# What is the symplectic form for rigid body systems?

I was pondering rigid body mechanics and the non-vanishing Poisson brackets $$\left\{J_x,J_y\right\}=J_z$$ etc. However in arbitrary coordinates $$q^i$$ with conjugate momenta $$p_i$$ we are supposed to have $$\left\{q^i,q^j\right\}=\left\{p_i,p_j\right\}=0$$ and $$\left\{p_i,q^j\right\}=\delta_i^j$$, so what is going on here? Can we express the poisson bracket in terms of $$\phi^i,J_j$$ coordinates?

The symplectic manifold phase space for the classical version of the angular momentum algebra is the two-sphere $$S^2$$. The symplectic form , in terms of spherical polar angles $$\theta$$, $$\phi$$, is $$\omega = \sin\theta\, d\theta\wedge d \phi,$$ i.e. the area 2-form for $$S^2$$. Then we take $$J_x= \sin\theta \cos \phi\\ J_y= \sin\theta \sin\phi\\ J_z= \cos\theta$$ as the moment maps. Using hamilton's equations $$dH= -\omega(V_H,-)$$ with $$H= J_x$$, etc gives us $$V_{J_x}=\sin\theta \partial_\theta+\cos\phi \cot \theta \partial_\phi\\ V_{J_y}=-\cos\theta \partial_\theta +\sin\phi \cot \theta \partial_\phi\\ V_{J_z}= \partial_\phi.$$ So from $$\{J_x,J_y\}= \omega(V_{J_x},V_{J_y})$$ we read read off (after a bit of algebra) that $$\{J_x,J_y\}= \cos\theta =J_z.$$ The other Poisson brackets work the same.
• Isn't the configuration space $SO\left(3\right)$? Also, I didn't phrase very well, but what I was really wondering is whether angular momentum is canonical momentum after a particular choice of coordinates, and if not, what is? Dec 20, 2019 at 16:03
• @Joshu Tilley. The configuration space of a rigid body top is $SO(3)$ and the associated phase space is 6 dimensional and has infinite volume. Both classical and the quantum version of the top can have any value of total angular momentum $J$. The two dimensional $S^2$ is the phase space for a spin with a definite angular momentum $J$. The spin phase space must have finite volume becsue the quatum Hilbert space is finite dimensional. I discussed the spin case in my answer because if you just want the Poisson brackets its symplectic geometry is simpler than the top. Dec 20, 2019 at 17:01