Narrow Results

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity.

We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum.

We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the noncovariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger and Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.

The properties of the D-brane fluctuations are investigated using the two types of deformation of the Dirac structure, based on the B-transformation and the β-transformation, respectively. The former gives the standard gauge theory with two-form field strength. The latter gives a nonstandard gauge theory on the Poisson manifold with bivector field strength and the vector field as a gauge potential, where the gauge symmetry is a diffeomorphism generated by the Hamiltonian vector field. The map between the two gauge theories is also constructed with the help of Moser's Lemma and the Magnus expansion. We also investigate the relation to the gauge theory on the noncommutative D-branes.

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of $\beta$-diffeomorphisms and $\beta$-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, $R$-fluxes are formulated as a twisting of this Courant algebroid by a local $\beta$-transformations, in the same way as $H$-fluxes are the twist of the generalized tangent bundle. It is a three-vector classified by Poisson three-cohomology and it appears in a twisted bracket and in an exact sequence.

It is known that the topological T-duality exchanges H- and F-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced [T. Asakawa, H. Muraki, S. Sasa and S. Watamura, Int. J. Mod. Phys. A30, 1550097 (2015), arXiv:1408.2649 [hep-th]] to define R-fluxes as field strength associated with $\beta$-transformations. We propose a definition of Q-flux associated with $\beta$-diffeomorphisms, and show that the topological T-duality exchanges R- and Q-fluxes.

We discuss some geometrical and dynamical properties of Rabinovich systems.

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints.

Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

We construct certain new classes of isochronous dynamical systems based on the recent constructions of Calogero and Leyvraz. We show how a Poisson structure can be ascribed to such equations in ℝ³ and indicate their connection with the Nambu structures.

In this paper, we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.

First we show that, associated to any Poisson vector field $E$ on a Poisson manifold $(M,\pi )$, there is a canonical Lie algebroid structure on the first jet bundle $J^{1}M$ which, depends only on the cohomology class of $E$. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and $L\infty$-algebras.

We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen.39(33) (2006) 10449–10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl.144 (2001) 145–154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a $3$-form and a $2$-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background.

Let be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on of a large infinite-dimensional reparametrization group SV with Lie algebra 𝔰𝔳[8, 10], called the Schrödinger–Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on . More precisely, the infinitesimal action of 𝔰𝔳 appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, 𝔤, of which 𝔰𝔳 is a quotient, while the Poisson structure is inherited from the corresponding Kirillov–Kostant–Souriau form.