On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations
- 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.
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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  and Sutherland [48,49]. The surveys [6,19,22] and the literature cited therein may be consulted in this regard.
In recent work in , 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]
In connection with many-body problems, a nine-body system has recently been investigated in detail in . 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  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 .
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 .
Jacobi and centre of mass co-ordinates are now introduced according to
Under this linear transformation, the invariance property
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  and in Cattaneo-type hyperbolic nonlinear heat conduction . Reciprocal transformations have also been applied in the theory of discontinuity-wave propagation . 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
On introduction of the translational change of variables
The sub-system (3.4), on the other hand admits the Hamiltonian
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 ℋI ···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
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. ). 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  that the class of Ermakov-Ray-Reid systems.
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
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 -