🚀 go-pugleaf

In article <1992Oct14.085015.150@news2.cis.umn.edu> burchard@horizon.math.utah.edu (Paul Burchard) writes:
>In article <1992Oct13.202915.29039@galois.mit.edu> "John C. Baez"  
><jbaez@bourbaki.mit.edu> writes:
>>
>> Laplacian u = f(u,Du)
>> 
>> where Du is the gradient.  (Actually she is doing much fancier stuff,
>> but I'll keep it simple here.)  She was trying to find a variational
>> principle good for the latter kind.  Is there any functional of u and
>> Du that when extremized gives you the latter PDE?
>
>Although there are certain obvious costs to this, you can always use |P(u)|^2
>as an energy, when solving P(u)=0, and look for absolute minima.

This is getting close but is still not quite what
I was looking for, since it requires finding the absolute minimum.  I
think my question may not have been loudly stated enough the first time,
so here it is again: for which if any equations of the above form is
there a functional of u and Du that when extremized gives you the
equation??  It's easy when there is no Du, of course.  

I am not looking for other methods of solving such PDE, general results
about such PDE, and so on - Mary knows lots of these and is in fact
trying prove existence in an extremely delicate case, which she has
managed to do using variational techniques in the case when there is no
Du.

In article <1992Oct13.202915.29039@galois.mit.edu>, jbaez@bourbaki.mit.edu (John C. Baez) writes:
|>                                                 there doesn't
|> seem to be any Lagrangian that gives the equations of motion
|> 
|> x'' = -x'
|> 
|> (which describes a particle with friction).  
|> 

There is : L = exp(t) * ( x')^2 
where t is the "time" or independant variable. The corresponding
Hamiltonian is H = (1/4) exp(-t) * p^2, which describes the decaying
of the kinetic energy due to the friction.

--
Martin Ouwehand, Swiss Institute of Technology in Lausanne