\diF \dot{q}(t) + \nabla_{\diF q (t)} \xi(t)_* \Bigr)
            \\
            &= \left(\diF q(t), \DifFrac{}{t} (\diF q) + \nabla_{\diF q (t)} \xi(t)_* \right).
    \end{split}$$ Thus, using , variation of the path in $ Q $ yields $$\begin{aligned}
        &\int_0^T \left( \dualPair*{\difpFrac{L}{q}}{\diF q} +  \dualPair*{\difpFrac{L}{\dot{q}}}{\DifFrac{}{t} (\diF q)} + \dualPair*{\difpFrac{L}{\dot{q}}}{\nabla_{\diF q} \xi_*} \right) \dif t = 0,
        \label{eq:clebschLagrange:proof:geom}
        \intertext{which after integration by parts on the second term is equivalent to}
        & \int_0^T \left(\dualPair*{\difpFrac{L}{q}}{\diF q} - \dualPair*{\DifFrac{}{t} \left(\difpFrac{L}{\dot{q}}\right)}{\diF q} + \dualPair*{\difpFrac{L}{\dot{q}}}{\nabla_{\diF q} \xi_*} \right)  \dif t = 0.
    \end{aligned}$$ Since the variations $ \diF q $ are arbitrary, we get the evolution equation .

Similarly, let $ t \mapsto \diF \xi(t) \in \LieA{g} $ be a tangent vector to $ t \mapsto \xi(t) $ in $ \sFunctionSpace(I, \LieA{g}) $. Variation of $ \xi $ yields $$\begin{aligned}
        \dualPair*{\difpFrac{L}{\dot{q}}}{\diF \xi \ldot q} + \kappa\left(\difpFrac{L}{\xi}, \diF \xi\right) = 0,
    \end{aligned}$$ which clearly gives .

\[rem:clebschLagrange:optimalControl\] Suppose the Clebsch–Lagrange problem is supplemented by the optimal control constraint $ \dot q = - \xi \ldot q $. Then,  reduces to the variational principle $ \diF s = 0 $ for the action $$s[q, \xi] = \int_0^T l(q, \xi) \dif t,$$ where the effective Lagrangian is given by $ l(q, \xi) = 