three-particle problem is considered, that leads to a modification of the. Delves coordinates. A type of the kinetic and potential energy is obtained for the system.
Few-Body Systems 2, 71-80 (1987)
ystcms sFgWav
0 by Springer-Verlag 1987
Modification of Hyperspherical Coordinates in the Classical Three-Particle Problem P. P. Fiziev* and Ts. Ya. Fizieva Laboratory of Theoretical Physics, JINR, Dubna, P. O. Box 79, SU-101000 Moscow, USSR Abstract. A new version of the hyperspherical coordinates in the classical three-particle problem is considered, that leads to a modification of the Delves coordinates. A type of the kinetic and potential energy is obtained for the system. The problem is reduced in these coordinates for different cases of the classical motion. From geometrical reasonings a formula is chosen for the reduced mass of the three-body system. The triangle of masses and the relevant basic quantities and relations are introduced.
1 Introduction The recent decades were marked by an intensive use of different versions of the hyperspherical coordinates in the quantum three-body problem (see refs. in [1 ]). One should emphasize an important step obviously first made by Delves [2] and implying a nontrivial renormalization of the Jacobi-particle radial vectors; this procedure has been also used in a number of papers [3]. In recent years the use of the Delves coordinates led to a considerable advance in the study of the quantum problem of three particles on a straight line interacting through phenomenological two-body potentials [4]. As far as we know, except for [5], there were no attempts to use these coordinates in studying the classical three-body problem with arbitrary masses rni and pair potentials Vii proportional to r~ 1 where rii is the distance between particles with numbers i, ] = 1, 2, 3. Probing of the coordinates is to some extent easier within a classical version of the problem. It results in a further modification of coordinates and may provide some new results in the classical region, which are interesting by themselves. Most of these results may also be transformed to a quantum version of the problem. It is also useful to study the classical problem for constructing a quasi-classical approximation and a path integral for the quantum case. We intend to study these problems in a sequel of papers beginning from this one.
* On leave from Department of Physics, University of Sofia, Bulgaria
72
P.P. Fiziev and Ts. Ya. Fizieva
2 Laboratory and Moving Frames-of-Reference In the laboratory reference-frame E l after introducing the Jacobi coordinates (Fig. 1) the kinetic energy of the three-particle system is T = I 7N1,2R4 2 + 89
4 1 -;2 r2 +,2MRcrn,
(1)
where -1 #1,2
=
-1 #12,3 = (ml
m ? 1 + m21,
+ m 2 ) -1 +
m~ 1
(2)
are the Jacobi quasi-particle masses, and M is the system's total mass. rn 3
rrl 1
Fig. 1. The triangle A(ml, m2, m3)
R
The moving reference-frame E is introduced so that therein
Rcm = 0,
(3)
= k,
(4)
where/~ = (0, 0, 1) is the unit vector towards a positive direction of the axis OZ. At the angular velocity c~ of rotation E with respect to E l we have T = 89
z + R2(c~ X/~)z] + 89
+ c~ X 3) 2.
(5)
We shall describe the motion of the three-body system in the standard way (see, e.g., [1, 5, 6]) using the solid-state m o t i o n [translation and rotation of the triangle A(m 1, m2, m3)] and in addition deformation (of the same triangle). 3 Coordinates of the Triangle Deformation The triangle deformation A(ml, m2, ma) will be described by the coordinates (p, ~, 0) introduced according to the formulae
R = (#/#1,2) 1/210
cos
r = (#/#12,3) 1/z p sin cos 0 -
9
c ( - - ~ , ~),
(6)
E (-- oo, ~,),
(7)
E [ - 1, 11,
(8)
where p C [0, ~),
~ E [0, 27r),
0 C [0, lr].
(9)
Our choice o f a region o f values o f the variables differs from the standard one [ 1 - 4 ] . Its advantages are a continuous description of the whole space o f variables
Modification of HypersphericaI Coordinates in the Classical Three-Particle Problem
73
{R, r} = IR(2), what is essential for a continuous description o f all configurations of the system, while moving along a straight line. Moreover, the angle r differs from the Delves angle 7r
c~= - - ~ , 2
(10)
what turns out to be convenient in the following. For brevity we introduce the notation s = sin r
.~=~r/r.
c = cos r
(11)
Then, the kinetic energy in the new variables takes the form T = 89
z + p 2 [ ~ + s2(~ + 03 • ~)2 + c2(03 •
(12)
One can easily derive a formula for a m o m e n t o f inertia I~r of the system (refs. [3, 5]), I = trllI~ll = pp2.
(13)
The simplicity o f the expressions (12) and (13) in comparison with the relevant ones in other variables (see, e.g., refs. [8, 9]) enables one to better understand the meaning o f the hyperradius p and the still undetermined parameter p: the reduced mass o f the three-particle system 9 4 Coordinates o f the Triangle Rotation To describe the triangle rotation A(m 1, m2, m3) as a solid state, we use the standard Euler angles ~ , ~I,, (9 [7]. Then COx = ~ sin 0 sin 'Is + 6 cos q,, (a)y = + sin O cos q~ - O sin ~P,
coz = 6 cos | + ~.
