# Hamiltonian for non-conservative forces?

We know if a force is conservative we can use a potential function. Assume there are non-conservative forces in our problem. For example the air resistance force exerting on a oscillating mass-spring system. How should we write the hamiltonian for this case?

Demystifier
Gold Member
I guess you want to have a Hamiltonian in order to quantize the system. Unfortunately the Hamiltonian cannot be introduced in a completely satisfactory way. Nevertheless, some attempts exist:
H. Bateman, Phys. Rev. 38, 815 (1931).
Y.N. Srivastava, G. Vitiello, and A. Widom, Ann. Phys. 238, 200 (1995).
M. Blasone, P. Jizba, G. Vitiello, Phys. Lett. A 287, 205 (2001).
W. E. Brittin, Phys. Rev. 77, 396 (1950).
R.W. Hasse, J. Math. Phys. 16, 2005 (1975).
N.A. Lemos, Phys. Rev. D 24, 2338 (1981).
S.A. Hojman and L.C. Shepley, J. Math. Phys. 32, 142 (1991).
V.E. Tarasov, Phys. Lett. A 288, 173 (2001).
for an attempt that avoids the use of a Hamiltonian.

Thanks

We know if a force is conservative we can use a potential function. Assume there are non-conservative forces in our problem. For example the air resistance force exerting on a oscillating mass-spring system. How should we write the hamiltonian for this case?
I studied this several years ago. I haven't looked at the references above but I remember that there are some cases where you could find solutions to the inverse Lagrangian problem, i.e., you know the equations of motions (eqm) and you want to find the Lagrangian (from where you could derive the Hamiltonian). The most simple case is the Langevins equation for which we have the solution:

$$L=L_0\exp(-\eta t)$$

where L0 is the conservative (normal) Lagrangian, but this gives you a time-dependent Hamiltonian, like

$$\hat{H}=\hat{T}\exp(\eta*t)+\hat{V}\exp(-\eta t)$$

You get the correct Ehrenfest eqm -the Langevin equation in terms of the expectation value <x>!, but it is not so funny to deal with in QM since of this time-dependence.

Generally, the effect of dissipation in QM has been investigated by for example Caldeira and Legget in an article about "Quantum Brownian Motion". I think they integrate out the interaction by the environment in terms of coupled harmonic oscillators (some Green functions). They also found a Hamiltonian where the temperature appear explicit (interesting limiting case). Even more generally dissipation in systems which are connected to some surroundings could be thought of as waves propagating outwards the systems, where the waves contains packages of energy.

It may look like this is of a purely scholastic interest, but seriously people in the "ab inito" field are interested in this and are debating about the reason about why resonating AFM-tips give rise to dissipation (AFM=atomic force microscope). Also dissipation appears in applied superconductivity, for example in the RJC-model (thats where I first came across this problem and the article of Caldeira and Legget and Andersson quantization etc.)