Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic orbits. It is known that on a dense set the (reconstructed) trajectory of $X$ in $M$ is quasiperiodic on 2-tori (or an $(r+1)$-torus in general for a $G$-action where $r$ is the rank of $G$, the dimension of a maximal torus), see e.g [1, 2].
- Is it possible to define local coordinates on $M$ (or on a neightborhood of one of those 2-torus) of the form $(\theta_1,\theta_2, J_3,...,J_m)$, where the $J_3,...,J_n$ are first integrals of $X$ with flow quasiperiodic in the coordinates $(\theta_1,\theta_2)$?
As a start I am using a local basis of $TM$ given by $\{ Y_1, Y_2, Y_3 , X_4,...,X_n\}$, where the $Y_i$ generate the "vertical" space (tangent to the $SO(3)$-orbits) and the $X_\alpha$ project to a basis of sections on $T(M/SO(3))$. Informally I want to relate the above local coordinates $(\theta_1,\theta_2, J_3,...,J_m)$ with the local coordinates $(\alpha,\beta,\gamma)$ of the vertical space (fiber) and $(\theta, J_5,...,J_n)$ of the base for the $SO(3)$-bundle $M \to M/SO(3)$, i.e. supposing that $J_5,...,J_n$ are known $SO(3)$-invariant first integrals, $J_3$ and $J_4$ are first integrals depending on $(\alpha,\beta,\gamma)$ and $\theta$ is the parameter of the periodic trajectories in the base $M/SO(3)$.
In fact, there is more structure in [2] (a non-poisson bivector field $\pi$ on $M$) but at first my goal is to find some kind of "angle" coordinates as above in the general context; and to describe the "two additional integrals of motion" $J_3$ and $J_4$ (cf. Section 5 in [2]) imposing the quasiperiodic motion to lie on a 2-torus.
[1] Field M., Local structure of equivariant dynamics, in Singularity Theory and Its Applications, Part II (1988/1989, Coventry), Lecture Notes in Math., Vol. 1463, Springer, Berlin, 1991, 142–166.
[2] Fassò, F., Giacobbe, A.: Geometry of invariant tori of certain integrable systems with symmetry and an application to a nonholonomic system. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 051, 12 pp (2007).
