On the exact discretization of the classical harmonic oscillator equation

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.

💡 Research Summary

The paper presents a rigorous construction of an exact discretization scheme for the classical harmonic oscillator, covering the homogeneous, inhomogeneous, and multidimensional cases, with a particular emphasis on the preservation of the energy integral. The authors begin by recalling the continuous equation (\ddot{x}+\omega^{2}x = f(t)) and its analytical solution expressed through the matrix exponential of the system matrix (A = \begin{pmatrix}0 & 1\ -\omega^{2} & 0\end{pmatrix}). By sampling the exact solution at uniform time steps (h), they derive a discrete update formula

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