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.

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