# Rigid-body in a central field: orbital and attitude motion

## Question

I would like to find a nice set of explicit coordinates for the family (parametrised by angular momentum) of reduced systems representing a rigid-body in a central field in which the orbital and attitude motion of the rigid body are clearly distinguishable in the Hamiltonian, and separable by considering different orders of the inverse of the radius of the orbital motion, when this is large.

That is some (non-canonical) coordinates in which the Hamiltonian looks like

$H(z) = H_{orb}(x,p_x) + H_{att}(q,p) + H_{coup}(x,q,p,L)$,

where $H_{orb}(x,p_x) = \frac{1}{2m} p_x + \frac{1}{2m} \frac{\mu^2}{x^2} + \bar{U} (x)$ is the 'central field' part, with $\mu^2 = \vert L \vert^2$ and $\bar{U}$ is the first term of the expanded potential, $H_{coup}(r,q,p,L)$ will contain a kinetic part and the rest of the potential.

I have not found this in the literature, but I believe it should be out there somewhere, and do not think that choosing the body-frame does the trick.

## Background and literature

Note: A rigid-body in a central field can be seen as the simplest case of two interacting rigid bodies when one of the two has spherical symmetry and the translational symmetry has been reduced.

Take as a potential the gravitational one say, or a function of the inverse of the radius of the orbital motion. Assuming it is very large compared with the dimension of the orbiting body, the potential can be expanded in a Taylor series, as nicely done by Wang et al. 1991. This separates $\bar{U}(x)$ and the rest $\tilde{U}(x,q)$.

Most of the literature concerning such problems, or those of $n$ interacting rigid-bodies, starting from Duboshin 1958, either doesn't remove the rotational symmetry, or chooses to use the rotating frame of the rigid body (see e.g. Maciejewski 1995 and Koon et al. 2004), which I find arbitrary, for multiple bodies, and unsatisfactory because the effective potential term is not the desired 'central field' one, and the coupling is messier than necessary.

## Idea

I think that using the Lagrangian formulation and choosing a rotating frame such that the Jacobi vector is along the $x$-axis say, and the body's attitude about the axis is fixed (Euler angle $\alpha_1=0$), and then using Serret-Andoyer coordinates (see e.g. Arnol'd et al., Dynamical Systems III, Sect.3.2.3; and Sansaturio and Vigueras's Appendix for their history) for the angular momentum should work, but cannot figure out what coordinates to choose for $(q,p)$ on $T^*S^2$ (or find a reference in which this is done).

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@Austen: I removed my previous\related question, because it was formulated badly, or maybe just wrong. But wanted to thank you again for your answer. –  Dayal C Strub Nov 2 '12 at 19:56