If we hang a second particle below the first one we get a more interesting problem than that dealt with in the previous section. In the example, the particles and springs are identical. The springs have a natural length 3d, and stretch a distance d under the weight of a particle. In the first case considered, the particles are released from rest with the upper spring at its natural length and the lower spring supporting the weight of the lower particle. The extension of the Feynman algorithm to deal with the displacements and velocities of two particles is straightforward. In the first case, the particles execute a complicated motion after they are released. The other two cases try different initial positions for the lower particle to try to find a configuration that gives simple harmonic motion when the particles are released. In the second case, the initial trial has the lower spring unstretched, and both particles start moving in the same direction. In the third case, the initial trial has the lower spring stretched d beyond its equilibrium position, and the particles start moving in opposite directions. In both cases, the time it takes each particle to reverse velocity is calculated, and the initial position is varied to minimize the difference between the times for the two particles. Once this is done, the initial positions are displayed, the amplitude ratio is printed, and the motion can be plotted. The motion in the second case is called the low-frequency normal mode, and in the third case, the high-frequency normal mode.
The analytic procedure for finding the amplitude ratio inserts harmonic displacements (from equilibrium, with amplitudes and common frequency to be determined) into the equations of motion to get algebraic equations that can be divided to eliminate the frequency and give a quadratic equation for the amplitude ratio values. It is an interesting exercise to do the calculation and see how accurate the numerical search procedure is.