zbMATH — the first resource for mathematics

Dirac manifolds. (English) Zbl 0850.70212
Summary: A Dirac structure on a vector space \(V\) is a subspace of \(V\) with a skew form on it. It is shown that these structures correspond to subspaces of \(V\oplus V^*\) satisfying a maximality condition, and having the property that a certain symmetric form on \(V\oplus V^*\) vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of \(O(V)\). Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace.
Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold \(P\), which are defined to be smooth subbundles of the bundle \(TP\oplus T^*P\) satisfying pointwise the properties of the linear case. If a bundle \(L\subset TP \oplus T^*P\) defines a Dirac structure on \(P\), then we call \(L\) a Dirac bundle over \(P\). A \(3\)-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are \([B,B]\) and \(d\Omega\), respectively. A function \(f\) on a Dirac manifold is called admissible if there is a vector field \(X\) such that the pair \((X,df)\) is a section of the Dirac bundle \(L\); the pair \((X,df)\) is called an admissible section. The set of admissible functions is shown to be a Poisson algebra.
A process is given for passing Dirac structures to a submanifold \(Q\) of a Dirac manifold \(P\). The induced bracket on admissible functions on \(Q\) is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.

53D17 Poisson manifolds; Poisson groupoids and algebroids
53D05 Symplectic manifolds (general theory)
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H45 Constrained dynamics, Dirac’s theory of constraints
Full Text: DOI
[1] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. · Zbl 0393.70001
[2] Gloria Alvarez-Sanchez [1986], Geometric methods of classical mechanics applied to control theory, Ph.D. thesis, University of California at Berkeley.
[3] Coste, Dazord and Weinstein [1986], (to appear).
[4] Pierre Dazord, Feuilletages à singularités, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 1, 21 – 39 (French). · Zbl 0584.57016
[5] A. M. Dirac [1964], Lectures in quantum mechanics, Yeshiva University. · Zbl 0141.44603
[6] J. Gotay and J. E. Nester [1979], Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 129.
[7] J. Gotay [1983], Coisotropic imbeddings, Dirac brackets and quantization, preprint.
[8] Mark J. Gotay, James M. Nester, and George Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys. 19 (1978), no. 11, 2388 – 2399. · Zbl 0418.58010
[9] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 14. · Zbl 0364.53011
[10] J. Hanson, T. Regge, and C. Teitelboim [1976], Accad. Naz. Lincei Rome 22.
[11] Robert Hermann, Lie algebras and quantum mechanics, W. A. Benjamin, Inc., New York, 1970. · Zbl 0206.26901
[12] André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253 – 300 (French). · Zbl 0405.53024
[13] Robert G. Littlejohn, A guiding center Hamiltonian: a new approach, J. Math. Phys. 20 (1979), no. 12, 2445 – 2458. · Zbl 0444.70020
[14] Robert G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids 24 (1981), no. 9, 1730 – 1749. · Zbl 0473.76123
[15] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, 1987. · Zbl 0683.53029
[16] Jerrold E. Marsden and Tudor Ratiu, Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), no. 2, 161 – 169. · Zbl 0602.58016
[17] Jean Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 95 – 178 (French, with English summary). · Zbl 0189.10001
[18] Richard Montgomery [1985], personal communication.
[19] Yong-Geun Oh, Some remarks on the transverse Poisson structures of coadjoint orbits, Lett. Math. Phys. 12 (1986), no. 2, 87 – 91. · Zbl 0601.58040
[20] Stephen Omohundro, Geometric Hamiltonian structures and perturbation theory, Local and global methods of nonlinear dynamics (Silver Spring, Md., 1984) Lecture Notes in Phys., vol. 252, Springer, Berlin, 1986, pp. 91 – 120. · Zbl 0611.70014
[21] -[1985], Geometric perturbation theory and plasma physics, Ph.D. thesis, University of California at Berkeley.
[22] Spyros N. Pnevmatikos , Singularities & dynamical systems, North-Holland Mathematics Studies, vol. 103, North-Holland Publishing Co., Amsterdam, 1985. · Zbl 0547.00033
[23] Spyros N. Pnevmatikos, Structures hamiltoniennes en présence de contraintes, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 16, A799 – A802 (French, with English summary). · Zbl 0441.53030
[24] Spyros N. Pnevmatikos, Structures symplectiques singulières génériques, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 201 – 218 (French, with English summary). · Zbl 0523.58014
[25] Jȩdrzej Śniatycki, Dirac brackets in geometric dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 20 (1974), 365 – 372. · Zbl 0295.70010
[26] Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171 – 188. · Zbl 0274.58002
[27] M. Vinogradov and I. S. Krasilshchik [1975], Russian Math. Surveys 30, 177-202.
[28] Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523 – 557. · Zbl 0524.58011
[29] Jȩdrzej Śniatycki and Alan Weinstein, Reduction and quantization for singular momentum mappings, Lett. Math. Phys. 7 (1983), no. 2, 155 – 161. · Zbl 0518.58020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.