Journal of Nonlinear Mathematical Physics

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

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

Colin Rogers
School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW2052,
Received 16 July 2018, Accepted 12 August 2018, Available Online 6 January 2021.
10.1080/14029251.2019.1544792How to use a DOI?
Many Body; Ermakov; Reciprocal

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.

© 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 (

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]

and admit a distinctive integral of motion, namely, the invariant
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, 1316, 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 3k 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 Ji(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

of the type introduced in [3] with

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

is seen to hold. Moreover,
’In extenso’ on re-labelling, (2.2) yields

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 [12,3538], 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, 4145]. 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

such that ℝ*2 = I is introduced wherein ρ is governed by the base equation
under ℝ*, the nine-body system becomes
together with

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

the system (3.4) produces the two-component Ermakov system
while addition of the constituent members of (3.3)(3.4) shows that
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


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

that is,
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 I ···ℋIV 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

so that
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

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.

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

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.


[25]C. Rogers. Multi-component Ermakov and non-autonomous many-body system connections, Ricerche di Matematica to be published (2018)
[46]C. Rogers and W.K. Schief, Ermakov-type systems in nonlinear physics and continuum mechanics, N. Euler (editor), Nonlinear Systems and Their Remarkable Mathematical Structures, CRC Press, Boca Raton Fl, 2018.
Journal of Nonlinear Mathematical Physics
26 - 1
98 - 106
Publication Date
ISSN (Online)
ISSN (Print)
10.1080/14029251.2019.1544792How to use a DOI?
© 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 (

Cite this article

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  -
DO  - 10.1080/14029251.2019.1544792
ID  - Rogers2021
ER  -