What does it mean when the eom of a field is trivially satisfied?

[Baela]

If a Lagrangian has the fields $a$, $b$ and $c$ whose equations of motion are denoted by $E_a, E_b$ and $E_c$ respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c
\end{align}
where $f_1$ and $f_2$ are some functions of the fields, if $E_b$ and $E_c$ are satisfied, then $E_a$ is automatically satisfied.

Does this tell us anything particular about the nature of field $a$?

 

[vanhees71]

Without clear definitions of your symbols there's no way to answer your question. Where do you get this from?

 

[Baela]

Which symbol do you need clarification for? My question is pretty general. I can't see what part you are confused about.

 

[vanhees71]

You don't give any definition of your symbols. How can you expect that anybody can understand what they mean?

 

[renormalize]

You don't give any definition of your symbols. How can you expect that anybody can understand what they mean?

For background, the OP also started this thread: https://www.physicsforums.com/threa...t-gauge-transformations.1051286/#post-6871749 (although they never returned to it as promised). My understanding of their notation is: $a(x), b(x), c(x)$ are spacetime fields individually satisfying the 3 Euler-Lagrange (field) equations $E_{a}(a(x))=0, E_{b}(b(x))=0, E_{c}(c(x))=0$. I think they want to know the consequences if $E_{a}(a(x))$ happens to be a linear-combination, with field-dependent coefficients, of $E_{b}(b(x)),E_{c}(c(x))$. My answer is that it simply means only 2 of the 3 fields are dynamically independent.