Is it true that Euler-Lagrange equations associated to a functional determine the functional?

Suppose I give you an equation and I claim that it is an Euler-Lagrange equation of some functional. Can you tell me what was the functional?

Of course, there is always more than one functional whith prescribed E-L equations, since the critical points of $E$ and of $\phi(E)$ where $\phi:\mathbb{R} \to \mathbb{R}$ is smooth and stirclty monotonic are identical. (By the chain rule $ (\phi \circ f)'(x)=\phi'(f(x))\cdot f'(x)$).

Is it true that there is a functional $E$ whose E-L equations are the prescribed ones, and every other functional with the same E-L equations is a function of $E$?

One can think on different ways to formalize this question like different choices for the domain of the functional: paths in a manifold, real valued functions on $\mathbb{R}^n$, mappings between Riemannian manifolds etc, but at this stage of the game I don't want to choose a specific form yet. (Although I am particularly interested in the latter case).

  1. It is well-known that

    • adding boundary/total divergence terms and/or
    • overall scaling

    of a functional preserve the Euler-Lagrange (EL) equations.

  2. On the other hand, there is in general no classification of possible functionals that lead to a given set of EL equations.

  3. An instructive example from Newtonian point mechanics is given in this Phys.SE post, where two Lagrangians $L=T-V$ and $L=\frac{1}{3}T^2+2TV-V^2$ both have Newton's second law as their EL equation.

  4. Another example: The functional in this Math.SE post has the same EL equation as the functional $F[y]=\int_0^3 \! \mathrm{d}x~y^{\prime 2}.$


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