Description of the physical problem

Suppose that there is an indexed system of particles, each bound to a position of repose by an elastic spring obeying Hooke's law while its excursions from equilibrium follow Newton's law:

$$\begin{eqnarray*} m_i \frac{d^2 x_i}{dt^2} & = & - k_i x_i. \end{eqnarray*}$$

Subscripting all three, displacements from rest, masses, and elastic constants, implies that in full generality, every particle has its own characteristics and environment. The solutions to differential equations such as these are well known to be sines and cosines, or phased cosines, or even complex exponentials taken up in real combinations.

An easy way of letting some of these particles be influenced by their neighbors is to suppose the existence of additional elastic springs whereby the displacement of one particle from its equilibrium exerts a force on some other particle - in the direction of the displacement, in contrast to the force felt from a particle's own displacement, which urges it back to whence it came. If the influence extends to neighboring particles and no farther, a collection of differential equations describing the motion would be

$$\begin{eqnarray*} m_i \frac{d^2 x_i}{dt^2} & = & k_{i-1}x_{i-1} - (k_{i-1} + k_i +k_{i+1}) x_i + k_{i+1}x_{i+1}. \end{eqnarray*}$$

In these equations, signs have been chosen so that the symbols themselves represent positive quantities. The middle terms are negative because all springs restrain a moved particle if the others are fixed at their origins.

The system can be placed in a more agreeable form by transformations which can either be made at the outset, or incorporated later by suitable matrix transformations, but the present interest is in the matrix, not the physics. Suffice it to say that multipying by square roots of masses solves the immediate problem (or just suppose they all equal one gram, or adjust the unit of time to compensate) and get on with the differential equation which still lets the elastic constants vary. There is a vector of coordinates and a matrix of elestic constants satisfying the differential equation

$$\begin{eqnarray*} \frac{d^2 }{dt^2} \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 ... ...gin{array}{c} x_1 \\ x_2 \\ x_3 \\ \ldots \end{array} \right] \end{eqnarray*}$$