@ĐàoThanhOai Hi, yes I agree. Let me know what information you would need from me.

@issoroloap I have a feeling that Fan Zheng may have a point. If you look at the proof of Darboux's theorem in Arnold's book on mechanics, the symplectically "flat" coordinates there seem to have been constructed exactly by finding the local Hamitonians in involution that generate the local tangent Lagrangian submanifold.

So basically you have a time-dependent periodic perturbation of a integrable Hamiltonian written in action-angle coordinates? And you want to apply some averaging technique? By the way, the homogeneity of the Hamiltonian suggests a conservation law, which leads to another Hamiltonian that commutes with this one, so you may be able to reduce the system a bit... What does Arnold's book on mechanics say? He has discussed averaging there...

If $\dot{\phi} = \omega(q)$ doesn't that imply that $K$ does not depend on $p_{\phi}$? Isn't your Hamiltonian $H(q,\phi, p, p_{\phi}) = p_{\phi} \omega(q) + K(q,\phi, p)$?

No, I think such index "commutativity" does not hold in general. I simply mean that the notation is, as you said it yourself, taking the $k$-th component of the covariant derivative $(\nabla_{F_4}F_3)^k = (F_4^l \nabla_l F_3)^k = F_4^l (\nabla_l F_3)^k$ is written as $F_4^l \nabla_l F_3^{\, k}$. Simply the parentheses are omitted. In general $$(\nabla_l v)^k \partial_k = \nabla_l ( v^k ) \partial_k + v^j \Gamma_{ l j }^k \partial_k$$

I am not convinced about the vector field decomposition $X = \langle X, N \rangle N + Y$ because, if I understand correctly, $N$ is not Lorentz orthogonal to your frame of vectors $e_1,..,e_{n-1}$. I would say $X = \lambda N + Y$ where $Y$ is light-like. Then $\langle X, Y \rangle = \lambda \langle N, Y \rangle + \langle Y, Y \rangle = \lambda \langle N, Y \rangle $ and solve for $\lambda$.

I don't know if it helps but these chains can be transformed to Poncelet's porism: the points of tangencies between pairs of black circles lie on a common circle orthogonal to all black ones. Furthermore the centers of the black circles lie on an ellipse. So the poligonal line formed by connecting pairs of centers of adjacent black circles is inscribed in the ellipse and tangent to the orthogonal circle.

Yeah, the philosophy is somewhat similar, when you think of Veronese embedding as a tool for studying incidence between points and conics... I guess...

It seems to me that you shouldn't have points like S and maybe R (if I'm seeing the picture correctly) because the curvature of the surface at those points will be positive after adding the transverse vertical circles...