On explicit solutions for completely integrable classical Calogero-Moser systems in external fields

11 Pages·0.09 MB·English

by  D. V. Meshcheryakov

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On explicit solutions for completely integrable
classical Calogero-Moser systems in external fields
D.V.Meshcheryakov, T.D.Meshcheryakova
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2 Physics Department, Moscow State University,

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Leninskie gory, 119899, Moscow, Russia
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0 Abstract
1
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2 Explicit solutions for one completely-integrable system of Calogero-
0
/
h Moser type in external fields are found in case of three and four interacting
t
-
p
particles. Relation between coupling constant, initial values of coordinates
e
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: and time of falling into the singularity of potential is derived.
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1. Introduction
Completely integrable finite-dimensional many-particle systems still
attract considerable attention[1-6]. By now different methods of solving
equations of motion have been developed for many potentials giving the
property of complete integrability [1-3]. One of such methods for Calogero-
1
Moser systems in external fields was found in [5]. Here we apply this method
in some special cases of two-parameter Calogero-Moser type potential to
obtain explicit solutions of the equations of motion. We investigate three
and four particle systems. We analyze explicit solutions and estimate the
time of falling particles into the singularities of the potentials for different
initial conditions.
Complete integrability is guaranteed by the existence of N involutory
constants of motion. One of these constants is Hamilton function which has
the form
N 2 N
p
H = i + W(x ) + V (x x ) (1)
i i j
(cid:18) 2 (cid:19) −
Xi=1 Xi>j
System (1) is completely integrable when and only when the external
field potential W(x) and pair interaction potential V (x) are the following:
a
4 2
V (x) = , W(x) = γ x + γ x + γ x (2)
1 2 3
x2
where a,γ ,i = 1,2,3 are arbitrary real constants.
i
In [5] a connection of N-particle system with Hamiltonian (1) and po-
tentials (2) with non-homogeneous Burgers–Hopf equation was established.
This connection led to the method of solving equations of motion for systems
(1)-(2). If the constants defining potentials satisfy the following constraints
2
α
2
a = A , γ = , γ = αβ, γ = αA(N 1) (3)
1 2 3
− − 2 − − −
where A,α,β are arbitrary real constants, then N coordinates of the
particles are N real zeros of the following function
2
N
φ(x,t) = xk[exp(tTˆ)~c(o)] (4)
k
kX=0
where
ˆ ˆ ˆ ˆ ˆ
T = β, T = a, T = Nβ, T = 2α, T = α
00 02 N−1N N−1N−2 NN−1
− − −
(k + 1)(k + 2)
ˆ ˆ ˆ
T = (N + 1 k)α, T = (k + 1)β, T = a,
kk−1 kk+1 kk+2
− − 2
1 k N 2 (5)
≤ ≤ −
and
c (0) = ( 1)k x (0)...x (0), c (0) = 1.
N−k − λ1 λk N
1≤λ1<X...<λk≤N
ˆ
Other elements of the matrix T are zeros. In these expressions x (0),i =
i
1,...N are the initial coordinates of N particles.
2. Two-parameter class of potentials.
Further we apply the method briefly described to obtain explicit solu-
tions for the following two-parameter class of potentials of the type (2)-(3).
Consider the following constraints
2
β = 0, α = A k (6)
−
where k,A are arbitrary constants. The case k = 1 was considered in
[6].Therefore we have two parameters A and k in the potentials V (x), W(x).
Constant A represents the intensity of pair interaction, so it can be called a
coupling constants for the system in question. Parameter k defines the ratio
of the pair interaction intensity to the intensity of the external field. At
3
k 0 particles are influenced by the pair interaction only, and at k
→ → ∞
we have no pair interaction. In this case particles are influenced by the
external field only.
3. Two particles
Here we apply the method to the simplest case of two particles. For
ˆ3 3 2
N = 2 from (4)-(5) one gets T = ( 2a k )E, where E is a unit matrix.
