Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE
$$(1)\quad \ddot{q}+V^\prime(q)=0, $$
where $V \colon \mathbb{R} \to \mathbb{R}$ is some smooth function (potential energy). We then easily see that any solution of this equation must satisfy
$$\frac{\dot{q}^2}{2} + V(q)=\text{constant}.$$
In other words, if we put
$$E(q, \dot{q})=\frac{\dot{q}^2}{2} + V(q)$$
(energy), then the image of every solution of (1) must lie in a level set of $E$.
Question Can it be that a solution of (1) is projected properly in a connected level set of the energy?
More formally, can there exist a function $u \colon I \to \mathbb{R}$ that solves (1) and such that the set
$$T_u=\{(u(t), \dot{u}(t)) \mid t \in I\}$$
is a proper subset of
$$E_u=\{(a, b) \in \mathbb{R}^2\mid E(a, b)=E(u_0, \dot{u}_0)\}?$$
Of course $I$ must be the maximal interval of existance for the solution $u$. Require also $E_u$ to be connected (thanks to AlbertH who pointed this out).
By physical considerations I guess that this cannot happen. Also, I found in some physics textbooks phrases like: "finding the level sets of the energy is equivalent to finding the trajectories of the system", which corroborate this conjecture.
I'm not necessarily after a fully rigorous answer. Even a good intuition will suffice. Thank you.