characterization of variational symmetries
I have the following definition of variational symmetry, this is from <PERSON> Calculus of Variations book
Suppose we have a functional of the form $$ J(y)= \int_{x_{0}}^{x_{1}} f(x,y,y')dx $$ and a smooth transformation $X=\theta (x,y: \epsilon)$ , $Y=\psi (x,y: \epsilon)$ where $\epsilon$ is a parameter.
Definition. The functional $J$ is invariant under the above transformation if, for all $\epsilon$ sufficiently small, in any subinterval $[a,b] \subset [x_{0},x_{1}]$ we have that $$ \int_{a}^{b} f(x,y,y')dx= \displaystyle \int_{a_{\epsilon}}^{b_{\epsilon}} f(X,Y,Y')dX $$ for all smooth functions $y$ defined on $[a,b]$. Here, $a_{\epsilon}=\theta (a,y(a): \epsilon)$, $b_{\epsilon}=\theta (b,y(b): \epsilon)$
And, $ \epsilon$ is a parameter such that $\theta (x,y; 0)=x$ and $\psi (x,y; 0)=y$.
Once said that, I would appreciate your help for a rigorous proof of the converse of the following theorem
Theorem. 669
Let $$ J(y)= \int_{x_{0}}^{x_{1}} f(x,y,y')dx $$ The transformation above with infinitesimal generators $\xi$ and $\eta$ is a variational symmetry for $J$ if and only if $$ \xi \displaystyle \frac{\partial f}{\partial x} + \eta \frac{\partial f}{\partial y} + ( \eta ' - y' \xi ' ) \frac{\partial f}{\partial y'} + \eta ' f=0 $$
For any smooth function $y$ on $ [x_{0},x_{1}] $. Here,
$$ \xi (x,y)= \frac{\partial \theta}{ \partial \epsilon } \Big |_{(x,y;0)}^{} $$
$$ \eta (x,y)= \frac{\partial \psi}{ \partial \epsilon } \Big |_{(x,y;0)}^{} $$
I have been thinking about it , but, I believe that, at least under this definition of invariance the converse is false. The author refers us to the book Introduction to the calculus of variations and its applications by <PERSON> for a proof, however, I see nowhere that the converse is proved.
Any help? Thanks in advance.
Remark:
The reason for which I think the converse it's false is because if we consider the functional $$ \displaystyle \int_{x_{0}}^{x_{1}} xy'^2 dx $$ And the transformation
$$ X=x+2 \epsilon x ln(x), Y=(1+ \epsilon ) y $$
Then, equation $(9.38)$ holds , but, I think there is no way that the above transformation is a variational symmetry since
$$ XY ' ^2 dX= \left( xy'^2+ \frac{(x \epsilon ^2+ 2 \epsilon ^3 x ln(x))y'^2}{(1+2 \epsilon ln(x)+ 2 \epsilon)} \right) dx $$
And this , definitely does not agree with the definition. Or am I wrong? If so, why? :( 669