In this work we consider the Riemannian geometry associated with the differential equations of one dimensional simple and damped linear harmonic oscillators. We show that the sectional curvatures are completely determined by the oscillation frequency and the friction coefficient and these physical constants can be thought as obstructions for the manifold to be flat. Moreover, equations of simple and damped harmonic oscillators describe nonisomorphic solvable Lie groups with nonpositive scalar curvature.
Harmonic oscillator, Riemannian geometry, constant scalar curvature, oscillation frequency, damping force, metric Lie groups
BAYRAKDAR, ZAHİDE OK and BAYRAKDAR, TUNA
"A geometric description for simple and damped harmonic oscillators,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 37.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/37