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Total-time-derivative invariance of a Lagrangian

Replacing a Lagrangian by

L^{'} (q, \overset{q}{˙}, t) = L (q, \overset{q}{˙}, t) + \frac{d}{d t} F (q, t)

adds only the endpoint term

F (q (t_{2}), t_{2}) - F (q (t_{1}), t_{1})

to the action. Fixed-endpoint variations therefore give the same Euler--Lagrange equations.

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