(14)
Next we constrain the definition of the system X by the requirement that lies in the plane OXZ at an angle 0 to 0Z, i.e. -~ = (sin O, 0, cos 0). Then (03 X/~)2 = O2 +~p2 sin 2 19, 9
.+
(15)
-+
(5 + co • 7) 2 = (0 + coy)2 + (COx cos 0 - COz sin 0) 2.
(16)
5 Hamiltonian Variables and the Total Angular M o m e n t u m of the System Performing a standard transition from generalized velocities {t5, ~b, O, ~, ~I,, (9) to generalized m o m e n t a (Po, P~, Po, Pc, P,v, Po) we get for the kinetic energy o f the system T = (p~ + p-2A2)/2p,
(17)
where A 2 = p~ + s-2L21 +
c-2L~
(18)
74
P.P. Fiziev and Ts. Ya. Fizieva
is a large angular m o m e n t u m o f the system and
L2x = c-2p~ + sin -2 0 p~,,
(19)
L~ = sin -2 (9 p~ + (ctg z O - 2 sin ,I, ctg O ctg 0 + ctg 2 0 ) p ~ + p~ + 2 sin -1 19 (sin q~ ctg t9 - ctg 19)pcp,~ - 2 sin -~ 19 cos ~ PePo + 2 cos ~I, ctg 19 P,I, Po + 2 cos 9 ctg 0 pv Po + 2 sin q, Po Po.
(20)
F o r the total mechanical m o m e n t o f the system K,
-K1,2+K12,3=lal,zRX(R+wXR)+la12,3rX(-r+~oXr),
(21)
we get in the system
Kx=cos~p|
+ s i n -1 19 sin 9 (pc - p .
cos 19),
K y = - sin 9 Po + s i n - 1 0 cos 9 (pc - p~ cos 19), K z = p,~
(22)
and in the system Et
Kxl = sin rb Po + sin-x O cos q5 (cos O pr - p . ) , Kyt = - cos q~ Po + s i n - 1 0 sin q~ (cos O pr - p . ) , Kz~ = Pc.
(23)
It is seen that the components o f /s are independent o f the deformation coordinates (p, ~o, 0) and its square is K 2 = p~ + p~. + sin -2 0 (pr - cos 0 p.)~ = p~ + pr + sin-2 | (pc cos 0 - p.)2. (24) 6 Particle Interaction Potentials
Consider systems o f three particles with an interaction of the type V = V12 + V23 +
V31,
(25)
where the two-particle potentials Vii are
Vii = oti//ri/;
eie] oLq = _ mim]
for electricity, for gravitation.
(26)
The distances ri] between particles in our variables are r12 = (#/pl,2)l/2plcl,
r2a = [#/(ml + m2)]l/2p[s2M/ma - 2 ( m l M / m 2 m a ) l / 2 s c cos 0 + c2ml/m2] 1/2, r31 = [#/(m 1 + m2)]l/2p[saM/m3 + 2 ( m 2 M / m l m 3 ) l / 2 s c cos 0 + c2m2/ma] 1/2. (27) Hence we get
V = o~(~o, O)lp,
(28)
Modification of Hyperspherical Coordinates in the Classical Three-Particle Problem
75
where o~(~o, O) = ~
(29)
(oqjp/rr
i 0) p ~
kp,
p ~ k-21.t;
(44)
this leads to a physical uncertainty o f the scales p and p taken independently. In
Modification of Hyperspherical Coordinates in the Classical Three-Particle Problem
77
this connection Delves [2] (see also ref. [3 ]), based on formal reasonings (simplicity o f normalization o f the wave function), proposed to fix # by the formula # = #D = x / m l m 2 m 3 / ( m l + m'2+ m3).
(45)
In ref. [10] another reduced mass was suggested, # = laL = r n l m e m 3 / ( r n l + rn2 + m3) 2.
(46)
In the literature, the choice o f a value o f # is sometimes considered to be a nonphysical problem [1, 2]; this is verified by the arguments about the m o m e n t o f inertia o f the system. The identity r2a2/la3 + r~3/#1 + F231/U2 = P2#MM*/rnlmzrn3,
where #e = m i / M * and M* is some normalization mass, allows without limitation on/~ to require I~MM* = rnlrnzrn3.
Then, under the normalization M* = # we get (45) and under the normalization M* = M we obtain (46). The choice o f # is still a formal problem. The masses #z) and #L behave differently as one out o f m1,2, 3 tends to oo. For instance, as rn3 -+ oo, tzo -+ x / - ~ m 2 , whereas/~L -+ 0; this is inadmissible from tile physical point o f view. This property can be used, while choosing a correct expression for #. b) Based on formula (45) we introduce the following geometrical construction, namely, the triangle 2x(P12, P23, P3~) o f Fig. 2. It is uniquely determined by the masses mi that define its sides P31Pz3 = ma + rnz,
P12P31=
m 2 + m3,
P23Pl~ = m3 + r e x .
As rn i >1 O, these sides satisfy the inequality for the triangle and A(P12,/23, P31) always exists. Now we determine the three angles ~1,2,3 by the formula ~i = arctg(mi/#)
E [0, 7r/2]
(47)
and assume ti = tg ~i,
si = sin ~i,
ci = cos ffi.
