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For example, hen working with general relativity, one sees that Einstein equations can be derived from an action principle via the Einstein-Hilbert action. This occurs too in classical mechanics, optics, electrodynamics,...

Even in modified theories of gravity, or other advanced theories like string theory, qft in curved spacetimes, quantum cromodynamics,... the approach always is to define an action and derive the field equations from the least action principle.

In classical mechanics this can be intuitive, understanding the least action principle as conservation of energy. However, in this sofisticated theories, how does one have the security of deriving the field equations from an action? How does one knows that the field equations derived are the unique field equations of the theory? I can't see the guarantee even though I have read similar questions in the forum, but the arguments still not convince me.

In summary, what is the precise theoretical argument of why the principle of least action works in complete diferent scenarios?

ALPs
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  • Is your question why we should believe there always exists a Lagrangian to go with any set of field equations we might write down? – Richard Myers Jul 31 '21 at 02:12
  • Possible duplicates: https://physics.stackexchange.com/q/15899/2451 , https://physics.stackexchange.com/q/3500/2451 , https://physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Jul 31 '21 at 03:24

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In summary, what is the precise theoretical argument of why the principle of least action works in complete diferent scenarios?

It is possible to derive a theory from a Lagrangian and the theory is not right, in the sense that it doesn't fit some experiments. Example: Nordstrom theory of gravitation.

So, I don't think that it is possible to have a theoretical argument for the general validity of that principle.

Claudio Saspinski
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  • Or course, we can invent some lagrangian whose field equations become absurd. For example, we could give up Einstein-Hilbert action lagrangian and say that its lagrangian is proportional only to R^2 instead of R. This would give us field equations that don't represent physical reality. BUT, those equations would be the correct equations for that model on particular, why that process never fails? Thanks! – ALPs Jul 30 '21 at 23:04
  • @Axionlikeparticles On the question why that process never fails: On my profile page I provide a link to a resource that I created. It's an interactive diagram; moving a slider sweeps out variation. The diagram shows in three sub-panels how variation sweep affects the variables that are involved. The interactive diagram provides visualization of the process of interconversion between differential equation form and stationary action form. (I link to my profile page instead of direct to avoid suspicion of vanity linking.) – Cleonis Jul 31 '21 at 04:18