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

- DOI
- 10.1080/14029251.2019.1544792How 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]

*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

*J*

_{1}(

*y/x*)

*, J*

_{2}(

*y/x*) and

*J*

_{3}(

*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

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

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

Under this linear transformation, the invariance property

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

## 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

^{*2}= I is introduced wherein

*ρ*is governed by the base equation

^{*}, the nine-body system becomes

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

*R*.

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

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

*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

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

^{*}*x*(

*t*)

*, y*(

*t*)

*,..., q*(

*t*) are given parametrically via

*t*by the residual reciprocal relations

^{*}## 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.

*J*(

*β/α*), admits the Hamiltonian

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

*J*

_{1}(

*y*

^{*}*/x*)

^{*}*, J*

_{2}(

*v*

^{*}*/z*) and

^{*}*J*

_{3}(

*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.

^{*}## References

*Ricerche di Matematica*to be published (2018)

### Cite this article

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 UR - https://doi.org/10.1080/14029251.2019.1544792 DO - 10.1080/14029251.2019.1544792 ID - Rogers2021 ER -