Classification of Lagrangians with given Euler-Lagrange equations In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of variables.
As far as I understand in physics usually the equations of motion are experimentally determined first, and $L$ is chosen a posteriori to satisfy these equations and may be some other natural symmetries. In classical (not quantum) physics $L$ has no physical meaning other than the Euler-Lagrange equations.

Hence the problem of uniqueness of the Lagrangian arises: Can one classify all Lagrangians $L$ such that its Euler-Lagrange equations are equivalent to a given system of equations of motion? A reference would be very helpful.

Remark. (1) My question is somewhat vague since I am not sure what does it mean exactly "equivalent". To make it precise is a part of the question.
One obvious option to define the equivalence is to say "the two systems of differential equations have exactly the same solutions".
(2) It is well known that the following two kinds of transformations of $L$ lead to equivalent (in any sense) systems of equations:
(a) $L\to aL+b$ where $a,b$ are constants.
(b)$L\to L+\frac{\partial F}{\partial x}\dot x+\frac{\partial F}{\partial t}$ where $F=F(x,t)$ is a function.
(3) I am far from this field and may not aware even of basic results in this direction.
 A: In a sense, all the Lagrangians giving the same Euler-Lagrange equations are exhausted by transformations of your type (b), which adds a total derivative/total divergence/boundary term/...  Transformations of your type (a) can alter the Euler-Lagrange equations, for instance if $a\ne 1$, then the EL equations get rescaled by the same constant $a$. Perhaps you don't care about such a rescaling. In that case, why care about any kind of violence that can be done to the EL equations, as long as they keep the same solutions. This point of view changes the question substantially, to the point where the general answer is not known.
The name given to your question is the "inverse problem of the calculus of variations" (also just mentioned in a comment by Robert Bryant!). The literature is vast, but there is one core result, which corresponds to the first part of the previous paragraph: (a) A Lagrangian $L(x,t)$ has vanishing equations of motion iff it is locally (in both independent and dependent variables) a total divergence. (b) $F(x,t)=0$ is locally an EL equation for some Lagrangian iff the linearization of $F(x,t)$ is a formally self-adjoint linear differential operator. Condition (b) is known as the Helmholtz condition.
To find references, a good start is typing in "inverse problem of the calculus of variations" into Google. Previously, it has come up on MO here, where you can also find some references, but they are not very up to date, since this field is still evolving:

*

*Which differential equations allow for a variational formulation?

*https://mathoverflow.net/a/38827

*https://mathoverflow.net/a/81846
A: This problem is discussed in Bryant, Griffiths, Hsu, Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, for Lagrangians for scalar fields.
