Suppose we have a system of two dipoles, each with dipole moment $\mathbf{p}=2aq$ each aligned at angles $\theta$ and $-\theta$ with the horizontal. I’m thinking of an angle bracket shape, essentially. The direction of the electric field is along the horizontal in the positive x-axis. The intersection point is also a sort of hinge, bearing charge $-2q$ while the two angled ends bear charge $q$ each.

I wanted to find the evolution of this system with time. I suppose it’ll stop accelerating horizontally at some point (gravity free space, and all three points are freely mobile).

I began to blindly apply the Lagrangian for this system. (I don’t really understand it well, though. I’m still in high school :p)

$$V = -4aEq\cos{\theta}$$ Where $2a$ is the length of each light in elastic rod connecting the dipole, and $E$ is the magnitude of uniform field. $$T = \frac{1}{2} (2m\dot{x}^2 + m(\dot{x}-2a\dot{\theta}\sin{\theta})^2 + 2m(2a)^2\dot{\theta}^2)$$

The first two terms are the translational kinetic energies. The last term is the rotational kinetic energy of the two ends with charge $q$ about the hinge bearing $-2q$.

I don't really know what to do from here on. I can't really stand typing out all the derivatives in mathjax so I won't. I did try to impose the initial constraints $\dot{x}=0$ and $\dot{\theta}=0$.

From the E-L equations with the above constraints, I obtain an expression for the initial $\ddot{\theta}$ in terms of trigonometric ratios of initial $\theta$.

However, my question is:

a) can I analyse this further without much calculus, and b) what will the approximate time evolution of this system be?

As to (b): I reason that at some point the horizontal translation should end, because of the field - when all three bodies are in a straight line (when dipole moments are antiparallel). What happens to the angular acceleration then? I think there'll be damped oscillations while $\theta$ shifts around $\frac{\pi}{2}$. Is this accurate? I don't know.

Any help with formatting and tags would be appreciated ;)


1 Answer 1


In terms of part (a) of your question, the next step in the Lagrangian procedure would be to obtain the Euler-Lagrange equations based on your Lagrangian $L=T-V.$

This means you compute the following:

$$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{q}_{j}} \right)-\frac{\partial L}{\partial q_{j}}=0 $$

where the $q_{j}$ and $\dot{q}_{j}$ are the generalized positions and velocities in your Lagrangian (like $x$ and $\dot{x}$). This is done for each $j$ separately.

So this means that some additional calculus will be necessary in general to continue the analysis. Following that, you'll end up with some differential equations of motion for the system which will need to be solved, either analytically if possible (more calculus), or numerically. There's probably a Python differential equation solver that you can try to use for that step.

I'm not saying that your Lagrangian is necessarily correct as written, but the above is the process.


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