arXiv subjects

Katarzyna Grabowska

Publications and source records attributed to Katarzyna Grabowska.

46 records · Page 3Linked to original sources

arXiv2008

Variational calculus with constraints on general algebroids

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.

math-ph↗

arXiv2008

The Schroedinger operator as a generalized Laplacian

The Schroedinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schroedinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -- the Schroedinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U(1)-principal bundle with an analogous list of trivializations -- the Schroedinger principal bundle. For the Schroedinger principal bundle a natural differential calculus for `wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudo-Riemannian metric. The free Schroedinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schroedinger principal bundle. The presented framework is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory.

math-ph↗

arXiv2006

AV-differential geometry and calculus of variations

The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.

math-ph↗

arXiv2006

The Schroedinger operator in Newtonian space-time

The Schroedinger operator on the Newtonian space-time is defined in a way which is independent on the class of inertial observers. In this picture the Schroedinger operator acts not on functions on the space-time but on sections of certain one-dimensional complex vector bundle over space-time. This bundle, constructed from the data provided by all possible inertial observers, has no canonical trivialization, so these sections cannot be viewed as functions on the space-time. The presented framework is conceptually four-dimensional and does not involve any ad hoc or axiomatically introduced geometrical structures. It is based only on the traditional understanding of the Schroedinger operator in a given reference frame and it turns out to be strictly related to the frame-independent formulation of analytical Newtonian mechanics that makes a bridge between the classical and quantum theory.

math-ph↗

arXiv2006

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 i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is uded. Lagragian and hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed.

math-ph↗

arXiv2006

AV-differential geometry: Euler-Lagrange equations

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.

math.DG↗

arXiv2004

AV-differential geometry: Poisson and Jacobi structures

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.

math.DG↗

arXiv2004

Frame-independent formulation of Newtonian mechanics

A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values 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.

math-ph↗

arXiv2003

Lie brackets on affine bundles

Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of Poisson structures is established as well as affine versions of the complete lift of a section and the Cartan exterior calculus.

math.DG↗

arXiv2002

The Lie algebra of a Lie algebroid

Results describing Lie ideals and maximal finite-codimensional Lie subalgebras of the Lie algebras associated with Lie algebroids with non-singular anchor maps are presented. It is also proved that every isomorphism of such Lie algebras induces a diffeomorphism of base manifolds respecting the generalized foliations defined by the anchor maps.

math.DG↗