Periodic solutions of a many-rotator problem in the plane

We consider the many-rotator system whose motions in the plane are characte rized by the Newtonian equations of motion ((r) over right arrow )over dot(n) = w (k) over cap boolean AND ((r) over r ight arrow )over dot(n) +2 Sigma (N)(m=1,m not equaln) r(nm)(-2)(alpha (nm) + alpha ' (nm)(k) over cap boolean AND). x[((r) over right arrow )over dot(n)(((r) over right arrow )over dot(m)(r) over right arrow (nm)) + ((r) over right arrow )over dot(m)(((r) over right arrow )over dot(n)(r) over right arrow (nm)) - (r) over right arrow (nm) ( ((r) over right arrow )over dot(n) ((r) over right arrow )over dot(m))], (r) over right arrow (nm) = (r) over right arrow (n)- (r) over right arrow (m), where superimposed arrows denote three-vectors living in the plane in which all motions take place, k- is a unit vector orthogonal to that plane, the symbol A denotes the usual three-dimensional vector product and omega, alph a (nm), alpha ' (nm) are 2N(N - 1) + 1 arbitrary real constants (without lo ss of generality omega > 0). This model is invariant under rotations and tr anslations (in the plane); it is Hamiltonian provided alpha (mn) = alpha (n m), alpha ' (mn) = alpha ' (nm); it is not known to be integrable Mn nm (fo r N > 2), unless all the coupling constants alpha ' (nm) vanish (alpha ' (n m) = 0) and the coupling constants Un,, either also all vanish (a,. = 0; tr ivial case, all motions completely periodic, with period T = 2 pi/omega)) o r are all equal to unity (alpha (nm) = 1; integrable/solvable case, all mot ions completely periodic, with period at most T ' = TN!). We prove that thi s model generally possesses a large class of solutions (corresponding to a set of initial conditions containing a non-empty open set) which are comple tely periodic with period T = 2 pi/omega. Analogous results also hold for m ore general evolution equations, interpretable as appropriately (linearly) deformed versions of those characterizing geodesic motions in N-dimensional space.