We show that the three-body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one variable.
REFERENCES
1.
Adler
, M.
, “Some finite dimensional integrable systems and their scattering behavior
,” Commun. Math. Phys.
55
, 195
–230
(1977
).2.
Benenti
, S.
, Chanu
, C.
, and Rastelli
, G.
, “The super-integrability of the three body inverse-square Calogero system
,” J. Math. Phys.
41
, 4654
–4678
(2000
).3.
Calogero
, F.
, “Solution to a three-body problem in one dimension
,” J. Math. Phys.
10
, 2191
–2196
(1969
).4.
Calogero
, F.
, “Solution to the one-dimensional -body problems with quadratic and/or inversely quadratic pair potentials
,” J. Math. Phys.
12
, 419
–436
(1971
).5.
Evans
, N. W.
, “Superintegrability in classical mechanics
,” Phys. Rev. A
41
, 5666
–5626
(1990
).6.
Fehér
, L.
, Tsutsui
, I.
, and Fülöp
, T.
, “Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables
,” Nucl. Phys. B
715
, 713
–757
(2005
).7.
Friš
, I.
, Mandrosov
, V.
, Smorodinsky
, Ya. A.
, Uhliř
, M.
, and Winternitz
, P.
, “On higher order symmetries in quantum mechanics
,” Phys. Lett.
16
, 354
–356
(1965
).8.
Gravel
, S.
, “Hamiltonians separable in Cartesian coordinates and third order integrals of motion
,” J. Math. Phys.
45
, 1003
–1019
(2004
).9.
Gravel
, S.
, and Winternitz
, P.
, “Superintegrability with third order integrals in quantum and classical mechanics
,” J. Math. Phys.
43
, 5902
–5912
(2002
).10.
Hartmann
, H.
, “Die Bewengung eines Körpers in einem ringförmigen Potentialfeld
,” Theor. Chim. Acta
24
, 201
–206
(1972
).11.
Hietarinta
, J.
, “Classical vs quantum integrability
,” J. Math. Phys.
25
, 1833
–1840
(1989
).12.
Horwood
, J. T.
, McLenaghan
, R. G.
, and Smirnov
, R. G.
, “Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
,” Commun. Math. Phys.
259
, 679
–705
(2005
).13.
Jacobi
, C. G. J.
, “Sur l’élimination des noeuds dans le Problème des Trois Corps
,” J. Reine Angew. Math.
26
, 115
–131
(1843
).14.
Kalnins
, E. G.
, Kress
, J.
, and Miller
, W.
, Jr., “Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory
,” J. Math. Phys.
46
, 1
–28
(2005
).15.
Kibler
, M.
, and Winternitz
, P.
, “Dynamical invariance algebra of the Hartmann potential
,” J. Phys. A
20
, 4097
–4108
(1987
).16.
Makarov
, A. A.
, Smorodinsky
, Ya. A.
, Valiev
, Kh.
, and Winternitz
, P.
, “A systematic approach for nonrelativistic systems with dynamical symmetries
,” Nuovo Cimento D
52
, 1061
–1084
(1967
).17.
McLenaghan
, R. G.
, Smirnov
, R. G.
, and The
, D.
, “Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the -symmetric Yang-Mills theories of Yatsun
,” J. Math. Phys.
43
, 1422
–1422
(2002
).18.
McLenaghan
, R. G.
, Smirnov
, R. G.
, and The
, D.
, “An extension of the classical theory of invariants to pseudo-Riemannian geometry and Hamiltonian mechanics
,” J. Math. Phys.
45
, 1079
–1120
(2004
).19.
Nekhoroshev
, M. N.
, “Action-angle and their generalizations
,” Trans. Mosc. Math. Soc.
26
, 180
–198
(1972
).20.
Rañada
, M. F.
, “Superintegrability of the Calogero-Moser system: Constants of motion, master symmetries and time-dependent symmetries
,” J. Math. Phys.
40
, 236
–247
(1999
).21.
Rauch-Wojciechowski
, S.
, and Waksjö
, C.
, “What an effective criterion of separability says about the Calogero type systems?
,” J. Nonlinear Math. Phys.
12
, 535
–547
(2005
).22.
Rodriguez
, M. A.
, and Winternitz
, P.
, “Quantum superintegrability and exact solvability in dimensions
,” J. Math. Phys.
43
, 1309
–1322
(2002
).23.
Tempesta
, P.
, Turbiner
, A. V.
, and Winternitz
, P.
, “Exact solvability of superintegrable systems
,” J. Math. Phys.
42
, 4248
–4257
(2001
).24.
Tempesta
, P.
, Winternitz
, P.
et al. (editors), Superintegrability in Classical and Quantum Systems
, CRM Proceedings and Lecture Notes
, Vol. 37
(American Mathematical Society
, Providence, RI, 2004
).25.
Winternitz
, P.
, and Friš
, I.
, “Invariant expansions of relativistic amplitudes and subgroups of the proper Lorenz group
,” Sov. J. Nucl. Phys.
1
, 636
–643
(1965
).26.
Wojciechowski
, S.
, “Superintegrability of the Calogero-Moser systems
,” Phys. Lett.
95A
, 279
–281
(1983
).© 2006 American Institute of Physics.
2006
American Institute of Physics
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