# Illustration Minimization of an Action Functional 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.