June 2025 - N Coupled Oscillators

The simplest coupled oscillator is equivalent to masses on a spring. Solving such a system is a classic exercise in using the Lagrangian method. For an abbreviated version,

It's not a periodic system at first sight, but the chaotic oscillations can be decomposed into normal modes via a diagonalization A = RDR^-1 with D a diagonal matrix with eigenvalues λ_i and R a matrix with the eigenvectors of A. R can transform X to a new set of variables X' = R^-1, and -D also acts as the double-derivative operator on X'. As such, each double derivative of x_i' adds -λ_i as a factor. The standard solution for this is a sine/cosine solution with a frequency of ω_i² = λ_i ≥ 0. This is generally assumed to be true for the way A is constructed. The solution for X can be written as a sum over the normal modes: X(t) = Σ_iξ⁽ⁱ⁾x_i'(0)cos(ω_it). The Eigenvector represents the direction of each mode, x the amplitude and ω its oscillation frequency. The generalized case for N coupled oscillators give a generic term.