Embeddings of Dynamic Systems into Lie Algebras

In mechanics, many classical equations are written as systems of differential equations on Euclidean space. As we have seen, for instance, in Ch. 1, the equations of motion of a three-dimensional rigid body fixed in the centre of mass are written in R (x,y, 2r) as follows  

As has already been noted (Ch. 1), this system is closely connected with the three-dimensional Lie algebra so(3) of the rotation group SO(3). The point is that the space R (x, y, z) can be naturally identified with the space so(3) of skew-symmetric real matrices X = (x y). To this end, it suffices to put X12 = is = y) 23 = that is, to associate with the vector with coordinates (x, y, z) the matrix 

in the Lie algebra so(3), we consider the (co)adjoint representation (action) of the group SO (3). It is readily seen that the orbits of this action are standard two-dimensional spheres centered at the origin. 

Identifying with so(3) we interpret the Euler equations (see above) as a certain vector field v on the Lie algebra so(3). Now make a simple transformation induced by the following change of coordinates in  

Claim 4.1.1 A field t described by the Euler equations of motion of a rigid body fixed in the centre of mass is tangent to the orbits 0 of the adjoint representation in the Lie algebra so(3) and is Hamiltonian on these orbits (which are homeomorphic to the spheres S ). 

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