Rigid motions: action-angles, relative cohomology and polynomials with roots on the unit circle

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a variable $r^2$ depending on the inertia moments. Normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without the use of ellitptic integrals (unlike the derivation of the action-angles). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.

💡 Research Summary

The paper revisits the classical problem of a rigid body rotating near a uniform spin and provides a complete canonical integration in terms of action‑angle variables. By separating the phase space into two dynamical regimes—hyperbolic (near the unstable equilibrium) and elliptic (near the stable equilibrium)—the author constructs two families of action‑angle coordinates. In each regime the Hamiltonian is expanded as a power series in the action variable, and the coefficients of the series are polynomials (P_n(r^2)) in a single dimensionless parameter (r^2) that encodes the ratios of the three principal moments of inertia.

A central methodological innovation is the use of relative cohomology (H^*(M,N)) to obtain the normal form of the Hamiltonian without invoking elliptic integrals. Here (M) denotes the full configuration manifold of the rigid body and (N) the energy level set (the invariant torus or hyperbolic surface). The coboundary map between the cohomology groups yields a set of linear relations that uniquely determine the polynomial coefficients (P_n(r^2)). Consequently the canonical transformation to action‑angle variables can be written explicitly as a convergent series, and the usual cumbersome elliptic‑integral expressions are replaced by elementary algebraic recursions.

The paper then turns to the algebraic properties of the polynomials (P_n(x)). By comparing the recursion obtained from the cohomological analysis with the three‑term recurrence of Chebyshev polynomials, the author shows that (P_n(x)) satisfies a relation of the form
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