In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary.

Now all derivations I've come across up to now, carry out the variation of the funtional over a finite time, which I understand fine. But how far can we reduce this time interval before we start to lose the generality of the Euler-Lagrange equations?

For example, consider the functional $$J[y] = \int_{t1}^{t2}L[t, y(t),y'(t)]dt$$ where $t1$ and $t2$ are constants, and there is zero variation at the end points $y(t1)$ and $y(t2)$

For a start, we need at least two $dt$ intervals; the first for the first variation of $y'(t1)$, and the second for a compensating variation of $y'(t1+dt)$ to satisfy the end points condition.

Can we derive the Euler-Lagrange equations over just these two $dt$ intervals, or do we need more?


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