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This paper concerns the existence of multiple rotating quasi-periodicsolutions for second order Hamiltonian systems with sub-quadratic potential.Such solutions have the form $x(t+T)=Qx(t)$ for some orthogonal matrix $Q$. Todeal with such quasi-periodic solutions, we introduce the $\mathcal{Q}(s)$index which is a development of the well known $S^1$ index. Applying the$\mathcal{Q}(s)$ index, we give an estimate of the number for rotatingquasi-periodic orbits with a fixed period.

Rotating Quasi-periodic Solutions of Second Order Hamiltonian Systems with Sub-quadratic Potential