Coupled small-oscillation eigenmodes

Consider a stationary natural Lagrangian with kinetic energy quadratic in rates and a configuration-only potential. Choose an equilibrium q₀ where the potential gradient vanishes and the system can remain at rest. Write q=q₀+η. To second order in small displacement and velocity, evaluate the mass matrix at q₀ and retain the Hessian K of the potential. Terms from the coordinate variation of M multiply two velocities and are third order in a uniform small-amplitude expansion, so they do not enter this leading quadratic model.

The approximation assumes a regular positive-definite mass matrix and a smooth potential in a neighborhood of the equilibrium. A moving reference solution, a rotating frame with gyroscopic terms, or a velocity-dependent magnetic coupling requires a more general linearization. Do not delete such terms by copying the rest-equilibrium formula. Check dimensions as well: for length coordinates, M has mass units and K has spring-constant units. For angular coordinates the corresponding coefficients carry different units but produce the same dimension of inverse time squared.

Insert a trial collective motion with a fixed vector of relative amplitudes and a sinusoidal time dependence. The equation becomes K a=ω² M a. Nonzero amplitude vectors exist when the determinant of K−ω²M vanishes. If M is positive definite and K is symmetric, the problem can be converted to a symmetric ordinary eigenproblem by mass whitening. This explains why the squared frequencies are real even though M inverse times K need not look symmetric in ordinary coordinate components.

For distinct squared frequencies, the eigenvectors are orthogonal in the M-weighted inner product, not generally in the unweighted Cartesian dot product. Normalize them so aᵀ M a equals one when convenient. The normalization changes modal-coordinate units and amplitudes, but not the physical displacement pattern. Degenerate frequencies permit a choice of mass-orthogonal basis within their eigenspace. Such freedom does not create additional independent frequencies or make one particular drawing of a degenerate mode physically unique.

A positive squared frequency gives an oscillatory linear mode. A negative one gives exponential growth and decay directions, indicating a linear instability of this rest equilibrium. A zero value needs additional analysis: it can represent an exact symmetry direction, free drift, or an equilibrium whose leading restoring term appears beyond quadratic order. A nonnegative spectrum alone is not a complete nonlinear stability proof. The smallness of the displacement and the neglected terms must remain compatible over the time interval being interpreted.

The oscillator lab represents one already-decoupled positive-frequency mode. Set damping to zero and predict how increasing natural frequency from one to two changes the period. It has no mass-matrix controls and does not compute eigenvectors or coupled frequencies. Its normalized initial displacement is one, so interpreting it as a physical small mode requires a separate small amplitude scale. Use the worked matrix example to derive frequencies first, then use the lab only to inspect the time dependence of one allowed positive-frequency mode.