Tulczyjew triple on the Atiyah algebroid with connection

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.