# Geometrical Mechanics on algebroids

@inproceedings{KGrabowska2005GeometricalMO, title={Geometrical Mechanics on algebroids}, author={K.Grabowska and J.Grabowski and P.Urba'nski}, year={2005} }

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler-Lagrange-type…

## 73 Citations

### JACOBI VECTOR FIELDS FOR LAGRANGIAN SYSTEMS ON ALGEBROIDS

- Mathematics
- 2013

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler–Lagrange (EL) equations are themselves solutions of the EL…

### The Supergeometry of Loday Algebroids

- Mathematics
- 2011

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure…

### Variational calculus with constraints on general algebroids

- Mathematics
- 2008

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…

### Dirac Structures and Hamilton-Jacobi Theory for Lagrangian Mechanics on Lie Algebroids

- Mathematics
- 2012

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of…

### Loday algebroids and their supergeometric interpretation

- Mathematics
- 2011

A concept of Loday algebroid (and its pure algebraic version – Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a…

### Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

- Mathematics
- 2008

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a…

### Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

We introduce the concept of a higher algebroid, generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as vector bundle…

### Lagrangian Mechanics and Reduction on Fibered Manifolds

- Mathematics
- 2017

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases,…

## References

SHOWING 1-10 OF 22 REFERENCES

### Hamiltonian systems, Lagrangian systems, and the Legendre transformation

- Symposia Math
- 1974

### Derivations of differential forms on jet bundles

- Mathematics
- 1987

SummaryThe theory of derivations of differential forms, originally formulated by Frölicher and Nijenhuis [2], is generalized to include derivations which connect differential forms on two different…

### Lagrangian submanifolds and higher-order mechanical systems

- Mathematics
- 1989

Higher-order Lagrangian and Hamiltonian systems (time dependent or independent) are interpreted in terms of Lagrangian submanifolds of sympletic higher-order tangent bundles. The relation between…

### Field and particle equations for the classical Yang-Mills field and particles with isotopic spin

- Physics
- 1970

SummaryA complete system of equations describing the interaction between the Yang-Mills field and isotopic-spin-carrying particles in the classical limit is extracted from the equations of motion for…

### Lagrangian Mechanics on Lie Algebroids

- Physics, Mathematics
- 2001

A geometric description of Lagrangian Mechanics on Lie algebroids is developed in a parallel way to the usual formalism of Lagrangian Mechanics on the tangent bundle of a manifold. The dynamical…

### Geometric objects defined by almost Lie structures

- Mathematics
- 2001

The aim of this paper is to extend from manifolds to vector bundles some classical geometric objects, associated with Lagrange and Hamilton metrics. Considering vector bundles endowed with almost Lie…

### Lie algebroids and mechanics

- Mathematics
- 1996

We give a formulation of certain types of mechanical systems using the structure of groupoid of the tangent and cotangent bundles to the configuration manifold $M$; the set of units is the zero…