Right. Well, there is one interesting exception. If 2
Lagrangians, say L_1 and L_2 have the same Euler-Lagrange
equations then their difference,

L_0 = L_1 - L_2

must have identically vanishing Euler-Lagrange equations. L_0 is
sometimes called a "null Lagrangian". Locally, null Lagrangians
can be expressed as a total derivative (or total divergence in a
field theory) just as you say, but this may not be true globally
if the configuration space of the theory has some topology. I
believe there is a theorem to the effect that, for a field
theory on an n-dimensional manifold (n=1 means mechanics), to
every representative of a degree n cohomology class on the
bundle of independent and dependent variables (i.e., the bundle
of fields) one can construct a Lagrangian that is not a total
divergence, but nevertheless has identically vanishing
Euler-Lagrange equations. To get an interesting example one
probably needs some cohomology in "field space" (rather than
just in spacetime). Probably I could cook up some examples if
you are perverse enough to really be interested in this
phenomenon. Anyway, this wasn't really what I was thinking of
when I made the comment about different Lagrangians and
quantization. As you say...

Excellent point. The more interesting possibility is that one
could have two Lagrangians whose Euler-Lagrange (EL) equations
are *equivalent* instead of identical. (I had inadvertently
drifted into this point of view when I made the comment about
inequivalent Lagrangians and quantum theory. Thanks for keeping
me honest.) This point of view gives a much more useful (and
much harder) form of the inverse problem in the calculus of
variations: when is there a Lagrangian whose EL equations are
*equivalent* (rather than identical) to a specified set of
equations. I say that this point of view is more useful since
one often times does not have equations expressed in just the
right form to be EL equations, even though there is an
underlying Lagrangian for the dynamical system of interest. For
example, the vacuum Einstein equations G_ab=0 (G is the Einstein
tensor) are not the EL equations of any Lagrangian. (Wait! Don't
shoot until after you read the next two sentences.) But they are
equivalent to a system of equations E_ab=0 which ARE EL
equations. Here E is the Einstein tensor multiplied by the
square root of the determinant of the metric.

The paper by Anderson and Thompson that I cited earlier in this
thread gives, I think, a pretty near state of the art treatment
of this more general type of inverse problem for ODEs. The PDE
version of this inverse problem is, I think, in a much more
primitive state. As I recall, in that paper you will find examples
of DEs which admit more than one Lagrangian such that the
various Lagrangians do not differ by a total derivative or
constant rescaling. These examples do NOT arise because of the
topological subtleties that I mentioned earlier, but rather
because of the freedom to choose alternative, but equivalent,
equations of motion. (Sorry. I don't have the paper available so
I can't whip out one of their examples. )

Charles Torre