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Re: Models, etc./Higher derivatives



Regarding:
The PDEs in nonlinear elasticity are fourth-order, i.e., have the fourth
derivative as the lead term, but I don't have Stoker's book and I've
forgotten the equation. I will look for the notes. Regards / Tom

As I recall the 4th order PDEs of elacticity theory are only 4th order *in
space* involving biharmonic functions I think. Elasticity theory does not
involve equations which are 4th order in the dynamical time parameter.

Off the top of my head I can only think of two instances of dynamical
equations in physics being greater than 2nd order *in time*.

1) The Abraham-Lorentz model which tries to include radiation damping in
the dynamics of charged particles without explicitly invoking dynamical
electromagnetic fields, i.e. the model only involves the particle's own
degrees of freedom. In this model the radiation damping term enters as a
contribution proportional to the particle's jerk rate and is 3rd order in
time.

2) Higher order models of gravitation (beyond general relativity) that have
terms in the gravitational Lagrangian density involving the square of the
Riemann tensor and/or the square of the Weyl tensor also yield higher
order derivative equations. These theories generate dynamical equations of
motion that are 4th order in both space and time. This is because these
extra terms in the Lagrangian density are already nonlinear in 2nd order
derivatives before the action is minimized. When the action is minimized
the generalized Euler-Lagrange equations of motion tend to have an order
which is twice the order of the highest order derivatives that appear
nonlinearly in the Lagrangian density.

If you have no idea what I'm taking about here that's okay. As an analog
with ordinary mechanics recall that Newton's 2nd law is 2nd order in time.
This is because it is the dynamical Euler-Lagrange equation generated by a
Lagrangian which is nonlinear (quadratic) in the velocity , i.e. a first
order deriviative, because the Lagrangian always contains the kinetic energy
term (1/2)*m*v^2. If the acceleration appeared nonlinearly in the Lagrangian
then the corresponding equation of motion would be 4th order in time. If the
jerk rate appeared nonlinearly in the Lagrangian then the corresponding
equation of motion would be 6th order in time. Etc.

Ordinary general relativity is generated by the Hilbert action whose pure
gravitational Lagrangian density involves only a single term proportional to
the curvature scalar R (i.e. the contraction of the Ricci tensor) which is
*linear* in all its 2nd order terms. Therefore, in the ordinary GR case the
Hilbert action generates the usual Einstein equations of GR as the dynamical
equations of motion which end up being only 2nd order in time and space as
is the usual custom of physical dynamical equations.

David Bowman
dbowman@gtc.georgetown.ky.us