−
In this case the row in (4) can be summed up and the coordinates are the
two solutions of the following equation
2 k
x
S (θ)T¯k~c(o)] = 0 (7)
Xk=0(cid:18)−a(2k2)13 (cid:19) k k
where aT¯ = Tˆ, θ = at(2k2)1/3,
−
1 √3
S (θ) = (eθ + 2e−θ/2cos( θ)),
0
3 2
1 √3 π
S (θ) = (eθ 2e−θ/2cos( θ + ))
1
3 − 2 3
1 √3 π
S (θ) = (eθ 2e−θ/2cos( θ ))
2
3 − 2 − 3
At k = 1 the asymptotic form of this equation for arbitrary initial
values is the following x2 21/3x+22/3 = 0. Thus we come to the conclusion
−
that in some finite time interval t the particles fall into the singularity of
0
the pair interaction. However at k = 1 for arbitrary initial values x (0) and
1
6
x (0) there exist such values of parameter k that particles do not fall into
2
the singularity of the potential. For example, in case x (0) = c, x (0) = c
1 2
−
particles do not fall into the singularity if the value of k satisfies the following
inequality
4
1
2
> c 1
k2 −
For arbitrary initial conditions the system does not collapse under the
condition that both the initial coordinates and k satisfy the following in-
equality
3 2 2 2
k 8x (0)x (0)(x (0) + x (0)) + k 4(x (0) x (0)
1 2 1 2 1 2
−
2
(x (0)+x (0)) 1)+k4(x (0)+x (0))+4(x (0)x (0) 1) (x (0)+x (0)) < 0
1 2 1 2 1 2 1 2
− − − −
This inequality means that at some k the attraction of the particles due
to the pair potential is compensated by the repelling due to the external
field so that the particles do not fall into the singularity in any finite time
interval.
4. Three particles
In this section we consider N = 3 system. Under constraints (6) one
ˆ4
can get T = 0 , and the algebra of T-matrix becomes nilpotent. Further
we consider symmetrical initial conditions
x (0) = x , x (0) = 0, x (0) = x (8)
1 0 2 3 0
−
2
Suppose r = 1/( A t). Then (8) has the following Cardano form:
−
2 2
r (3 + x r)
3 0
y + uy + q = 0, x = y (9)
− 3Z
where
4 4 2 2 2
r (x k + 9r + 3x r )
3 2 2 2 0 0
Z = r x k r k , u = − ,
− 0 − 3Z2
5
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k C rm
q = m=0 m ,
P Z3
4 4 2 4 4 2 2 6
C = k , C = 3k x , C = 3k x , C = k (k x 3),
0 1 0 2 0 3 0
−
2
2 2 2 4 2 6
C = 6k x , C = 3k x , C = 4 k x ,
4 0 5 0 6 0
− − − 27
2
2 4
C = 4x , C = x
7 0 8 0
3
Solutions of (9) are determined by the value of the discriminant Q =
3 2
108((u/3) + (q/2) ) [8]. At 0 t < t we thus have Q > 0 and as
0
− ≤
a consequence three real roots being the three coordinates of interacting
particles:
N + M N M
y = N + M, y = i√3 −
1 2,3
− 2 ± 2
where
q Q q Q
1 1
N = ( + − )3, M = ( − )3,
−2 r108 −2 − r108
provided MN = u/3. At t = t we have Q = 0 , which leads to two
0
−
equal coordinates of two particles. In other words at this moment two of
the particles fall into the singularity of the potential of pair interaction.
For arbitrary initial conditions one can find from (3)–(5) the asymptotic
form of equation (9) to be (provided A = 0 ) :
6
1
3
(1 kx (0)x (0)x (0))(x ) = 0 .
1 2 3
− − k
From this relation one can easily conclude that the system in question
collapses in finite time interval for any initial conditions.
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Consider a limiting procedure which enables us to find exact expression
2
for t . Suppose A = z, k = 1/z . Suppose z 0. This limiting procedure
0
→
corresponds to infinitely weak pair interaction at finite external field. In
this case it is easy to find t explicitly. Consider initial conditions of the
0
form x (0) = x ,x (0) = x ,x (0) = 0. Without pair interaction the third
1 1 2 2 3
coordinate remains equal to zero. Two other solutions are the following
1 2
2 123(x1 + x2) 123
U(t,k,x ,x ) = t t +
1 2
− kx x k2x x
1 2 1 2
1 1
1 123 123
x (t) = − (t )
1,2
U(t,k,x ,x ) k − kx
1 2 2,1
From these expressions one can get for t
0
1 (x + x ) 1 |x1−x2|
3 3 1 2 − |x1+x2|
t = (cid:12) (cid:16) (cid:17)(cid:12)
0
(cid:18)2(cid:19) (cid:12) kx x (cid:12)
1 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Using the discriminant on(cid:12)e can get for general in(cid:12)itial conditions the
following upper estimate for t :
0
1
t = ((1 k) + 9(x (0)x (0) + x (0)x (0) + x (0)x (0))+
0 1 2 1 3 2 3
4A2(k + k−1/4)| −
1
3(k 1)x (0)x (0)x (0) 21(k 1)(x (0)+x (0)+x (0))
1 2 3 1 2 3
− − − kx (0)x (0)x (0)|
1 2 3
(10)
This estimate is valid for x (0) = 0, i = 1,2,3. It is easy to see that t
i 0
6
2
increases as A and/or k decrease. For some initial conditions there exists
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local maximum of t at finite k > 0. For example, for x (0) = x , x (0) =
0 1 0 2
−
x , x (0) = x , if
1 3 0
1 24x2(12 + √15) 43
0
x > −
1 2
129 x + 7
0
or
1 24x2(12 √15) 43
0
x < − −
1 2
129 x + 7
0
there is a local maximum of t at k . We do not write down the exact
0 0
expression due to its complexity. A simple upper estimate reads as follows
1
k = 9(W 8x2) + 129(W2 13x2W + 41x4)
0 0 0 0
8W (cid:18) − q − (cid:19)
2
where W = 1 + 3x (7 + x ).
