How can the constraint condition be used to define generalized coordinates?

[intervoxel]

Homework Statement

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.

Homework Equations

Lagrange function
$L=T-V$

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

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?

 

[member 553083]

Lagrangian of the systemL=\sum\limits^{N}_{i=1}\frac{1}{2}m\,v_i^2-\sum\limits^{N/2}_{i=1}\frac{kq}{r_i}+\sum\limits^N_{i=N/2+1}\frac{kq}{r_i}+\lambda(r_j-r_i-l^2)Lagrange's equations \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x_k}}\right)-\frac{\partial L}{\partial x_k}=0\frac{d}{dt} (m\dot{x_k}) - \frac{\partial V}{\partial x_k} - \lambda \frac{\partial (r_j-r_i-l^2)}{\partial x_k} = 0 Difference equationsx_{n+1}=x_n+\Delta t \left(\frac{\partial V}{\partial x_k} + \lambda \frac{\partial (r_j-r_i-l^2)}{\partial x_k} \right)