Norbert Dragon <dra...@itp.uni-hannover.de writes:Right. Well, there is one interesting exception. If 2
> * Charles Torre to...@cc.usu.edu writes:
>> What's perhaps a little more amusing in this regard are the systems
> The correspondence of Euler-Lagrange equations and Lagrangean is
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
Excellent point. The more interesting possibility is that one
> However, it may turn out that different systems of equations have
> An example of two different, local functionals with the same set of
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
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