P~2
Fig. 2. The triangle A(PI: , P23, P31)
(48)
78
P.P. Fiziev and Ts. Ya. Fizieva Then from (45) and the summation rule of tangents, one can easily infer ~1 + ~b2 + ~a = 7r.
(49)
It is clear from the elementary geometry that ~i are the angles between bisectors in A(P12, P2a, Pal). The reduced mass #z) is the radius of the circle inscribed into A(P12, P2a, Pal) (Fig. 2). This construction can easily be generalized to the problem with N particles, when the triangle is changed to a structure in the relevant multi-dimensional space (tetrahedron, etc.) and the circle by the inscribed (hyper-)sphere with radius ta = M ( r n l / M "'" m N / M ) II(N-1) [2] where M = m l + " " + raN. With the quantities introduced above we can write the distances re,. between the particles in a simpler form, rl= = O(sa/s,s2)l/=[cl,
r2a = O(sl/s2sa)l/Z(s~s 2 - 2szczsc cos O + C2C2) 1/2, raa = P($2/s1s3)1/2($2s 2 + 2SlClSC cos t9 + c~c2)a/2;
(50)
likewise the quantity a(~0, O) from (28), (29) can be expressed as = 3nlc1-1 + 32a(s~s = - 2s=c2sc cos 0 + c~c=) -1/= + [3ai(s]s 2 + 2 s l c l s c cos 0 + c2c2) -1/2,
(51)
where (3q = (sisjsk)l/2oqj ;
i 4=] 4= k 4= i.
(52)
The simple, but exact, relations of geometric nature (47)-(52) are new evidence in favour of the choice of # in the form of (45). 9 Comparison with Other Choices of Hyperspherical Angles We have arrived at a version of hyperspherical angles that differs somewhat from the ones described in the literature (see refs. [1, 3, 5, 6] and refs. therein). We consistently kept the description with a separated particle rn3 both in choosing the system of coordinates s together with the Euler angles {r * , O} and in choosing the angles (% O} for the description of the triangle deformation. Our expressions for the kinetic energy of the system are more convenient for solving the classical problem as they do not lead to the discontinuous changes of angles described in ref. [5 ]. This will be shown elsewhere. The introduction of angular hyperspherical coordinates for the classical problem o f three particles, democratic with respect to the three particles, is presented in ref. [5]. R is based on a modification of the hyperspherical coordinates of ref. [3 ] and differs from ours both by the choice of a rotating system of coordinates and the choice of the triangle-deformation angles. For a detailed comparison of the two choices of the triangle-deformation angles, it is convenient to represent them as angles on the two-dimensional sphere $(21 (Fig. 3). In order to put points (r = 0, ~r/2; O-arbitrary} (see Eqs. (6)-(9) and Fig. 1) in the poles of the sphere $(2) one should double the angle ~o, i.e. pass to the angle ~ = 2so.
Modification of Hyperspherical Coordinates in the Classical Three-Particle Problem
79
N
Fig. 3. The sphere $(2)
Then through variation in the interval ~bE [0, 2~r] the point on $(2) moves along the circumference thereby covering the sphere once, while 0 varies in the interval [0, zr]. The choice of (9) for ~ E [0, 2zr] leads to ~bE [0, 4~r], i.e. to the two-fold covering of the sphere. This results in a continuous description of configurations of the system, and as we shall see elsewhere, this amounts to the construction of some two-sheet Riemann surface for the classical motion. On the sphere $(2) there is a physically separated vertical circumference C(0-= 0) corresponding to collinear configurations of particles. On C there are three points with angles ~bl, ~2, ~b3 that correspond to pair collisions of particles. Under our choice of angles {~b, 0} the poles N and S of the spheres are on the circumference C, as is shown in Fig. 3. A democratic introduction of angles on the sphere $(2) corresponds to the rotation of axis NS to NS, i.e. the introduction of new angles {~1,e} by the formulae (see Fig. 3) cos 2~ = sin ~1 cos e,
cos ~1= sin 0 sin 2~.
They are simply related by the angles of the triangle deformation in refs. [1,3, 5, 13 ] and lead to 3 =
ijv
[1 -
sin n cos (e -
k=l
for the function a from (28). A drawback of such a choice is the occurrence of nonphysical coordinate singularities at points N, S on $(2) with respect to which the circumference of collinear configurations C becomes the equator. In a subsequent paper we shall apply the afore-mentioned results to a further study of the classical and quantum three-particle problem, in particular, the motion along the straight line when all dependences are extremely simplified.
80
Modification of Hyperspherical Coordinates in the Classical Three-Particle Problem
Acknowledgement. The authors are deeply indebted to A. V. Matveenko for numerous discussions and elucidation of various questions in the three-particle problem. His interesting paper [14] has initiated the present study. Thanks are also due to B. M. Barbashov, A. D. Donkov, V. B. Belyaev, Yu. A. Smorodinsky and to V. A. Mescheryakov for fruitful discussions, help, and support in preparing this paper.
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