Abstract: Damped oscillating system is an extremely important topic in the study of molecular dynamics and classical dynamics, so the analytical result is the key problem on this system. But the exact expressions are always to obtain in many-body system. Motivated to get the two-bodys analytic expressions, Laplace transform, residual theory and Cardano formula are applied to the calculation. Accurate computations show that this system could be distinguished into three situations: critical damping oscillation, overdamped oscillation and underdamped oscillation, which is quite analogous to simple damped harmonic oscillator. Similarly, the three modes are also classified via the discriminant Δ. Finally, this study sheds some light into the topics on more complicated systems.

Key words: two-body oscillating system, analytic solutions, residue theorem, Critical phenomenon