Physics assignment
Classical mechanics
Problem statement
Particles with two different masses m and M are located along a linear harmonic chain
of infinite length. The chain has a force constant k (see the picture below). The
distance between two particles with the same mass is equilibrium and equals to a.
q_{j} and r_{j} are the deviations of particles m_{j} and M_{j} from their equilibrium positions respectively.

 Find both potential and kinetic energies of the system and the Lagrangian of the
system. Write down the equations of motion for q_{i} and r_{i}.  If the equations and are applied, what are the equations for the amplitudes Q_{j} and R_{j}?
 Let and , where and the sum on j is over all particles of mass m for Q(k) or over all particles of mass M for R(k). Perform the sum over all of the amplitudes in part 2) above and determine the equations for Q(k) and R(k) using the above definitions for Q(k) and R(k).
 Determine the normal mode frequencies for the system, where .
 Find both potential and kinetic energies of the system and the Lagrangian of the
Solution
 The kinetic energy of the system can be determine from the equation below:
The potential energy of the system can be found from the equation below:
The Lagrangian can be found as follows:
Where T is the kinetic energy and V is the potential energy, determined above.
The equations of motion for j^{th} points can be found as the partial differentials from the
Lagrangian:
 Let with and with .Then and .
Therefore,  In order to find the solution for part 3, multiplication by exp[i(jka)] should be
performed.
Therefore, Applying the differentiation for Q(k), R(k) enables to sum over j:  For nontrivial solutions Q(k) and R(k), the determinant, as shown below, should
be equal to zero.
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