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

AI Thread Summary
The discussion centers on the implications of a Lagrangian with fields a, b, and c, where the equation of motion for field a is expressed as a linear combination of the equations of motion for fields b and c. It is established that if equations E_b and E_c are satisfied, then E_a is automatically satisfied, indicating a dependency among the fields. Participants emphasize the need for clear definitions of symbols to facilitate understanding, as the original question lacks clarity. The conclusion drawn is that this relationship suggests that only two of the three fields are dynamically independent. This highlights the interconnected nature of the fields within the Lagrangian framework.
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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##?
 
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Without clear definitions of your symbols there's no way to answer your question. Where do you get this from?
 
Which symbol do you need clarification for? My question is pretty general. I can't see what part you are confused about.
 
You don't give any definition of your symbols. How can you expect that anybody can understand what they mean?
 
vanhees71 said:
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.
 
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