Subtleties about Hamilton's equation

[Harry Mason]

Hello everybody,
from a non-relativistic point of view , taking into account an N-point particle isolated system, in which interacting with each others in principle we can describe the time-evolution of the system, defined by hamilton's equations:

Where H is the total internal energy of the sistem.

The question is: if forces between particles does not satisfy the 3rd principle of dynamic (like electromagnetic forces) and they're not conservative are the hamilton's equation always true? What's the new physical interpretation of the hamiltonian?

Thank you.

 

[andresB]

Not sure for dissipative system (but I found a link that might be helpful http://www.m-hikari.com/ams/ams-2010/ams-17-20-2010/biswasAMS17-20-2010.pdf)

But for the case of electrodynamics the thing is, in principle, like the following. For each particle You have to replace

$P^2/2m$

for

$(P-eA(r,t))^2/2m+eV(r,t)$

Where $V (r,t)$ and $A(r,t)$ are the electric and magnetic potentials and their dynamics is given by Maxwell's equations.

So, in principle to find the complete dynamics you have to solve at the same time the Maxwell's equations plus the Hamilton equation (they will be Newton equations with the Lorentz force), but this approach have a lot of problems that I think have not been solved so far.