Consider a second order Hamiltonian system of the type $$ \ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N. $$ Under very natural assumptions it is possible to prove the existence of a non constant $T$-periodic orbit $\varphi^T=(\varphi_{1}^{T},\ldots,\varphi_{N}^{T})$ for any given $T>0$. If $N \ge 2$ it may as well happen that $\varphi_{i}^{T} \equiv \text{ constant }$ for some index $i$.

Is there any result that guarantees the existence of a $\varphi^T$ for which all the components are non constant functions?