نگاشت تکانی و عمل گروه خارج قسمتی روی فضای کاهش یافته - K
Subject Areas : هندسهMohammad Dara 1 , اکبر دهقان نژاد 2 *
1 - School of Math., Iran University of Science and Technology, Tehran, Iran
2 - School of Mathematics, Iran University of Science and Technology,Narmak, Tehran 16846 -13114, Iran.
Keywords: فروکاست, نگاشت تکانی, دستگاههای همیلتونی, هندسه همتافته,
Abstract :
In Hamiltonian systems, symmetries on the phase space are important. The symmetry of the Hamiltonian system is describable by the action of a Lie group on the phase space which leads to conservation laws on the system. The conserved quantities have been used by the founders of classical mechanics to eliminate degrees of freedom in the particular systems under investigation. These procedures are called reduction theory. The moment map is a mathematical expression of the concept of the conservation associated with the symmetries of a Hamiltonian system and is a fundamental tool for the study of symplectic reduction. In this theory, given the Hamiltonian action, the quotient of a level set of the moment map by the freely and properly action of an isotropy subgroup to form a new symplectic manifold. By considering the topological properties of the moment map and generalization of the codomain of these maps we are led to the definition of the $k$-moment maps. These maps originated in the study of invariant $k$-covariant tensor on a $G$-manifold $M$. In this paper; we prove that if $G$ has a closed normal subgroup $N$, and $M_{\nu}$ is reduced space by $k$-moment map $\mu$ in regular value $\nu$, then $M_{\nu}$ with quotient group action $G_{\nu}/N_{\nu}$ has a $k$-moment map.
[1] Arnol’d, Vladimir Igorevich. Mathematical methods of classical mechanics, vol. 60. Springer Science & Business Media, 2013.
[2] Greiner, Walter. Classical mechanics: systems of particles and Hamiltonian dynamics. Springer Science & Business Media, 2009.
[3] Abraham, Ralph, Marsden, Jerrold E, and Marsden, Jerrold E. Foundations of mechanics, vol. 36. Benjamin/ Cummings Publishing Company Reading, Massachusetts, 1978.
[4] Audin, Michèle, Da Silva, Ana Cannas, and Lerman, Eugene. Symplectic geometry of integrable Hamiltonian systems. Birkhäuser, 2012.
[5] Ortega, Juan-Pablo and Ratiu, Tudor S. Momentum maps and Hamiltonian reduction, vol. 222. Springer Science & Business Media, 2013.
[6] Marsden, Jerrold E and Ratiu, Tudor S. Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, vol. 17. Springer Science & Business Media, 2013.
[7] Hofer, Helmut and Zehnder, Eduard. Symplectic invariants and Hamiltonian dynamics. Birkhäuser, 2012.
[8] Dwivedi, Shubham, Herman, Jonathan, Jeffrey, Lisa C, and Van den Hurk, Theo. Hamiltonian Group Actions and Equivariant Cohomology. Springer, 2019.
[9] Ma, Xiaonan and Zhang, Weiping. Geometric quantization for proper moment maps. Comptes Rendus Mathematique, 347(7-8):389–394, 2009.
[10] Kostant, B. Orbits, symplectic structures and representation theory proc. in USJapan Seminar in Differential Geometry (Kyoto, 1965)”, Nippon Hyoronsha, Tokyo, p. 71, 1966.
[11] Souriau, JM. Structure des systèmes dynamiques, dunod, paris, 1970. MR, 41:4866, 1970.
[12] Dara, M and Dehghan Nezhad, A. On the -ary moment map. International Journal of Industrial Mathematics, 13(1):53– 61, 2021.
[13] Meyer, Kenneth R. Symmetries and integrals in mechanics. in Dynamical systems, pp. 259–272. Elsevier, 1973.
[14] Marsden, Jerrold and Weinstein, Alan. Reduction of symplectic manifolds with symmetry. Reports on mathematical physics, 5(1):121–130, 1974.
[15] Marsden, Jerrold E, Misiolek, Gerard, Ortega, Juan-Pablo, Perlmutter, Matthew, and Ratiu, Tudor S. Hamiltonian reduction by stages. Springer, 2007.
[16] De Nicola, Antonio and Esposito, Chiara. Reduction of pre-hamiltonian actions. Journal of Geometry and Physics, 115:178–190, 2017.
[17] Karshon, Yael. Moment maps and noncompact cobordisms. Journal of Differential Geometry, 49(1):183–201, 1998.
[18] Guillemin, Victor, Ohsawa, TL, Karshon, Yael, and Ginzburg, Viktor L. Moment maps, cobordisms, and Hamiltonian group actions. no. 98. American Mathematical Soc., 2002.
[19] Guillemin, Victor W and Sternberg, Shlomo. Supersymmetry and equivariant de Rham theory. Springer Science & Business Media, 2013.
[20] Tu, Loring W. Introductory Lectures on Equivariant Cohomology. Princeton University Press, 2020.