Shlomo Sternberg

Bio: Shlomo Sternberg is an academic researcher from Harvard University. The author has contributed to research in topics: Symplectic manifold & Symplectic geometry. The author has an hindex of 39, co-authored 92 publications receiving 11177 citations. Previous affiliations of Shlomo Sternberg include Massachusetts Institute of Technology & Tel Aviv University.

Symplectic Techniques in Physics

Abstract: Preface 1 Introduction 2 The geometry of the moment map 3 Motion in a Yang-Mills field and the principle of general covariance 4 Complete integrability 5 Contractions of symplectic homogeneous spaces References Index

Lectures on Differential Geometry

Abstract: Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.

Convexity properties of the moment mapping. II

Abstract: be its associated momen t mapping. (See w for definitions.) The set, @(X), is the union of co-adjoint orbits. The main result of this paper is a description of the orbit structure of this set. To describe this result, we first note that the coadjoint orbits in ,q* can be parametr ized as follows: Let t be a Caf tan subalgebra of g and let B be a positive-definite G-invariant bilinear form on $. By means of B we get a G-equivariant identification, , q ~ * . Let t* be the subspace of ~q* corresponding to t. Then every co-adjoint orbit intersects i* in a Weyl-group orbit; so there is a one-one correspondence between co-adjoint orbits in ,q* and Weyl-group orbits in t*. To get an even sharper parametrization, let t* be a Weyl chamber in {*. Then every Weyl-group orbit intersects t* in a single point : so t* is a "modul i space" for the co-adjoint orbits. Two orbits in X are said to be of the same type if they are isomorphic as homogeneous G-spaces. Since X is compact, there are at most a finite number of different types of orbits. (See [1113 Let H be a closed subgroup of T of codimension r and let Cit be the set of all orbits in X of type G/H.

Geometric Quantization and Multiplicities of Group Representations.

Abstract: The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true "points"; see Weinstein [17] Now let G be a compact Lie group and G x X ~ X a Hamiltonian action of G on X (see w for definitions) It is well-known that the fixed points of this action form a symplectic submanifold of X (See for instance Guillemin and Sternberg [5]) However, what can one say about the fixed "points" of G? We will show that they are also the "points'" of a symplectic manifold, Xc This manifold is the Marsden-Weinstein reduction of X with respect to the zero orbit in g*, and will be described in Sect 2 (It was introduced in a completely different context fl'om ours by Marsden and Weinstein [12]) Problems in classical mechanics can often be reduced to the study of Hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on Hilbert space This fact has inspired a number of efforts to "quantize" symplectic geometry by devising schemes for associating Hilbert space to symplectic manifolds The "no-go" theorems of Groenwald and Van Hove impose some embarrassing limitations on all such schemes; however, it seems to be a useful idea heuristically to think of every symplectic manifold, X~a~ic, 1, as being symbiotically associated with a Hilbert space, Xq,a, t in such a way that the classical observables on the first space correspond to quantum observables on the second space The heuristics further suggests that if G is a group of symmetries of X~la~,i~, j, it should also be a group of symmetries of Xq,a,,tum In this heuristic spirit, we will state the main conjecture of this paper:

Differentiable dynamical systems

Abstract: This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

Fourier Integral Operators. I

Abstract: Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

Abstract: Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

Topological methods in hydrodynamics

Abstract: A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.