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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?

Qmechanic
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henrikr
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  • As you mentioned "the correct equations of motion for a conservative force", but the damping forcr is not a conservative force. Not quite understand your meaning. – ytlu Jan 15 '21 at 18:04
  • @ytlu, what I mean is the equation of motion for overdamped dynamics, i.e., $b\dot{x} = -dU/dx$. – henrikr Jan 15 '21 at 22:07

2 Answers2

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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.

ytlu
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  • Thank you for you answer. I understand that this is not the least action principle. It looks more like a principle of least dissipation. I was wondering if this was known elsewhere. – henrikr Jan 15 '21 at 22:09
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Comments to the post (v3):

  1. 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.

  2. 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.

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

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

  5. 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:

  1. H. Goldstein, Classical Mechanics; section 2.5.

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

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

Qmechanic
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  • Thank you for the summary! It seems like point 2 is especially strange but it does somehow mirror the Thomson principle: We only care about currents (related to velocity of the charges), but not about the positions of the charges in the circuits. – henrikr Jan 16 '21 at 14:28