arXiv subjects
Publications and source records attributed to Richard Cushman.
This note shows that the module of smooth vector fields on ${\mathbb{R}}^n$, which are invariant under the linear action of a compact Lie group $G$ is finitely generated by polynomial vector fields on ${\mathbb{R}}^n$ which are invariant under the action of $G$.
In this paper we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz-Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated $n$-gon in the complex plane.
We construct a natural transformation between the category of Aronszajn subcartesian spaces and the category of subcartesian differential spaces, which is a subcategory of Sikorski differential spaces.
In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems we interpreted shifting operators as quantization of functions ${\mathrm{e}}^{ \pm iθ_j}$ , where $(I_j , θ_j )$ are action angle coordinates. The aim of this paper is to show how these operators occur in geometric quantization.
We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfies Smith's version of de Rham's theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.
We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.
This paper uses differential spaces to obtain some new results in integrable Hamiltonian systems
The goal of this paper is to investigate a class of algebras called sandwich algebras, which are certain complex Lie algebras with a nilpotent radical whose elements are sandwiches. We present a classification of all very special sandwich algebras.
We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map.
This paper extends the Bohr-Sommerfeld quantization of the spherical pendulum to a full quantum theory. This the first application of geometric quantization to a classical system with monodromy.
This paper gives a slight refinement of a theorem of Hamilton, which shows that the velocity of a Keplerian motion moves on a circle.
Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$ than the companion matrix. The computation of this normal form uses only operations from $\mathrm{k}$ and does not require finding roots of any polynomial.
This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples.
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.
Based on a recent classification of coadjoint orbits of the full Poincaré group, we give a new group theoretic interpretation for the mass of a classical relativistic particle.