I just find two different ways of formulating the Euler-Lagrange equation on Riemannian manifolds. Let's start with a Riemannian manifold $(M,g)$ and a Lagrangian $L:\mathbf R\times TM\to\mathbf R$. We use the generalized coordinates $(q^i,\dot q^i)$ on $TM$.
The first one is the common one, which can be found in Wikipedia: \begin{equation}\tag{1} \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q^i} (t,q(t), \dot q(t)) \right) = \frac{\partial L}{\partial q^i} (t,q(t), \dot q(t)). \end{equation} It makes use of the integration by parts in coordinates.
The second one can be found in Page 153 of Villani's book: \begin{equation}\tag{2} \frac{D}{dt}\Big( \nabla_{\dot q} L(t,q(t), \dot q(t)) \Big) = \nabla_q L(t,q(t), \dot q(t)), \end{equation} where $\frac{D}{dt}$ stands for the covariant derivative along $q(t)$; $\nabla_q$ and $\nabla_{\dot q}$ are the gradient with respect to the position variable $q$ and velocity variable $\dot q$ respectively, under the Riemannian tensor $g$. This way adopts the integration by parts for covariant derivatives.
Obviously, $(1)$ and $(2)$ do not coincide. So I am really confused about which one is the right one? And what is the relation between these two equations?
TIA...