The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid body motion. It seems to me that there should be a straightforward generalization of this from $SO(3)$ to any (compact) Lie group. (Compact, presumably, because we want the Hamiltonian to be bounded below.) Would someone be so kind as to point me to some literature that discusses the configuration space, Lagrangian, canonical momenta, Hamiltonian and equations of motion in this more general case? I would also like to gain some insight as to how the left and right actions of the Lie group on itself are related to the physical concepts of inertial and rotating frames. My initial attempts took me immediately into some areas of integrable systems and algebraic geometry that, while interesting, assume that the simpler question(s) I am asking have already been understood by the reader.



Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups
Here is the link that should satisfy you:
<http://ncatlab.org/nlab/show/Hamiltonian+dynamics+on+Lie+groups> .


It is also interesting to note that if you consider in this context infinite dimensional Lie group (eg. group of volume-preserving diffeomorphisms of some manifold), you'd recover hydrodynamics of the ideal fluid:
<http://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/> .


Except for above two references I'd recommend Arnold's "Mathematical methods of classical mechanics", "Topological methods in Hydrodynamics". Also, take a look at "Symplectic techniques in physics" by Sternberg and Guillemin. 


