  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 $Q$ invariant
convex energy surface under a pinching condition. As a result, it is proved
that if the symmetric energy surface admits a nonsymmetric periodic solution,
it has infinitely many periodic orbits. In order to prove the result, we
introduce a new index on rotating periodic orbits.
