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?
Concerning #3, it is not true that all systems of differential equations have a variational formulation, cf. the inverse problem for Lagrangian mechanics.