q(t) \in Q $ and $ t \mapsto \xi(t) \in \LieA{g} $, where the physical action $ S $ is of the form $$\label{eq:clebschLagrange:action}
    S[q, \xi] = \int_0^T L\bigl(q(t), \dot q(t) + \xi(t) \ldot q(t), \xi(t) \bigr) \dif t$$ and where the variations of $t \mapsto q(t) $ are restricted to vanish at the endpoints. Here, $t \mapsto (q (t), \dot q (t) + \xi(t) \ldot q (t)) $ is viewed as a curve in $ \TBundle Q $. We will refer to this variational principle as the and to $ L $ as the .

In order to derive the associated equations of motions, we first recall some notions and results from geometric mechanics (see, [@AbrahamMarsdenEtAl1980; @RudolphSchmidt2012]). In particular, we need the theory of linear connections in vector bundles. A linear connection in a vector bundle $ \pi: E \to M $ is a vector bundle homomorphism $ K: \TBundle E \to E $ over $ \pi: E \to M $ and over $ \TBundle M \to M $ such that the following digram commutes:

E & E\
E \_M E & E,

where the vertical tangent bundle $ \VBundle E = \ker \tangent \pi $ is identified with 