# Lagrangian of dipole set

1. Jan 18, 2014

### intervoxel

1. The problem statement, all variables and given/known data

Build the lagrangian of a set of N electric dipoles of mass m, length l and charge q.
Find the equations of motion.
Find the corresponding difference equations.

2. Relevant equations
Lagrange function
$L=T-V$

Lagrange's equations
$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x_k}}\right)-\frac{\partial L}{\partial x_k}=0$

3. The attempt at a solution
Electrostatic potential
$V=\sum\limits^{N/2}_{i=1} \frac{kq}{r_i}-\sum\limits^N_{i=N/2+1} \frac{kq}{r_i}$

Kinetic energy
$T=\sum\limits^{N}_{i=1} \frac{1}{2}m\,v_i^2$

Constraints
$(r_j - r_i)-l^2=0$

The system has N/2 degrees of freedom. (???)

how the constraint condition defines the generalized coordinates?

Last edited: Jan 19, 2014