42 references, page 1 of 5 [1] J.Schwinger, J. Math. Phys. 2, 407 (1961); P.M.Bakshi and K.Mahanthappa, J. Math. Phys. 4, 12 (1963); L.V.Keldysh, Zh. Eksper. Teoret. Fiz. 47, 151 (1964) [English translation: Sov. Phys. JETP 20, 1018 (1964)].

[2] A standard physics text on the subject is L. Landau and E. Lifshitz, Fluid Mechanics (2nd ed. Pergamon, Oxford UK 1987). A mathematical treatment is V. Arnold and B. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag, Berlin 1988). The relation between Langrange and Euler descriptions of a fluid is discussed by R. Salmon, Ann. Rev. Fluid. Mech. 20, 225 (1988); see also I. Antoniou and G. Pronko, arXiv: hep-th/0106119.

[3] These brackets were first posited (in somewhat imprecise form) by L. Landau, Zh. Eksper. Teoret. Fiz. 11, 592 (1941) [English translation: J. Phys. USSR 5, 71 (1941)]; see also P. Morrison and J. Greene, Phys. Rev. Lett. 45, 790 (1980), (E) 48 569 (1982).

[4] S0(2,1) together with the Galileo group, equivalently the “Schr¨odinger” group, provides the maximal symmetry group for the kinetic term of non-relativistic dynamics. Various interactions in various dynamical systems preserve the S0(2,1) symmetry. In particle dynamics the inverse square potential is S0(2,1) invariant [R. Jackiw, Physics Today 25(1), 23 (1972); U. Niederer, Helv. Phys. Acta 45, 802 (1972); C.R. Hagen, Phys. Rev. D 5, 377 (1972).] In fluid mechanics the equation of state that follows from (1.2.34b): pressure =∝ ρ1+2/d enjoys Schr¨odinger group invariance; see M. Hassa¨ıne and P. Horvathy, Ann. Phys. (NY) 282, 218 (2000); L. O'Raifeartaigh and V. Sreedhar, Ann. Phys. (NY) 293, 215 (2001).

[5] S. Chaplygin, Sci. Mem. Moscow Univ. Math. Phys. 21, 1 (1904). [Chaplygin was a colleague of fellow USSR Academician N. Luzin. Although accused by Stalinist authorities of succumbing excessively to foreign influences, unaccountably both managed to escape the fatal consequences of their alleged actions; see N. Krementsov, Stalinist Science (Princeton University Press, Princeton NJ 1997).] The same model (1.2.37) was later put forward by H.-S. Tsien, J. Aeron. Sci. 6, 399 (1939) and T. von Karman, J. Aeron. Sci. 8, 337 (1941); see also K. Stanyukovich, Unsteady Motion of Continuous Media (Pergamon, Oxford UK 1960), p. 128.

[6] For an introduction to the properties of the Abelian and non-Abelian Chern-Simons terms, see S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (NY) 140, 372 (1982), (E) 185, 406 (1985). In fluid mechanics or in magnetohydrodynamics the Abelian Chern-Simons term is known as the fluid or magnetic helicity, and was introduced by L. Woltier, Proc. Nat. Acad. Sci. 44, 489 (1958), and further studied in M. Berger and G. Field, J. Fluid Mech.147, 133 (1984); H. Moffatt and A. Tsinober, Ann. Rev. Fluid Mech.24, 281 (1992).

[7] L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988). For a detailed exposition, see R. Jackiw in Constraint Theory and Quantization Methods, F. Colomo, L. Lusanna, and G. Marmo, eds. (World Scientific, Singapore 1994), reprinted in R. Jackiw, Diverse Topics in Theoretical and Mathematical Physics (World Scientific, Singapore 1995).

[8] D. Bak, R. Jackiw and S.-Y. Pi, Phys. Rev. D 49, 6678 (1994), appendix.

[20] D. Bazeia, Phys. Rev. D 59, 085007 (1999).

[23] R. Jackiw and A.P. Polychronakos, Comm. Math. Phys. 207, 107 (1999).