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I'd like to point out that knowing $2n$ quantities and the equations of motion are not enough to determine the solution. Even at the level of $L=T-U$ for just one particle ($n=1$). Consider $T=\frac{1}{2}m\left(\frac{dx}{dt}\right)^2,$ and $U=-C\frac {9m}{2}x^{4/3}.$ Then, for your equations of motion, you get ...


Observe that, knowing $\ddot{q}$, to get $\dot{q}$ and then $q$ you have to integrate twice. This introduces $2n$ integration constants you have to know to fully describe the system, which is the same amount of freedom you get when solving the Euler-Lagrange equations, where you need initial conditions for $q$ and $\dot{q}$.

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