Solved Problems in Physics: Classical Mechanics

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.
qj and rj are the deviations of particles mj and Mj from their equilibrium positions respectively.

    1. Find both potential and kinetic energies of the system and the Lagrangian of the
      system. Write down the equations of motion for qi and ri.
    2. If the equations q_{j}=Q_{j}\exp{[i{\omega}t]} and r_{j}=R_{j}\exp{[i{\omega}t]} are applied, what are the equations for the amplitudes Qj and Rj?
    3. Let Q(k)=\sum_{j=-\infty}^{j=+\infty}{Q_{i}\exp[i(jka)]} and R(k)=\sum_{j=-\infty}^{j=+\infty}{R_{j}\exp[i(jka)]}, where i=\sqrt(-1) 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).
    4. Determine the normal mode frequencies \omega (k) for the system, where k\in[-\pi/(2a),+\pi/(2a)].

phisics

Solution

  1. The kinetic energy of the system can be determine from the equation below:
    t0-5
    The potential energy of the system can be found from the equation below:
    v
    The Lagrangian can be found as follows:
    lWhere T is the kinetic energy and V is the potential energy, determined above.
    The equations of motion for jth points can be found as the partial differentials from the
    Lagrangian:
    d
  2. Let q_{j}=Q_{j}\exp{[i({\omega}t})] with Q(k)=\sum_{j=-\infty}^{j=+\infty}{Q_{i}\exp[ijka]} and r_{j}=R_{j}\exp{[i({\omega}t)]} with R(k)=\sum_{j=-\infty}^{j=+\infty}{R_{j}\exp[ijka]} .Then \"{q}_{j}=-\omega^2Q_{j}\exp[i(\omega{t})] and \"{r}_{j}=-\omega^2R_{j}\exp[i(\omega{t})] .
    Therefore, k
  3. In order to find the solution for part 3, multiplication by exp[i(jka)] should be
    performed.
    Therefore, resultApplying the differentiation for Q(k), R(k) enables to sum over j:result
  4. For non-trivial solutions Q(k) and R(k), the determinant, as shown below, should
    be equal to zero.
    result
    result

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