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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?

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# Periodic orbits of Hamiltonian systems

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?