Beltrami identity and E-L equation incompatible?

Question

$\newcommand{\dd}[2]{\frac{\partial {#1}}{\partial {#2}}}$ $\newcommand{\DD}[2]{\frac{d {#1}}{d {#2}}}$

Begin with the Euler-Lagrange equation $\dd Ly = \DD \ x [\dd L {y'}]$ where $L=L(y,y',x)$, (implicit dependence on $x$, so $\dd L x= 0$).

We get $\DD \ x L = \dd L {y'} y'' + \dd L y y' + [\dd L x = 0] \\= \DD \ x [\dd L {y'} y']$

so $$L - \dd L{y'}y'=Const.$$

This is a well known result.

My problem follows in this math:

$\dd L y = \dd \ y (\dd L {y'} y') =\dd { (\dd L {y'} )} y y' *$ From the equation just derived

but it also equals from the E-L equations $\dd L y=\DD \ x \dd L {y'} = \dd {\dd L {y'}} y y' + \dd {\dd L {y'}} {y'} y'' + [\dd {\dd L {y'}} x = 0] *$

Setting the two starred equations (*) equal to each other, this gives $$ \dd {\dd L {y'}} {y'} y'' =0$$

which is clearly wrong. What was the step that messed things up here?

Answer 1

The EL equations and the Beltrami identity are on-shell relations. There are no reason why they should still hold after a partial differentiation wrt. a dependent variable $y$ or $y^{\prime}$.

Example: Let $L=\frac{1}{2}y^2$. Then $y\approx 0$ on-shell. However we are not allowed to differentiate the relation $y\approx 0$ wrt. $y$. That would lead to the contradiction $1\approx 0$.