This paper concerns the existence of multiple rotating periodic solutions for
2n dimensional convex Hamiltonian systems. For the symplectic orthogonal
matrix Q, the rotating periodic solution has the form of z(t+T)=Qz(t),
which might be periodic, anti-periodic, subharmonic or quasi-periodic according
to the structure of Q. It is proved that there exist at least n
geometrically distinct rotating periodic solutions on a given convex energy
surface under a pinched condition, so our result corresponds to the well known
Ekeland and Lasry's theorem on periodic solutions. It seems that this is the
first attempt to solve the symmetric quasi-periodic problem on the global
energy surface. In order to prove the result, we introduce a new index on
rotating periodic orbits.Comment: arXiv admin note: text overlap with arXiv:1812.0583