# Subsequent motion (time evolution) of angled dipoles in electric field

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 ;)

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