The following problem has recently come up in the course of a project I'm working on, and I would be very appreciative if someone could point me in the right direction:


Let $X \colon I \to G$ be a smooth curve taking values in a Lie group $G$ with a surjective exponential map, and define another curve $v\colon I \to T\_e(G)$ according to $v(t) \triangleq \omega\_{X(t)}\left( \dot{x}(t) \right)$, where $\omega$ is the Maurer-Cartan form on $G$. Then intuitively, $v(t)$ represents a kind of "local" measurement of the velocity $\dot{X}(t)$, expressed in a local, "co-moving" coordinate system that is rigidly attached to the point $X(t)$ itself.


Given access to $v(t)$ and an initial condition of the form $X(t\_0) = X\_{t\_0} \in G$ for some $t\_0 \in I$, I would like to determine a curve $\gamma \colon I \to T\_e(G)$ such that $X = \exp \circ \gamma$.



Kinematics on Lie groups
$v(t)$ is widely used, even for a proof of existence of a Lie group for a given Lie algebra (see the book of Duistermaat and Kolk). 


It is also used a lot for infinite dimensional Lie groups where the exponential map is not a local diffeomorphism, in general (for diffeomorphism groups, for example). See the book "Younes: Shapes and diffeomorphisms" or the overview article [here](http://www.mat.univie.ac.at/~michor/shape-overview.pdf). 


As pointed out by Ben MacKay, even in finite dimensions, the exponential mapping might not be a diffeomorphism everywhere, or it might not be surjective. See the following very nice article for illustrations of this:


* MR0808680 Hilgert, Joachim; Hofmann, Karl H. Old and new on Sl(2). Manuscripta Math. 54 (1985), no. 1-2, 17–52.


