Re: Non-lagrangian from lagrangian

wandering.the.cosmos@xxxxxxxxx wrote:
Igor Khavkine wrote:

First, it doesn't really make sense to "integrate out" degrees of
freedom unless one is talking of a quantum theory, specifically in the
path integral formulation.

If you write S = -(1/4) int d^4x F_{ab} F^{ab} - int dx^c A_c +
S_{gauge fixing}, assume k^0 << || vec{k} || and taylor expand the
propagator 1/k^2 = -1/vec{k}^2 (1 + O[ (k^0/vec{k})^2 ]), and calculate
the effective action as you would in a path integral, you will be able
to reproduce the results in

Landau, L. D. and Lifschitz, E. M. "The Lagrangian to Terms of Second
Order." in The Classical Theory of Fields.

That is, by integrating out the photon at the classical level, you can
obtain an effective lagrangian for charged particles interacting with
each other as a function of only their coordinates. It is a
perturbative series, where the expansion parameter is the
characteristic speed of the particles. You can also do a similar thing
with general relativity, and the result is the (well known?)
post-Newtonian expansion.

This is all well and good. However, I would still object to the
terminology "to integrate out" in this case. But at the same time, I
don't know of an accepted terminology for this procedure. Sometimes
it's called "back substitution", in the sense that the equations of
motion for the EM field are solved and back substituted into the
action.

What I'm curious about is whether we can turn this around, and
introduce ficticious fields that, when integrated out, would give us
equations of motion corresponding to physical problems, say ones with
friction, which by themselves can't be obtained from a variational
principle.

No, again. When doing back substitution, as you described above, when
you solve the equations of motion for the field you want to eliminate,
you substitute the solution back into the action. As such, you have a
new effective action principle for the degrees of freedom that remain.
Thus the new effective equations of motion will again be derived from
varying the action or the corresponding Euler-Lagrange equations.

Hope this helps.

Igor

.

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