Particle moving on group manifold I'm trying to learn about particles / strings moving on group manifolds and am looking for a reference which introduces this idea. 
For example, in this paper the Lagrangian for a particle moving on $SU(2)$ is given by $$L=-\frac{1}{2}{\rm tr}(g^{-1}\dot{g}g^{-1}\dot{g})$$ where $g\in SU(2)$ and $\dot{g}=\frac{dg}{dt}$.
I see an Action of this type everywhere, but cannot find a source to actually derive / motivate it.
 A: This kind of Lagrangians are common in effective field theory descriptions. One example would be chiral sigma model. A beginning discussion is found in chapter 15 (4th ed) of J. Zinn-Justin's book Quantum Field Theory of Critical Phenomena.
A: The Lagrangian is motivated by the following observation. The Lagrangian of a QM particle moving in n-dimension with a coordinate $
\vec x(t)=(x_1(t),x_2(t),\dots,x_n(t))$ is given by
$$
\frac{1}{2}m 
\dot {\vec x}^2\;,
$$
Which is just the  kinetic energy of the particle. I will set $m=1$ below. Now if we want to have a Lagrangian for a particle moving on some constrained hypersurace, we would impose constraints on $\vec x$, but the Lagrangian would be unchanged. For example we can impose $\vec x^2=1$ So that the particle moves on a unit circle. 
By the same token, let $g$ be an $n \times n$ matrix of complex coordinates (meaning that there is $2n^2$ real coordinates), and let these coordinates not be constrained in any way for the moment. We would write the Lagrangian
$$
\text{tr }(\dot g^\dagger \dot g)\;.
$$
You can check that this is the same as before if you view as the real and imaginary components of the matrix as coordinates.
But now we can constrain these coordinates by requesting that $g$ be $SU(N)$. In this case we have that $g^\dagger=g^{-1}$ and it is simple to show that the Lagrangian above is equivalent to the one you quoted in your question if $g$ is a unitary matrix.
