# Physical origin of gradient-like system with nonautonomous damping term

I am in mathematics major, and I am considering the equation $$\ddot u(t) + \gamma(t) \dot u(t) + \nabla \phi (u(t)) = g(t),$$

where $\phi: \mathbb{R}^n \to \mathbb{R}$ is of class $C^1$, convex and bounded below; $\nabla \phi$ is locally Lipcshitz; $\gamma\in W^{1,1}_{\text{loc}} (\mathbb{R}^+, \mathbb{R}^+)$; and $g\in L^1$.

I would like to ask for the physical origin of the equation above, which is called the second-order gradient-like system with nonautonomous damping term. I am really if this question seems silly or trivial, since I do not have basics in physics.

• You could might be able to get something like that from the Navier-Stokes equations. – user121330 May 22 '17 at 17:21
• Looks similar enough to a damped harmonic oscillator to make me think of it, but doesn't seem to be an exact fit. – Pawr May 22 '17 at 22:49
• I think this covers a lot of possible scenarios. Effectively any motion of a particle in classical physics can be expressed in this from. You have from left to right: acceleration, damping, potential and forcing. Effectively all the models I've personally seen are simpler forms of this equation (i.e. haven't ever seen time dependence in damping), but one can probably come up with an obscure scenario where damping constnat is time dependant. – Ilya Lapan May 23 '17 at 8:38
• I agree with @IlyaLapan, your equation is very physical. The nonautonomous damping is not common only because it's complicated, but an object with non-constant shape, or oscillating inside a chamber with increasing pressure, for example, would have $\gamma=\gamma(t)$, and there's also quadratic damping: $\gamma=\dot{u}(t)$. – stafusa Aug 11 '17 at 20:41