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Illustration Minimization of an Action Functional Between Two Points

Minimization of an action functional between two points Small variation of the path between two points
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One-dimensional example of minimizing an action functional \( S[q] \) between two points \(t_1\) and \(t_2\). The goal is to find a function \(q(t)\) that makes the following action functional stationary, for example minimal:$$ S[q] ~=~ \int_{t_1}^{t_2} \text{d}t \, L(t, q,\dot{q}) $$

The searched function \(q(t)\) must take certain fixed values \( q(t_1) = q_1 \) and \(q(t_2) = q_2 \) at the edges. Between these edges all possible functions \(q(t)\) are considered and we search for a function which makes the action functional \( S[q] \) minimal.