How to derive the Lagrangian for a system of first order equations of motion?

Question

Please be lenient if this question has been asked before in a similar manner. My background is in CS, but I am working on the physics-based modeling of complex systems and I am studying physics as an undergraduate.

I am studying a dynamical system that is partially physical and partially complex (non-physical). For the purpose of this question, let us say that the system is described by three states: the spatial position $x$ and $y$, and money $m$, all measured at time $t$. You can think of a person who runs around and collects or throws money as he goes. This is not the real problem I am modeling, but it is a good enough approximation since the underlying mathematics is the same.

I have a set of differential equations that describes the evolution over time of these three states:

1. $\dot{x} = f(x)$
2. $\dot{y} = g(y)$
3. $\dot{m} = h(m)$ caveat: this might be a non-linear function of $x,y,m \to \dot{m} = h(m,x,y)$

These three equations were modeled in application of [1] on the basis of some datasets (synthetic, if it matters).

Under a hypothesis that derives from the literature and that we can challenge if necessary (but please assume that this is true for a second), the evolution of the system is such that there exists an unknown Lagrangian $L(x,y,m,\dot{x},\dot{y},\dot{m},t)$ for which the associated action is stationary to first-order. The corresponding Euler-Lagrangian equations should therefore be:

1. $\frac{\partial L}{\partial x} = \frac{d}{dt} \frac{\partial L}{\partial\dot{x}}$
2. $\frac{\partial L}{\partial y} = \frac{d}{dt} \frac{\partial L}{\partial\dot{y}}$
3. $\frac{\partial L}{\partial m} = \frac{d}{dt} \frac{\partial L}{\partial\dot{m}}$

Question 1: How can I derive the possible forms of $L$ that satisfy the constraints that derive from the known equations of motion? I am familiar with the method for deriving the equations of motion from a given Lagrangian in physical systems, as is commonly done in analytical mechanics, but I do not understand how to go the other way around. I understand that the Lagrangians that satisfy these constraints may be many, but it seems like not all possible functionals of these state variables will be Lagrangians: for me, it is sufficient to identify one, and ideally one with a few symbols, and discard some trivial cases. Since this is a problem that I will have to solve frequently in the future, I would like to have an idea on how to construct a Lagrangian given the equations of motion, in general, and not only in the application of this specific problem.

Question 2: This paper [2] estimates the polynomial representation of the Lagrangian that results in a minimised (not stationary) action given certain time series. If the Lagrangian can be approximated as a Taylor expansion, then this approach is reasonable. They do however make use of a control trajectory for which the action associated with the Lagrangian being approximated is not minimised, in order to avoid trivial trajectories (e.g. $\forall q,\dot{q}|L(q,\dot{q},t) = 1$). Does the knowledge of a control trajectory help in answering question 1? I have trajectories for which, by the same hypothesis deriving from the literature, the action should not be stationary, which means that those trajectories might provide further constraints on the shape of possible Lagrangians.

Please tell me if something is not clear or should be expressed better.

[1] Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15), 3932-3937. https://doi.org/10.1073/pnas.1517384113

[2] Hills, D. J., Grütter, A. M., & Hudson, J. J. (2015). An algorithm for discovering Lagrangians automatically from data. PeerJ Computer Science, 1, e31. https://peerj.com/articles/cs-31.pdf

Edit: I had written $dt$ instead of $t$ in the first expression of $L$

Answer 1

I am not sure if this is your problem ?.

with the non holonomic constraint equations

$$\left[ \begin {array}{c} {\dot x}-f \left( x \right) \\ {\dot y}-g \left( y \right) \\ {\dot m}-h \left( m,x,y \right) \end {array} \right] =\vec 0$$

and the kinetic energy

$$T=\frac 12\,(\dot x^2+\dot y^2+\dot m^2)$$

you can obtain with EL the equations of motion

$$\ddot x=\left( {\frac {d}{dx}}f \left( x \right) \right) {\dot x}\\ \ddot y=\left( {\frac {d}{dy}}g \left( y \right) \right) {\ddot y}\\ \ddot m=\left( {\frac {\partial }{\partial x}}h \left( m,x,y \right) \right) {\dot x}+ \left( {\frac {\partial }{\partial y}}h \left( m,x, y \right) \right) {\dot y}+ \left( {\frac {\partial }{\partial m}}h \left( m,x,y \right) \right) {\dot m} $$