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### Initial in time conditions and Lagrangian approach in classical mechanics

When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space....
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### How does Hamilton's Principle give us the path taken?

We defined the action as: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed. Hamilton's principle states that the path that is ...
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### If the path integral formulation includes future events, why doesn't that imply retrocausality?

I know that such events would cancel out in the math, but if an extreme event were to happen in the future (say a black hole forming or something on that par), would a particle in the present react to ...