Laplace-Lagrange secular theory linearizes the orbit-averaged disturbing function to second order in eccentricities and inclinations. It represents planetary eccentricity evolution by a constant matrix.
The eigenvalues of the Laplace-Lagrange secular matrix are apsidal precession frequencies, and its eigenvectors give the relative eccentricities and apsidal phases of the corresponding normal modes.
The complex eccentricity packages eccentricity and longitude of periapsis as , turning linear secular dynamics into a complex linear system.
A secular eigenmode is a normal mode in which all complex eccentricities precess at one eigenfrequency with fixed amplitude ratios and relative apsidal phases.
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