Doubts about "whether a given system has a Lagrangian" and "inverse problem of the calculus of variations"

Question

There has been extensive discussion in the literature and on this forum regarding the question of "whether a given system has a Lagrangian" (e.g. post1, post2, post3, and paper1, paper2).

The fundamental approach involves framing this question as the "inverse problem of the calculus of variations", wherein, given a set of second-order differential equations, one seeks the necessary and sufficient conditions for transforming them into the Euler-Lagrange equations. The results of this approach can be found on various platforms, including wiki.

However, I still harbor doubts about the underlying premise of this approach. Our understanding of a system may not necessarily begin with its differential equations; rather, we might only be aware of the components of the system (particles, interactions, etc., similar to the positive problem). From this perspective, there seems to be a gap between the question of "whether a given system has a Lagrangian" and the "inverse problem of the calculus of variations."

How can we address this gap?

Answer 1

Briefly speaking OP's question seems to be: Given a physical system, how to model it and find the equations of motion? That's a good but broad question whose answer presumably depends on the circumstances. However, ideally speaking$^1$, once the equations of motion are known, it then becomes the inverse problem for Lagrangian mechanics.

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$^1$It should perhaps be mentioned that model building in modern theoretical physics (say e.g. beyond the standard model) often takes the opposite route: Start with an action principle respecting certain symmetries and properties, and then work out the equations of motion.