In the context of small oscillations, which one of the following does ...
We begin with the one-dimensional case of a particle oscillating about a local minimum of the potential energy We'll assume that near the minimum, call it , the potential is well described by the leading second-order term, , so we're taking the zero of potential at assuming that the second derivative , and (for now) neglecting higher order terms.
To simplify the equations, we'll also move theorigin to , so
replacing the second derivative with the standard "spring constant" expression.
This equation has solution
(This can, of course, also be derived from the Lagrangian, easily shown to be )
The physical motion corresponding to the amplitudes eigenvector has two constants of integration (amplitude and phase), often written in terms of a single complex number, that is,
with real.
Clearly, this is the mode in which the two pendulums are in sync, oscillating at their natural frequency, with the spring playing no role.
In physics, this mathematical eigenstate of the matrix is called a normal mode of oscillation. In a normal mode, all parts of the system oscillate at a single frequency, given by the eigenvalue.
The other normal mode
where we have written . Here the system is oscillating with the single frequency , the pendulums are now exactly out of phase, so when they part the spring pulls them back to the center, thereby increasing the system oscillation frequency
The matrix structure can be clarified by separating out the spring contribution: