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Lagrangians, Nth-rank tensors, and Nth derivatives

At 03:25 AM 8/8/99 -0400, David Bowman wrote:

In general, if an interaction between sources is mediated by an
even-integer-spin field (e.g. scalar or 2-tensor potential field--like
gravity which has the |grad([phi])|^2 term keep the same sign between
the Hamiltonian and the Lagrangian) then it causes like-signed sources
to *attract* each other, and if it is an odd-integer-spin field (e.g.
vector potential field like E&M which has the |grad([phi])|^2 term
switch signs between the Lagrangian and the Hamiltonian) then it causes
like-signed sources to *repel* each other.

That's something that has intrigued and tantalized me since childhood.
Consider the following additional angle:

N=0) Potential energy depends on position, X.

N=1) The usual kinetic energy depends on velocity, X dot.

N=2) Electromagnetic radiated power and energy have source terms that
depend on acceleration, X dot dot.

N=3) The corresponding gravitational radiation formulas contain X dot dot dot.

In each case, when you apply Hamilton's principle of least action, you need
to integrate that term by parts N times, causing N sign-flips.

=========

Can somebody elucidate the relationship between the rank of the tensor
field and the order of the derivative? I suspect it is not merely a
coincidence.

Presumably the following is part of the story:

N=2) Two electric charges make a dipole moment mu. Two charges moving past
each other in uniform rectilinear motion (which obviously must *not*
radiate) create a mu dot, but not a mu dot dot. Therefore mu dot dot is
the lowest-order thing that can radiate electromagnetic waves.

N=3) Two masses make a quadrupole moment I. Two masses moving past each
other in uniform rectilinear motion (which obviously must *not* radiate)
create an I dot dot, but not an I dot dot dot. Therefore I dot dot dot is
the lowest-order thing that can radiate gravitational waves.

What's the rest of the story? Can anybody recommend a reference?

Thanks --- jsd