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Classical mechanics texts generally derive Lagrangian mechanics from Newton's laws, demonstrating how a particular Lagrangian, when its action is minimized, leads through Euler-Lagrange to differential equations that describe the evolution of a physical system. In some cases, this Lagrangian is introduced with motivation from prior knowledge of the system's equations of motion, and in others it’s introduced using symmetry arguments.

Our ability to find a Lagrangian in the first place for every system we encounter seems, to me, miraculous. This has led me to look for mathematical justification for universality of the Lagrangian method for solving equations of motion.

To what extent can we generalize Lagrangian methods to general dynamical systems and, thereby, understand the existence of the action and the principle of least action as mathematical, rather than physical, phenomena? In other words, can we rigorously reconstruct Lagrangian mechanics as a method for solving differential equations/ dynamical systems without any reference to physics?

This mathematical approach might contrast explanations like the one here, which attempt to justify Lagrangian mechanics through physical rather than mathematical foundations.

I suspect there's concepts from dynamical systems and the study of differential equations that should help us here, but haven't seen this connection made in any texts.

Formalizing the problem, I'm trying to find some formalism that roughly says:

1) A physical system at any point in time (ignore GR) is defined as a point on its phase space, represented mathematically by a manifold. This seems uncontroversial, as I'm unaware of any alternative representations.

2) We can choose a minimal set of equations of motion to assign every point to a time-parameterized trajectory on its phase space, represented mathematically by differential equations defining curves on this manifold. This also seems uncontroversial, though assuming all equations of motion, i.e., all physical laws, take the form of differential equations is a non-trivial assumption.

3) Any system of differential equations defining curves on our manifold can be mapped to a functional, such that this functional is minimized by the curves defined by that system of differential equations. Or, in physics terms, all possible equations of motion, not just “true” equations of motion of the real world, have an action and associated Lagrangian that they minimize. All dynamical systems, in this way, have a characteristic Lagrangians, closely associated like a matrices's characteristic polynomials.

Is something like #3 true? What texts/ references support such a project?

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  • $\begingroup$ One area that basically does this is "symplectic geometry", which studies manifolds which are equipped with enough structure to have a Hamiltonian. $\endgroup$ – pseudocydonia Oct 23 '19 at 23:35
  • $\begingroup$ On the other hand, I am not at all sure what it would mean to justify Lagrangian mechanics without reference to physics! In some sense the whole point is that it makes a certain class of physical predictions... $\endgroup$ – pseudocydonia Oct 23 '19 at 23:36
  • $\begingroup$ I see Lagrangian mechanics as a way of taking a given Lagrangian (of arbitrary form), and putting them into the Euler-Lagrange equations to produce equations of motion. All of the physics are embedded in the correct choice of the Lagrangian, and choosing a "bad" Lagrangian will yield unphysical equations. Implicit then to all of this, is the idea that there's some mapping between Lagrangians, actions and systems of differential equations, even before we choose a Lagrangian representing our physics. I'm wondering if there's mathematical grounding for the existence/uniqueness of this connection. $\endgroup$ – Dragonsheep Oct 23 '19 at 23:41
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    $\begingroup$ By the way, I think it is highly unlikely (i.e. false) that every dynamical system has an associated Lagrangian, for the reason that the Euler-Lagrange equations have a laundry list of special properties. $\endgroup$ – pseudocydonia Oct 23 '19 at 23:48
  • $\begingroup$ You seem to be correct. Further research suggests I'm really asking about the inverse problem in Lagrangian Mechanics. I.e., not all dynamical systems can be generated from application of Euler-Lagrange to a Lagrangian, but luckily all equations of motion in the real world have a Lagrangian, see physics.stackexchange.com/questions/357775/… . This answer is a bit dissatisfying and just compels me to ask why this should be true, what physical properties do real world equations of motion have to guarantee they have a Lagrangian. $\endgroup$ – Dragonsheep Oct 24 '19 at 0:13
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Concerning #3, it is not true that all systems of differential equations have a variational formulation, cf. the inverse problem for Lagrangian mechanics.

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