1 0
5. Four particles
ˆ5 2 3 ˆ2
For N = 4 systems equations (4)-(5) lead to T = 18k a T . In this
case it is possible to sum up the row in (4). After summation equation (7)
turns into the following expression
2
xlS (θ)T¯l~c(o)] = 0
l l
Xl=0
where aT¯ = Tˆ, θ = at(2)1/3,
−
S (θ) = 1, S (θ) = (18)−1/3k−2/3θ,
0 1
9 √3 π 1 √3
S (θ) = (eθ 2e−θ/2cos( θ )) , S (θ) = (eθ+2e−θ/2cos( θ) 3),
2 3
121/3k4/3 − 2 −3 54k2 2 −
(18)−1/3 √3 π
S (θ) = (eθ 2e−θ/2cos( θ + ) 3θ) (13)
4
54k8/3 − 2 3 −
8
Therefore the four coordinates of interacting particles at any given time
are defined by the real roots of
4
R xl = 0. (14)
l
Xl=0
For initial conditions of the form x (0) = 2x ,x (0) = 0,x (0) =
1 0 2 3
−
x ,x (0) = 2x , coefficients in (14) have the form
0 4 0
3 3 2 2 3 2
R = 24k x S 8k x S x (S + 30k S ) + (1 + 12k S ) ,
0 0 3 0 2 0 1 4 3
− −
3 3 2 2 2
R = 24k x S 8kx (S + 18k S ) x (1 + 30k S ) + 12kS
1 0 2 0 1 4 0 3 2
− −
3 3 2 2 2
R = 4x (3kS + 54k S ) 4x (1 + 18k S ) 15kx S + (6S + 108k S )
2 0 1 4 0 3 0 2 1 4
− −
3 2 2 2
R = 4x (1 + 6k S ) 8kx S 3x (S + 10k S ) + 12kS
3 0 3 0 2 0 1 4 3
− −
3 2 2
R = 12kx S + 4x (S + 18k S ) + 15kx S 6S
4 0 2 0 1 4 0 3 2
− −
At t the asymptotic form of (14) reads as follows
→ ∞
3 1
x4 + k−1/3(18)1/3x3 + k−2/3(12)1/3x2 + k−1x k−4/3(18)1/3 = 0
2 − 2
This equation has only two real roots. Thus we come to the conclusion
that for arbitrary x two particles fall into the singularity of the potential
0
in finite time interval t .
0
4
For arbitrary x (0) and k = 1 by reducing (14) to the form x +
i
2
ux + qx + r = 0 and using the general expression for the discriminant
9
4 3 2 2 2 2 2 3
[8] D = 16u r 4u q 128u q +144uq r 27q +256r , one can derive
− − −
the following upper estimate for t :
0
(18)1/3 4 x2(0)x2(0)
j<l j l
t = (15)
0 4A2 P 4 x2(0)
j=1 j
P
2
One can conclude from (15) that the smaller the coupling constant A
is, the greater t is.
0
By now at N > 4 such estimates have not been derived yet. For
example, at N = 5 one can obtain from (4)-(5)
ˆ6 ˆ3
R = R + bE,
where Rˆ = (2−1(9)−1/3k−2/3)Tˆ and b = (5/2)23−4, E– identity matrix. In
this case we have
bθ6 ∞ a θ9+3l
l
S (θ) = 1 + + ,
0
6! (9 + 3l)!
Xl=1
where
l−3
(l + m)!
a = b bm.
l
(l 1)!m!
mX=0 −
Therefore summation of rows in (4) becomes more complicated.
5.6 Summary
We have considered Calogero-Moser systems in external fields under
special constraint (6). In fact we have considered two-parameter subclass
of potentials defined by (1), (2). For two-, three- and four-particle systems
exact solutions have been derived and analyzed. We have derived estimates
10