I'm trying to retrieve the equations of motion for a free rigid body: $$ I(t)\dot{\omega}(t)+\omega(t)^T \times (I(t)w(t)) = 0 $$ where $$ I(t)=R(t)I_{0}R(t)^T $$
I know that Euler-Lagrange equations are expressed in a local coordinate system of the Tangent Bundle of the configuration manifold. In Introduction to Mechanics and Symmetry (by Marsden and Raitu), they state the Euler-Lagrange takes the form (Theorem 7.3.3), on a chart $U\times E \subset TQ$ where $Q$ is the configuration manifold $$ \frac{d}{dt} D_{2}L(u(t),v(t)).w = D_1L(u(t),v(t)).w, \forall w \in E $$ $$ \frac{d}{dt} u(t) = v(t) $$
I tried to apply this for rotations. The Lagrangian is defined on $TSO(3)$ by $$L(R(t),\omega(t)) = \frac{1}{2}w(t)^TR(t)I_{0}R(t)^Tw(t)=\frac{1}{2}w(t)^TI(t)w(t)$$
I tried to apply the above formula. I've only started self-studying Differential Geometry so my equations must be messy, not rigorous and completly false. To find $D_1L(R,\omega)$, I took a curve $\gamma(t)=R exp(t[\eta]_{\times})$ where $\eta$ is a vector in $\mathbb{R}^{3}$ and I computed $(L\circ \gamma)'(0)$. This gave me: $$ D_1L(R,\omega):\eta \in T_{R}SO(3) \mapsto \frac{1}{2}w^TR([\eta]_{\times} I_{0}-I_{0}[\eta]_{\times})R^Tw $$
Similarly: $$ D_2L(R,\omega): a \in T_{\omega}\mathbb{R}^3 \mapsto a^T I \omega $$
Now I plug time into these equations. Differentiating the second one gives: $$ \frac{d}{dt} D_2L(R(t),\omega(t)): a \in T_{\omega}\mathbb{R}^3 \mapsto a^T[\omega]_{\times} I(t) \omega(t) + a^T I(t) \dot{\omega}(t) $$
I can't see how I can get the original equations of motion from here.
