Abstract
Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket.
Similar content being viewed by others
References
Abraham, R. and Marsden, J., Foundations of Mechanics, 2nd edn., Addison-Wesley, 1978.
ArmsJ., MarsdenJ., and MoncriefV., Commun. Math. Phys. 78, 455–478 (1981).
ArnoldV., Ann. Inst. Fourier, Grenoble 16, 319–361 (1966).
ArnoldV., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
BayenF., FlatoM., FronsdalC., LichnerowiczA., and SternheimerD., Ann. Phys. 111, 61, 1978.
ChernoffP. and MarsdenJ., Properties of Infinite Dimensional Hamiltonian Systems, Springer Lecture Notes in Mathematics 425, Springer-Verlag, New York, 1974.
CushmanR. and RodD. L., Physica D 6, 105–112 (1982).
DepritA., Celestial Mech. 30, 181–195 (1983).
Dirac, P. A. M., Lectures on Quantum Mechanics, Belfer Grad. School of Sci., Monograph Series, Vol. 2, Yeshiva University (1964).
FlatoM., LichernowiczA., and SternheimerD., J. Math. Phys. 17, 17–54 (1976).
GuilleminV. and SternbergS., Invent. Math. 67, 515–538 (1982).
Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics, Cambridge University Press, 1984.
IwaiT., J. Math. Phys. 23, 1088–1092 (1982).
IwaiT., J. Math. Phys. 26, 885–893 (1985).
KazhdanD., KostantB., and SternbergS., Commun. Pure Appl. Math. 31, 481–508 (1978).
KirillovA. A., Russian Math. Surveys 31, 55–75 (1976).
KostantB., Quantization and Unitary Representations, Lecture Notes in Mathematics 170, Springer-Verlag, New York, 1970.
Krishnaprasad, P. S. and Marsden, J., Hamiltonian Structures and Stability for Rigid Bodies With Flexible Attachments, Arch. Rat. Mech. An. (to appear).
KummerM., Indiana Univ. Math. J. 30, 281–291 (1981).
LichnerowiczA., J. Diff. Geom. 12, 253–300 (1977).
MarleG. M., ‘Symplectic Manifolds, Dynamical Groups and Hamiltonian Mechanics’, in M.Cahen and M.Flato (eds.), Differential Geometry and Relativity, D. Reidel, Dordrecht, 1976.
Marsden, J., Lectures on Geometric Methods in Mathematical Physics, CBMS-NSF Regional Conference Series #37, SIAM, 1981.
MarsdenJ., RatiuR., and WeinsteinA., Trans. Am. Math. Soc. 281, 141–177 (1984).
MarsdenJ. and WeinsteinA., Rep. Math. Phys. 5, 121–130 (1974).
MarsdenJ. and WeinsteinA., Physica D 4, 394–405 (1982).
MarsdenJ. and WeinsteinA., Physica D 7, 305–323 (1983).
Marsden, J., Weinstein, A., Ratiu, T., Schmid, R., and Spencer, R. G., Proc. IUTAM-ISIMM Symposium on ‘Modern Developments in Analytical Mechanical’, Torino, June 7–11, 1982, Atti della Academia della Scienze di Torino 117, 289–340 (1983).
Meyer, K. R., ‘Symmetries and Integrals in Mechanics’, in Dynamical Systems, Academic Press, 1973, pp. 259–273.
MishchenkoA. S. and FomenkoA. T., Funct. Anal. Appl. 12, 113–131 (1978) and Math. USSR Izvestia 12, 371–389 (1978).
MontgomeryR., Lett. Math. Phys. 8, 59–67 (1984).
MontgomeryR., MarsdenJ., and RatiuT., Cont. Math. AMS 28, 101–114 (1984).
Nehoroshev, N., Trud. Mosk. Math. 26, 1972.
RatiuT., Involution Theorems in Springer Lecture Notes in Mathematics 775, Springer-Verlag, New York, 1980.
SmaleS., Inv. Math. 10, 205–331; 11, 45–64 (1970).
SniatyckiJ., Ann. Inst. H. Poincaré 20, 365–372 (1974).
SouriauJ. M., Structures des Systèmes Dynamiques, Dunod, Paris, 1970.
SternbergS., Proc. Nat. Acad. Sci. 74, 5253–5254 (1977).
TulczyjewW. M., ‘Poisson Reductions’, in J-P.Dufour (ed.), Singularités feuilletages et mécanique hamiltonienne, Hermann, Paris, 1985.
WeinsteinA., Adv. Math. 6, 329–346 (1971).
WeinsteinA., Lett. Math. Phys. 2, 417–420 (1978).
Woodhouse, N., Geometric Quantization, Oxford University Press, 1980.
Author information
Authors and Affiliations
Additional information
Research supported by DOE contract DE-AT03-85ER 12097.
Supported by an A. P. Sloan Foundation fellowship.
Rights and permissions
About this article
Cite this article
Marsden, J.E., Ratiu, T. Reduction of Poisson manifolds. Lett Math Phys 11, 161–169 (1986). https://doi.org/10.1007/BF00398428
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00398428