Let consider a hamiltonian system on $[0,T]$ (resulting from Pontryaging Maximum Principle) ($q=(q_1,q_2)$ is the state and $p=(p_1,p_2)$ is the co-state)
$$ \frac{dq}{dt} = \frac{\partial H}{\partial p}(q,p), \quad \frac{dp}{dt} = - \frac{\partial H}{\partial q}(q,p) $$
A solution of this system have to check some boundary conditions such as \begin{align*} &q_2(0)=0, \; q_2(T)=q_{2T}, \\ &q_1(0)=q_1(T), \; p_1(0)=p_1(T). \end{align*} (4 conditions in dimension 4)
Do you have some general/heuristic ideas to prove that on a fix level of energy $H$, there exists at most one solution ?
Thanks