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.

37 pages, minor corrections, final version to appear in J. Geom. Phys

References in corpus (5)

• Jacobi structures revisited
• The graded Jacobi algebras and (co)homology
• Lie algebroid structures and Lagrangian systems on affine bundles
• Lie algebroid structures on a class of affine bundles
• Frame-independent formulation of Newtonian mechanics

Cited by in corpus (25)

• Lagrangian submanifolds and dynamics on Lie algebroids
• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• A unified framework for mechanics. Hamilton-Jacobi equation and applications
• Variational calculus with constraints on general algebroids
• Higher order mechanics on graded bundles
• AV-differential geometry: Euler-Lagrange equations
• Lagrangian submanifolds and dynamics on Lie affgebroids
• The Tulczyjew triple for classical fields
• Tulczyjew Triples in Higher Derivative Field Theory
• Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds
• Tulczyjew triples: from statics to field theory
• Partial Differential Hamiltonian Systems
• Reduction of symplectic principal $\mathbb{R}$-bundles
• AV-differential geometry and Newtonian mechanics
• Isotropic submanifolds and the inverse problem for mechanical constrained systems
• Jacobi Structures in ${\mathbb{R}}^3$
• A general framework for nonholonomic mechanics: Nonholonomic Systems on Lie affgebroids
• Tulczyjew's Approach for Particles in Gauge Fields
• Wigner's Theorem and geometry of extreme positive maps
• The Schroedinger operator as a generalized Laplacian
• AV-differential geometry and calculus of variations
• Affine bundles are affine spaces over modules
• Frame-independent formulation of Newtonian mechanics
• The Schroedinger operator in Newtonian space-time
• Affine Tensors in Mechanics of Freely Falling Particles and Rigid Bodies