# Theory of conserved quantities in damped systems?

As is well-known, classical conserved systems have conserved quantities by virtue of continuous symmetries, which can be derived from Lagrangian mechanics. For example, two masses on a spring can swap momentum between the two, but translational invariance ensures that the total momentum is conserved.

But what if we introduce damping, specifically viscous damping, into the equations? If we consider a fluid medium, our two-mass system surely loses momentum to the fluid. But suppose that the spring itself is viscoelastic, so that it generates equal and opposite forces on the two masses proportional to their relative velocity. Then, the system is translationally invariant in some sense and conserves momentum, but is not Lagrangian (at least not in a simple way).

Is there a theory of how to calculate conserved quantities based on symmetries in damped systems, analogous to how it is done in undamped systems? I can do this for specific cases, but I'm not sure how to do it in general. A few attempts:

• Constructing a complicated Lagrangian consistent with damping
• Declaring that damping is a purely phenomenological effect coming from many microscopic degrees of freedom behaving conservatively, and so concluding that the system should have the same conserved quantities as if there were no damping (problem here is that some of the conserved quantities, like energy, must leak into the unobservable microscopic degrees of freedom).
• Considering the corresponding conservative system and showing that it has conserved quantities. Then arguing that this is true for each frequency even if the "spring constants" are complex and frequency-dependent, and so concluding that damping doesn't really change anything. Again, though, this seems to prove too much, since energy shouldn't really be conserved.

By way of motivation, I'm working with a system that has an unusually large and complicated number of symmetries, and I'm trying to determine how worried I should be about damping killing the conservation in a real experimental system.

• By a "Lagrangian consistent with damping" do you mean using the Rayleigh dissipation function? Commented May 24, 2023 at 3:09
• @Qmechanic I don't think I do mean that. I'm not terribly familiar with the concept, the Rayleigh dissipation function seems to be pretty close to just tacking on the damping terms into the EOM. I had in mind more a method I vaguely recall from a class, in which one generates an actual Lagrangian with explicit time-dependence (hence killing energy conservation, of course, as one wants) that recovers the damped EOM. Commented May 24, 2023 at 14:30
• But that trick is an unsystematic approach: It does not work for systems with several DOFs with different masses and different viscous damping coefficients. Commented May 25, 2023 at 10:44
• Yup, @Qmechanic. Perhaps it could be made systematic, but I don't know how to do it. Commented May 26, 2023 at 15:44