On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations

Colin Rogers

doi:10.1080/14029251.2019.1544792

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Volume 26, Issue 1, December 2018, Pages 98 - 106

On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations

Authors

Colin Rogers

School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW2052, Australiac.rogers@unsw.edu.au

Received 16 July 2018, Accepted 12 August 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1544792 How to use a DOI?
Keywords: Many Body; Ermakov; Reciprocal
Abstract: Here, a recently introduced nine-body problem is shown to be decomposable via a novel class of reciprocal transformations into a set of integrable Ermakov systems. This Ermakov decomposition is exploited to construct more general integrable nine-body systems in which the canonical nine-body system is embedded.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

The study of many-body problems with their established importance in both classical and quantum mechanics has an extensive literature motivated, in particular, by the pioneering models of Calogero [4,5], Moser [20] and Sutherland [48,49]. The surveys [6,19,22] and the literature cited therein may be consulted in this regard.

In recent work in [25], non-autonomous extensions of 3-body and 4-body systems incorporating those set down in [2, 7, 17] have been shown to be decomposible into integrable multi-component Ermakov systems. Two-component Ermakov systems adopt the form [23 , 24, 26]

α¨+ω(t)α=1α2βΦ(β/α),β¨+ω(t)β=1αβ2Ψ(α/β)(1.1)

and admit a distinctive integral of motion, namely, the invariant

ℰ=12(αβ˙−βα˙)2+∫β/αΦ(z)dz+∫α/βΨ(w)dw(1.2)

together with concomitant nonlinear superposition principles. Here, a dot indicates a derivative with respect to the independent variable t. Ermakov systems of the type (1.1) have diverse physical applications, notably in nonlinear optics [8, 13–16, 27, 28, 51]. There, in particular, they arise in the description of the evolution of size and shape of the light spot and wave front in elliptical Gaussian beams. In 2+1-dimensional rotating shallow water hydrodynamics, Hamiltonian two-component Ermakov systems of the type (1.1) have been derived in [29] which describe the time evolution of the semi-axes of the elliptic moving shoreline on an underlying circular paraboloidal basin. Ermakov-Ray-Reid systems have also been obtained in magnetogasdynamics in [30,50] and novel pulsrodon-type phenomena thereby isolated analogous to that observed in an elliptic warm-core oceanographic eddy context [31, 47]. Nonlinear coupled systems of Ermakov-type have also been shown in [32] to arise in the description of gas cloud evolution as originally investigated by Dyson [11]. In [28], it was shown that the occurrence of integrable Hamiltonian Ermakov-Ray-Reid systems in nonlinear physics and continuum mechanics, remarkably, extends to the spiralling elliptic soliton system of [9] and to its generalisation in the Bose-Einstein context of [1].

In connection with many-body problems, a nine-body system has recently been investigated in detail in [3]. The mode of treatment for this canonical system was thereby extended to a class of 3^k many-body problems. Here, an alternative approach to nine-body problems of the type in [3] is adopted wherein they are shown via a novel class of reciprocal transformations to be decomposible into an equivalent set of integrable Ermakov systems. This Ermakov connection is exploited here to embed the original nine-body system in a wide solvable class involving a triad of arbitrary functions J₁(y/x), J₂(y/x) and J₃(y/x) where x,y are Jacobi variables. These J_i(y/x), i = 1,2,3 are associated with a general parametrisation of two-component Hamiltonian Ermakov systems as originally introduced in [29].

2. A Class of Nine-Body Problems. Ermakov Reduction

Here, we consider a class of nine-body problems

x¨i=∂Z∂xi,i=1,2,…,9(2.1)

of the type introduced in [3] with

Z=∑1⩽i⩽j<3λ1,1(xi−xj)2+∑4⩽i⩽j⩽6λ2,1(xi−xj)2+∑7⩽i⩽j⩽9λ3,1(xi−xj)2 +∑1⩽i⩽j⩽33λ1,2(x3i−2+x3i−1+x3i−x3j−2−x3j−1−x3j)2−δ2(∑i=19xi)2μ∑i=19xi2.(2.2)

It is noted that δ = 0 in the system investigated in [3].

