\[prop:clebschLagrange:clebschEulerLagrangeDirect\] Let $ Q $ be a smooth Fréchet $ G $-manifold. Assume that $ Q $ can be endowed with a local addition[^2]. Moreover, assume that $ \TBundle Q $ and $ \CotBundle Q $ are endowed with dual torsion-free linear connections. Then, for a Clebsch–Lagrangian $ L: \TBundle Q \times \LieA{g} \to \R $ the following are equivalent:

The curves $ t \mapsto q(t) \in Q $ and $ t \mapsto \xi(t) \in \LieA{g} $ are solutions of the variational Clebsch–Lagrange problem .

The following equations hold:

\[eq:clebschLagrange:clebschEulerLagrangeDirect\]

( ) - - &= 0, \[eq:clebschLagrange:clebschEulerLagrangeDirect:evol\]\
+ (, ) &= 0, \[eq:clebschLagrange:clebschEulerLagrangeDirect:constraint\]

for all $ \zeta \in \LieA{g} $, where evaluation of all derivatives at $ (q, \dot q + \xi \ldot q, \xi) \in \TBundle Q \times \LieA{g} $ is understood. Furthermore, the second term in  denotes the functional $ T_q Q \ni X \mapsto \dualPair[\big]{\difpFrac{L}{\dot{q}}}{\nabla_X \xi_*} \in \R $ viewed as an element of $ \CotBundle Q $.

The equations  will be referred to as the .

Let $ I = [0, T] $ with $ T > 0 $. Since $ Q $ admits a local addition, the space $ \sFunctionSpace_{q_i q_f}(I, Q) $ of paths 