# Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups

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.

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Isn't the integer-spin case distinct from the half-integer case? For example, spin 1/2 can't be created by the motion of particles through space, so in the case of SU(2), it's not clear to me what would be meant physically by a frame of reference tied to the principal axes, as in Euler's equations for SO(3). –  Ben Crowell Mar 7 '12 at 6:18

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.

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"Symplectic techniques in physics" –  Ben McKay Mar 7 '12 at 8:35
Michal, Thank you. This is what I was looking for, with the added bonus of seeing the relevance of diffeomorphism groups to hydrodynamics. –  Mark Mueller Mar 7 '12 at 20:22
Abraham and Marsden's Foundations of Mechanics is quite good for this topic, the Arnold books are great, of course. –  alvarezpaiva Mar 14 '12 at 19:47
For those who really like classical mechanics from a classical point of view, don't neglect Herbert Goldsteins's now-nearly ancient book, Classical Mechanics. There one finds what H.G. calls the "Jabberwockian statement", "The polhode rolls without slipping on the herpolhode lying in the invariable plane." –  drbobmeister Mar 19 '12 at 23:05
A useful mnemonic which always makes me smile! Having said that, I remain a dedicated fan and student of A&H and Arnold. But I'm sure others in the mechanics community cut their teeth on Goldstein. Ah how those (we) Ph106 students used to suffer at old Caltech! A great place to learn mechanics in a truly coordinate dependent notation! –  drbobmeister Mar 19 '12 at 23:05