Paweł Urbański

Lie algebroids and Poisson-Nijenhuis structures

Abstract Poisson-Nijenhuis structures for an arbitrary Lie algebroid are defined and studied by means of complete lifts of tensor fields.

Tangent lifts of Poisson and related structures

The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

Algebroids — general differential calculi on vector bundles☆

Abstract A notion of an algebroid— a generalized of a Lie algebroid structure on a vector bundle is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T M can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid.

Tangent and Cotangent Lifts and Graded Lie Algebras Associated with Lie Algebroids

Generalized Schouten, Frölicher–Nijenhuis and Frölicher–Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebroid on the tangent bundle, a new common generalization of the Frölicher–Nijenhuis and the symmetric Schouten brackets, as well as embeddings of the Nijenhuis–Richardson and the Frölicher–Nijenhuis bracket into the Schouten bracket, are obtained.

AV-differential geometry: Poisson and Jacobi structures ☆

Abstract Based on ideas of W.M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems.

Lie Brackets on Affine Bundles

Natural affine analogs of Lie brackets on affine bundles are studied.In particular, a close relation to Lie algebroids and a duality withcertain affine analog of Poisson structure is established as well asaffine versions of complete lifts and Cartan exterior calculi.

AV-differential geometry: Euler–Lagrange equations☆

Abstract A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent Lagrangians and Hamiltonians. In our approach, Lagrangians and Hamiltonians are, in general, sections of certain R -principal bundles, and the solutions of analogs of Euler–Lagrange equations are curves in certain affine bundles. The correct geometrical and frame-independent description of Newtonian Mechanics is of this type.

AV-differential geometry and Newtonian mechanics

A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values (AV differential geometry), i.e. the differential geometry in which affine bundles replace vector bundles and sections of one-dimensional affine bundles replace functions on manifolds, is used. Lagrangian and Hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed.

Poisson-Jacobi reduction of homogeneous tensors

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold

Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

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