Practice question · True or false
In a variational problem, checking that the path is optimal at its start and end points is enough, the middle takes care of itself.
Hints
- The condition has to hold at every instant, not only at the ends.
- A wasteful middle segment can always be deformed into something better.
Show the answer
False
Why
Euler–Lagrange must hold at every instant: a path perfect at both ends but wasteful for one minute in the middle is beaten by deforming exactly that minute. Global scores punish local sloppiness anywhere.
Practise Calculus of Variations
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