that a priori are only in $ \LieA{g}' $ may actually be represented by elements of $ \LieA{g}^* $. This remark applies in particular to the partial derivative $ \difpFrac{L}{\xi} $.

In order to formulate the variational principle, we also need the following technical tool. A on $ Q $ is a smooth map $ \eta: \TBundle Q \supseteq U \to Q $ defined on an open neighborhood $ U $ of the zero section in $ \TBundle Q $ such that the composition of $ \eta $ with the zero section is the identity on $ Q $, , $ \eta(q, 0) = q $, and the map $ \pr_Q \times \eta: \TBundle Q \supseteq U \to Q \times Q $ is a diffeomorphism onto an open neighborhood of the diagonal, [@KrieglMichor1997 Section 42.4]. A local addition may be constructed using the exponential map of a Riemannian metric on $ Q $ but it also exists if $ Q $ is an affine space, a Lie group or a space of sections of a fiber bundle.

With these preliminaries out of the way, we are now able to state the equations of motion corresponding to the Clebsch–Lagrange variational principle.