Jacobi and centre of mass co-ordinates are now introduced according to

x=12(x1−x2), y=16(x1+x2−2x3),z=12(x4−x5), v=16(x4+x5−2x6),r=12(x7−x8), s=16(x7+x8−2x9)w=13(x1+x2+x3),p=13(x4+x5+x6),q=13(x7+x8+x9).(2.3)

Under this linear transformation, the invariance property

∑:=x12+x22+⋯+x92=x2+y2+z2+v2+r2+s2+w2+p2+q2(2.4)

is seen to hold. Moreover,

x1=12x+16y+13w, x2=−12x+16y+13w, x3=−23y+13w,x4=12z+16v+13p, x5=−12z+16v+13p, x6=−23v+13p,x7=12r+16s+13p, x8=−12r+16s+13q, x9=−23s+13q.(2.5)

’In extenso’ on re-labelling, (2.2) yields

Z=λ1(x1−x2)2+λ2(x2−x3)2+λ3(x3−x1)2 +μ1(x4−x5)2+μ2(x5−x6)2+μ3(x6−x4)2 +v1(x7−x8)2+v2(x8−x9)2+v3(x9−x7)2 +σ1(x1+x2+x3−x4−x5−x6)2+σ2(x4+x5+x6−x7−x8−x9)2 +σ3(x7+x8+x9−x1−x2−x3)2 −δ2(x1+x2+…+x9)2+μx12+x22+…+x92.(2.6)

In terms of the Jacobi and centre of mass co-ordinates, the 9-body system determined by (2.1) with Z given by (2.6) becomes

x¨+2μx∑2=−λ1x3+4λ2(3y−x)3−4λ3(x+3y)3,y¨+2μy∑2=43λ2(3y−x)3−43λ3(x+3y)3,z¨+2μz∑2=−μ1z3+4μ2(3v−z)3−4μ3(z+3v)3,v¨+2μv∑2=−43μ2(3v−z)3−43μ3(z+3v)3,r¨+2μr∑2=−v1r3+4v2(3s−r)3−4v3(r+3s)3,s¨+2μs∑2=−43v2(3s−r)3−43v3(r+3s)3,w¨+2μw∑2=23[−σ1(w−p)3+σ3(q−w)3]+δ3(w+p+q)3,p¨+2μp∑2=23[σ1(w−p)3+σ2(p−q)3]+δ3(w+p+q)3,q¨+2μq∑2=23[σ2(p−q)3−σ3(q−w)3]+δ3(w+p+q)3.(2.7)

3. Application of a Reciprocal Transformation

Reciprocal-type transformations have diverse physical applications in such areas as gasdynamics and magnetogasdynamics [33,34], the solution of nonlinear moving boundary problems [12,35–38], the analysis of oil/water migration through a porous medium [39] and in Cattaneo-type hyperbolic nonlinear heat conduction [40]. Reciprocal transformations have also been applied in the theory of discontinuity-wave propagation [10]. In soliton theory, the conjugation of reciprocal and gauge transformations has been used to link integrable equations and the inverse scattering schemes in which they are embedded [18, 21, 41–45]. Here, a novel class of reciprocal transformations is introduced in the present context of many-body theory. This is used to reduce the nine-body system (2.7) in Jacobi and centre of mass co-ordinates to an equivalent set of integrable two-component Ermakov systems of the type (1.1) augmented by a classical single component Ermakov equation in a centre of mass component.

Thus, the class of reciprocal transformations

x*=x/ρ,y*=y/ρ,z*=z/ρ,v*=v/ρ,s*=s/ρ,w*=w/ρ,p*=p/ρ,q*=q/ρρ*=ρ−1,dt*=ρ−2dt}ℝ*(3.1)

such that ℝ^*2 = I is introduced wherein ρ is governed by the base equation

ρ¨+2μρ∑−2=0(3.2)

under ℝ^*, the nine-body system becomes

xt*t**=−λ1x*3+4λ2(3y*−x*)3−4λ3(x*+3y*)3,yt*t**=43λ2(3y*−x*)3−43λ3(x*+3y*)3,zt*t**=−μ1z*3+4μ2(3v*−z*)3−4μ3(z*+3v*)3,vt*t**=43μ2(3v*−z*)3−43μ3(z*+3v*)3,rt*t**=−v1r*3+4v2(3s*−r*)3−4v3(r*+3s*)3,st*t**=−43v2(3s*−r*)3−43v3(r*+3s*)3,(3.3)

together with

wt*t**=23[−σ1(w*−p*)3+σ3(q*−w*)3]+δ3R*3,pt*t**=23[σ1(w*−p*)3+σ2(p*−q*)3]+δR*3,qt*t**=23[σ2(p*−q*)3+σ3(q*−w*)3]+δR*3,(3.4)

where

R*=Rρ,R=13(x1+x2+⋯+x9)=w+p+q.(3.5)

In the above, (3.3) constitute three copies of integrable Hamiltonian Ermakov systems of the same kind as obtained in [25] for the original 3-body system of Calogero [4].

On introduction of the translational change of variables

W*=w*−p*, P*=p*−q*(3.6)

the system (3.4) produces the two-component Ermakov system

Wt*t**=23[−2σ1W*3+σ2P*3−σ3(W*+P*)3],Pt*t**=23[σ1W*3+2σ2P*3−σ3(W*+P*)3](3.7)

while addition of the constituent members of (3.3)–(3.4) shows that

Rt*t**=δ3R*3(3.8)

namely, a single component Ermakov equation in R^*.

