Geodesic
Remarks on Multi-Dimensional Conformal Mechanics
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Recently, Galajinsky, Lechtenfeld and Polovnikov proposed an elegant group-theoretical transformation of the generic conformal-invariant mechanics to the free one. Considering the classical counterpart of this transformation, we relate this transformation with the Weil model of Lobachewsky space.
Keywords: conformal mechanics; integrability. 
@article{SIGMA_2009_5_a3,
     author = {Cestm{\'\i}r Burd{\'\i}k and Armen Nersessian},
     title = {Remarks on {Multi-Dimensional} {Conformal} {Mechanics}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a3/}
}
TY  - JOUR
AU  - Cestmír Burdík
AU  - Armen Nersessian
TI  - Remarks on Multi-Dimensional Conformal Mechanics
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2009
VL  - 5
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a3/
LA  - en
ID  - SIGMA_2009_5_a3
ER  - 
%0 Journal Article
%A Cestmír Burdík
%A Armen Nersessian
%T Remarks on Multi-Dimensional Conformal Mechanics
%J Symmetry, integrability and geometry: methods and applications
%D 2009
%V 5
%U http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a3/
%G en
%F SIGMA_2009_5_a3
Cestmír Burdík; Armen Nersessian. Remarks on Multi-Dimensional Conformal Mechanics. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a3/

[1] de Alfaro V., Fubini S., Furlan G., “Conformal invariance in quantum mechanics”, Nuovo Cimento A, 34 (1976), 569–612 | DOI

[2] Calogero F., “Solution of a three-body problem in one-dimension”, J. Math. Phys., 10 (1969), 2191–2196 ; Calogero F., “Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys., 12 (1971), 419–436 | DOI | MR | DOI | MR

[3] Polychronakos A. P., “Physics and mathematics of Calogero particles”, J. Phys. A: Math. Gen., 39 (2006), 12793–12845 ; hep-th/0607033 | DOI | MR | Zbl

[4] Gurappa N., Prasanta K., “Equivalence of the Calogero–Sutherland model to free harmonic oscillators”, Phys. Rev. B, 59 (1999), R2490–R2493 ; ; Ghosh P. K., “Super-Calogero–Moser–Sutherland systems and free super-oscillators: a mapping”, Nuclear Phys. B, 595 (2001), 519–535 ; ; Brzezinski T., Gonera C., Maslanka P., “On the equivalence of the rational Calogero–Moser system to free particles”, Phys. Lett. A, 254 (1999), 185–196 ; cond-mat/9710035hep-th/0007208hep-th/9810176 | DOI | DOI | MR | Zbl | DOI

[5] Galajinsky A., Lechtenfeld O., Polovnikov K., “Calogero models and nonlocal conformal transformations”, Phys. Lett. B, 643 (2006), 221–227 ; hep-th/0607215 | DOI | MR

[6] Freedman D. Z., Mende P. F., “An exactly solvable $N$-particle system in supersymmetric quantum mechanics”, Nuclear Phys. B, 344 (1990), 317–343 | DOI | MR

[7] Galajinsky A., Lechtenfeld O., Polovnikov K., “$N=4$ superconformal Calogero models”, J. High Energy Phys., 2007:11 (2007), 008, 23 pp., ages ; ; Galajinsky A., Lechtenfeld O., Polovnikov K., $N=4$ mechanics, WDVV equations and roots, ; Lechtenfeld O., WDVV solutions from orthocentric polytopes and Veselov systems, arXiv:0708.1075arXiv:0802.4386arXiv:0805.3245 | DOI | MR

[8] Galajinsky A. V., “Remark on quantum mechanics with conformal Galilean symmetry”, Phys. Rev. D, 78 (2008), 087701, 3 pp., ages ; arXiv:0808.1553 | DOI | MR

[9] Hakobyan T., Nersessian A., Lobachevsky geometry of (super)conformal mechanics, arXiv:0803.1293 | MR

[10] Landau L. D., Lifshits E. M., Mechanics, 5th ed., Nauka, Moscow, 2004

[11] Higgs P. W., “Dynamical symmetries in a spherical geometry. I”, J. Phys. A: Math. Gen., 12 (1979), 309–323 | DOI | MR | Zbl

[12] Schrödinger E., “A method of determining quantum-mechanical eigenvalues and eigenfunctions”, Proc. Roy. Irish Acad. Sect. A, 46 (1941), 9–16 ; Schrödinger E., “Further studies on solving eigenvalue problems by factorization”, Proc. Roy. Irish Acad. Sect. A, 46 (1941), 183–206 ; Schrödinger E., “The factorization of the hypergeometric equation”, Proc. Roy. Irish Acad. Sect. A, 47 (1941), 53–54 ; physics/9910003 | MR | MR | Zbl | MR | Zbl

[13] Otchik V. S., “Symmetry and separation of variables in the two-center Coulomb problem in three dimensional spaces of constant curvature”, Dokl. Akad. Nauk BSSR, 35 (1991), 420–424 (in Russian) | MR

[14] Nersessian A., Yeghikyan V., “Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems”, J. Phys. A: Math. Theor., 41 (2008), 155203, 11 pp., ages ; arXiv:0710.5001 | DOI | MR | Zbl

[15] Hakobyan T., Nersessian A., Yeghikyan V., Cuboctahedric Higgs oscillator from the Calogero model, arXiv:0808.0430

[16] Bellucci S., Krivonos S., Sutulin A., “$N=4$ supersymmetric 3-particles Calogero model”, Nuclear Phys. B, 805 (2008), 24–39 ; arXiv:0805.3480 | DOI | MR

[17] Arnold V. I., Mathematical methods in classical mechanics, Nauka, Moscow, 1973