= Solution
For fixed endpoint values and a smooth <variation> vanishing at the endpoints, the first variation is
$$
\delta T=\int_a^b(f_y\delta y+f_{y'}\delta y')\,dx
=\int_a^b\left(f_y-\frac d{dx}f_{y'}\right)\delta y\,dx.
$$
The <fundamental lemma of the calculus of variations> gives the <Euler-Lagrange equation>
$$
\boxed{\frac d{dx}\frac{\partial f}{\partial y'}-\frac{\partial f}{\partial y}=0}.
$$
If $f$ is independent of $y$, then $y$ is a <cyclic coordinate>, and integration gives
$$
\boxed{\frac{\partial f}{\partial y'}=C}.
$$
This is conservation of the corresponding <generalized momentum>.
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