Overdamped dynamics and minimization principles Most physicists are familiar with Hamilton's principle, which allows us to derive the equations of classical mechanics from the principle of stationary action, $$\delta S=0.\tag{1}$$
At the same time, there is Thomson's principle for circuits, which tells us that the current flowing in electric circuit that that which minimized the dissipated power,
$$ P = \sum R I^2. \tag{2}$$
In trying to couple an electrical circuit dynamics to an overdamped dynamics that must satisfy some constraints, I noticed that you can get the correct equations of motion for a conservative force if
you minimize
$$ P = \frac{1}{2} b \dot{x}^2 + \frac{d U}{dx} \dot{x}, \tag{3}$$
over $\dot{x}$, i.e. $$\partial P/\partial \dot{x} = 0,\tag{4}$$
where the first term is the dissipated kinetic energy, and the second term is the time derivative of the
potential $dU/dt$. This principle appears to work fine and can be used to get constrained overdamped dynamics using Lagrange multipliers. The first term looks a little bit like a Rayleigh dissipation function, but I haven't seen the second term in the textbooks I checked.
Have you seen this "principle" before, does it have a name, how is it related to other minimization/stationarity principles in physics?
 A: The term $P$ in your expression is an integral of Force by $d\dot{x}$ (by treating $x$ as a constant):
$$
  -\int d\dot{x} F_{total} = - \int d\dot{x} \{-bx - \frac{dU}{dx} \}
$$
Treat $x$ as a constant ans carry out this integral over $d\dot{x}$
$$
  =\frac{1}{2} b\dot{x}^2 + \frac{dU}{dx} \dot{x} = P
$$
Thus, it recovers the force if your partial derives $P$ with respect to $\dot{x}$.
The process here is some what different with the least action principle. The least action principe render the equation of motion $m\ddot{x} = F$ , not only the forces.
A: Comments to the post (v3):

*

*It seems that OP's principle (4) only reproduces the dynamical side $\sum_i{\bf F}_i$ of Newton's 2nd law, not the kinematic side $m{\bf a}$, so it only applies to creeping/overdamped motion.


*Also OP's principle (4) treats velocity $\dot{x}$ as independent of position $x$. In a conventional variational action principle, they are not independent, cf. e.g. this Phys.SE post.


*For a discussion of dissipative forces in a Lagrangian formulation, see e.g. this Phys.SE post.


*Goldstein [1] constructs Lagrange equations with Rayleigh dissipation function for $RCL$ circuits.


*What OP refers to as "Thomson's principle" (2) seems to be the following [2,3]:

Given a resistor network with fixed current sources $I_e$, assign a current $i_e$ to each edge, and impose KCL at each vertex. Then the actual currents minimize the dissipation $\sum_{e}R_e i_e^2$.

(Refs. [2] & [3] also consider a modified principle where the current sources are replaced by voltage sources.)
References:

*

*H. Goldstein, Classical Mechanics; section 2.5.


*D. A. Van Baak, Am. J. Phys. 67 (1999) 36.


*N.R. Sree Harsha, arXiv:1903.07197; eqs. (14)-(17).
