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Tulczyjew triple on the Atiyah algebroid with connection

Katarzyna Grabowska, Paweł Korzeb, Kuba Krawczyk

TL;DR

This work develops the Tulczyjew triple for a principal bundle endowed with a connection, providing a convenient trivialisation that separates horizontal and gauge components. It then performs a reduction by the tangent group to obtain a clear, trivialised Tulczyjew triple on the Atiyah algebroid, including the reduced canonical flip, Tulczyjew map, and symplectic-based isomorphism, together with explicit dynamics. The paper also furnishes detailed examples on frame bundles of the sphere, showing how the reduction yields tractable expressions when the structure group is abelian (e.g., so(2)). Overall, the results unify the geometric mechanism for deriving Lagrangian and Hamiltonian dynamics on Atiyah algebroids with connections, and demonstrate the practical computation of reduced dynamics in canonical rigid-body-like systems on curved configuration spaces.

Abstract

The Tulczyjew triple on a principal bundle with connection is constructed in a convenient trivialisation. A reduction by the structure group is performed leading to the triple on the trivialised Atiyah algebroid and a presentation of this algebroid via a double vector bundle morphism. The dynamics of physical systems with configuration manifolds having the structure of a principal bundle with connection or the Atiyah algebroid is discussed and applied to the example of an axially symmetric body confined to a sphere.

Tulczyjew triple on the Atiyah algebroid with connection

TL;DR

This work develops the Tulczyjew triple for a principal bundle endowed with a connection, providing a convenient trivialisation that separates horizontal and gauge components. It then performs a reduction by the tangent group to obtain a clear, trivialised Tulczyjew triple on the Atiyah algebroid, including the reduced canonical flip, Tulczyjew map, and symplectic-based isomorphism, together with explicit dynamics. The paper also furnishes detailed examples on frame bundles of the sphere, showing how the reduction yields tractable expressions when the structure group is abelian (e.g., so(2)). Overall, the results unify the geometric mechanism for deriving Lagrangian and Hamiltonian dynamics on Atiyah algebroids with connections, and demonstrate the practical computation of reduced dynamics in canonical rigid-body-like systems on curved configuration spaces.

Abstract

The Tulczyjew triple on a principal bundle with connection is constructed in a convenient trivialisation. A reduction by the structure group is performed leading to the triple on the trivialised Atiyah algebroid and a presentation of this algebroid via a double vector bundle morphism. The dynamics of physical systems with configuration manifolds having the structure of a principal bundle with connection or the Atiyah algebroid is discussed and applied to the example of an axially symmetric body confined to a sphere.
Paper Structure (34 sections, 1 theorem, 144 equations)

This paper contains 34 sections, 1 theorem, 144 equations.

Table of Contents

  1. Introduction
  2. Tulczyjew triple and algebroids
  3. Tulczyjew triple in mechanics
  4. Algebroids via Tulczyjew triple morphisms
  5. Tulczyjew triple on a Lie group
  6. Principal bundles and connections
  7. Tulczyjew triple on a principal bundle with connection
  8. Trivialisation of the tangent and cotangent of a principal bundle
  9. Trivialisation of TTP
  10. Trivialisation of the canonical flip
  11. Trivialisation of TT*P
  12. Trivialisation of the pairing of TT*P and TTP
  13. Trivialisation of T*TP
  14. Trivialisation of the pairing of T*TP and TTP
  15. Trivialisation of the Tulczyjew isomorphism
  16. ...and 19 more sections

Key Result

Proposition 4.3

Let $X$ and $Y$ be two vector fields on a manifold $M$. The following formula holds:

Theorems & Definitions (10)

  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4
  • Remark 5.5