

[9] Although we specifically consider a particle of charge - q , q > 0, in our discussion, obviously with the electron in mind, the discussion may be easily modified to cover the case of a positively-charged particle by allowing q to be negative

[10] We may note that, in general, a vector function A = A(t) which precesses around another constant vector C with the angular frequency ICl satisfies the differential equation d A / d t = C x A

[11] A M Portis, Electromagnetic fields: Sources and media (John Wiley, New York, 1978) pp. 245-246, 712-713

[12] E M Purcell, Electricity and magnetism: Berkeley physics course (Mc Graw Hill, New York, 1965) vol. 2, pp. 370-377

[13] J R Reitz, F J Milford and R W Christy, Foundations of electromagnetic theory, third edition (Addison-Wesley, New York, 1979) pp. 222-223

[14] A N Matveev, Electricity and Magnetism (Mir Publishers, Moscow, 1986) pp. 277-280 [15] Actually, we get tp = o~t + constant, but we have chosen the constant to be zero in (2.18) [16] Note the deviation in notation here: As per the convention adopted in arriving at equations

(2.6) to (2.8), we must rather have denoted (I, J, K) and X, Y, Z respectively by i*,j*, k* and x*, y*, z*. But we have deliberately chosen the former set which is simpler. With the exception of these, all other quantities in the Larmor frame are marked by an asterisk as before

[17] We consider the unit vector L as we are interested in studying the change in the orientation of L with time, rather than in its length

786 Pramana - J. Phys., Vol. 48, No. 3, March 1997