The triad (3.3) constitutes three de-coupled Ermakov systems of the type (1.1) and with associated Hamiltonians

ℋI=12[xt**2+yt**2]+1x*2[λ12+2λ2(3y*/x*−1)3+2λ3(3y*/x*+1)3],ℋII=12[zt**2+vt**2]+1z*2[μ12+2μ2(3v*/z*−1)3+2μ3(3v*/z*+1)3],ℋIII=12[rt**2+st**2]+1r*2[ν12+2ν2(3s*/r*−1)3+2ν3(3s*/r*+1)3].(3.9)

The sub-system (3.4), on the other hand admits the Hamiltonian

ℋIV=12[wt**2+pt**2+qt**2]−13[σ1(w*−p*)2+σ2(p*−q*)2+σ3(w*−q*)2]+δ6(w*+p*+q*)2(3.10)

that is,

ℋIV=16[2Wt**2+(Pt**+Rt**)2]−13[σ1W*2+σ2P*2+σ3(W*+P*)2]+σ6R*2(3.11)

wherein R^* is determined by the Ermakov equation (3.8).

Thus, the nine-body system encapsulated in (3.3)–(3.4) is seen to be reducible to a quartet of two-component Ermakov systems of the classical type (1.1), augmented by the single component Ermakov equation (3.8) in R^*. Moreover, the Ermakov-Ray-Reid systems in (3.3) and (3.7), in addition to their admittance of characteristic Ermakov invariants, also admit second integrals of motion, namely, the Hamiltonians ℋ_I ···ℋ_IV and hence are integrable.

The preceding determines x^*(t^*), y^*(t^*), z^*(t^*), v^*(t^*), r^*(t^*), s^*(t^*) together with w^* − p^*, p^*−q^*and R^* = w^* + p^* + q^* and hence w^*(t^*), p^*(t^*) and q^*(t^*). In addition, the base equation (3.2) in ρ, in the reciprocal variables becomes

ρt*t**−2μρ*[x*+y*2+⋯+q*2]2=0.(3.12)

The latter becomes determinate on insertion of x^*, y^*,...,q^* as obtained by means of the integrable Hamiltonian Ermakov systems. With ρ^*(t^*) to hand, t = t(t^*) is determined by the pair of reciprocal relations

ρ*=ρ−1,dt*=ρ−2dt(3.13)

so that

dt=ρ*−2dt*(3.14)

while the original Jacobi and centre of mass variables x(t), y(t),..., q(t) are given parametrically via t^* by the residual reciprocal relations

x=x*(t*)/ρ*(t*),y=y*(t*)/ρ*(t*), …, q=q*(t*)/ρ*(t*).(3.15)

4. An Extended Solvable Nine-Body System. Application of the Ermakov Connection

The isolation of integrable nonlinear systems of Ermakov-type is a subject of current interest (see e.g. [46]). Here, the Ermakov reduction of the nine-body system (2.1)–(2.2) may be exploited to embed it in a more general novel class of solvable nonlinear systems. Thus, it was shown in [29] that the class of Ermakov-Ray-Reid systems.

α¨=2α3J(β/α)+βα4dJ(β/α)/d(β/α)β¨=−1α3dJ(β/α)/d(β/α)(4.1)

parametrised in terms of arbitrary J(β/α), admits the Hamiltonian

ℋ=12[α˙2+β˙2]+1α2J(β/α)(4.2)

and Ermakov invariant

ℰ=12(αβ˙−α˙β)2+(α2+β2α2)J(β/α)(4.3)

which together allow the solution of the system (4.1) in an algorithmic manner.

In the present nine-body context, the Ermakov connection shows that the system (3.3) may be embedded in the integrable triad of Hamiltonian Ermakov-Ray-Reid systems

xt*t**=−2x*3J1(y*/x*)+y*x*4dJ1(y*/x*)/d(y*/x*),yt*t**=−1x*3J1(y*/x*),zt*t**=−2z*3J2(v*/z*)+v*z*4dJ2(v*/z*)/d(v*/z*),vt*t**=−1z*3J2(v*/z*),rt*t**=−2r*3J3(r*/s*)+s*r*4dJ3(r*/s*)/d(r*/s*),st*t**=−1r*3J3(r*/s*),(4.4)

parametrised in terms of arbitrary J₁(y^*/x^*), J₂(v^*/z^*) and J₃(r^*/s^*). These systems augmented by the integrable triad (3.4) determine a solvable nine-component system which is reciprocally associated to a novel class of nine-body problems in which the original system (2.1)–(2.2) is embedded.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 1
Pages: 98 - 106
Publication Date: 2021/01/06
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DOI: 10.1080/14029251.2019.1544792 How to use a DOI?
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TY  - JOUR
AU  - Colin Rogers
PY  - 2021
DA  - 2021/01/06
TI  - On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 98
EP  - 106
VL  - 26
IS  - 1
SN  - 1776-0852
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