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Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be analytically solved, so should we wish to generate a phase or something, our only option is to numerically integrate.

It's easy to start off with $q$ and $\dot{q}$, and use Euler's method or Runge-Kutta or whatever fits the desired specifications (speed, accuracy, etc.) But this answer indicates that it's possible to use an initial $q_i$ and final $q_f$ to solve this equation, and I'm not sure how that would work. Is it possible? If yes, how can implemented? If it isn't possible, are there any elegant counterexamples, where there are multiple equations of motion corresponding to a given Lagrangian wherein the value of the coordinate is a certain value at a particular time?

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It seems OP is talking about caustics/focal points/conjugate points, where the Jacobi field vanishes. In other words, special boundary values where a boundary value problem has infinite or no solutions.

Example: Simple harmonic oscillator, where $t_f-t_i\in \frac{T}{2}\mathbb{Z}$.

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