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In mechanics, the application of Euler Lagrange equations always feels a bit odd to me as they are a necessary condition on extrema of functionals on trajectories with fixed endpoints. This is weird as there seems to be no reason to fix endpoints. I would propose the following as a more natural variational problem:

Minimize $\int_a^b \mathscr{L}(t, x(t), x'(t))\, dt$ subject to $x(a) = x_0$, $x'(a) = v_0$.

Hopefully we still get the Euler Lagrange equation as a necessary condition in this case, but I don't know of a proof. Can anyone give a literature reference or prove/disprove the conjecture that we get the Euler Lagrange equation?

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At one step in the derivation of EL equation one needs to integrate by parts. In order for the boundary terms to vanish one needs boundary conditions. Initial conditions (which OP suggests to use instead) clearly won't cut it at the final end point.

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