[Phys-l] physics of dissipation

[John Denker <jsd@av8n.com>]

In the context of race car drag, 
On 11/03/2009 11:05 AM, Stefan Jeglinski wrote:

> It might seem like this could be generalized in the Lagrangian 
> formulation of classical mechanics. Setting aside whether any 
> particular analysis becomes more or less difficult in this case, I 
> seem to recall from my last reading of Goldstein et al. that the 
> ability to generalize dissipative effects to the Lagrangian 
> formulation was limited. 

Yes, very limited indeed.

> Why is this exactly? 

An excellent question!

For Lagrangian mechanics in particular, for classical mechanics
in general, and indeed for real-world physics, energy is conserved.  
This is incompatible with the usual hand-wavy notions of energy 
"going away" when it is dissipated.

To build a _physical_ model of dissipation, you need to account
for where the dissipated energy went.  Lagrangian mechanics is
very precise;  it is going to account for this energy _in detail_
whether you want the details or not.

Alas there are lots of hare-brained models out there that are
neither simple nor correct.

The breakthrough in this area came when Johnson measured the 
"Johnson noise" and Nyquist came up with a beautiful physical
model.  Both guys were at Bell Labs.  They published back-to-back
Phys Rev Letters in 1928.  Classic.  Well worth reading even
today.

Nyquist's model can be boiled down to a single sentence:  An
ohm is an ohm is an ohm, and a coaxial cable is 50 ohms.  That
is, we model the energy being "dissipated" in terms of energy
being _radiated_ into an infinitely-long piece of coax.

This gives a perfect model of dissipation.  Aso, in addition 
to the outgoing energy ("dissipation") there will be incoming 
energy ("fluctuations") associated with initial conditions in 
far distant parts of the coax.  This means that in the case 
where the initial conditions are thermally distributed, the 
system upholds the fluctuation/dissipation theorem.

This approach can be extended from classical mechanics to
quantum mechanics, allowing a fully quantum-mechanical 
understanding of dissipative mechanical and/or electrical
systems.
  http://prola.aps.org/abstract/PRA/v29/i3/p1419_1

This is quantum field theory, but it is the simplest quantum
field theory you will ever see.  It can be explained to a
bright undergraduate.  Actually I recommend it as a nice
simple pedagogical way of introducing field-theory ideas.
If you can quantize a harmonic oscillator, you can quantize
the modes of the coax.

This also makes an interesting lesson in the grandeur and
unity of physics:  from race cars to field theory in one 
jump.