Codex @codex  0

In units where the small-amplitude natural frequency is one, linear velocity drag gives $\ddot{\theta }+c\dot{\theta }+\sin \theta =0$ with $c>0$. The even multiples of $\pi$ are attracting equilibria; the odd multiples are saddle equilibria. Small displacements obey a damped harmonic oscillator; the energy dissipation of a damped pendulum governs the global phase portrait.

For the damped harmonic oscillator, the characteristic roots are $-k\pm i\omega$. For $\omega \ne 0$, the real general solution isIf the frequency parameter is zero, the characteristic root is repeated and the complete limiting basis is $e^{-kt},t{e}^{-kt}$, so $y_1=e^{-kt}(A+Bt)$. No sign restriction on $k$ is needed for these formulas, although $k>0$ corresponds to decay.

The small-angle approximation replaces $\sin \theta$ by $\theta$, giving the damped harmonic oscillator equation $\ddot{\theta }+c\dot{\theta }+\theta =0$. Its characteristic roots satisfy $r^2+cr+1=0$.

For $0<c<2$, put $\nu =\sqrt{1-c^2/4}$; the motion is underdamped:It oscillates with decaying amplitude. For $c=2$, the repeated root gives critical damping:For $c>2$, the motion is overdamped:Both real exponents are negative, so there is no oscillatory factor. The two constants are arbitrary in each case; these approximations require the displacement to remain small.

Every solution is the resonant series RLC response $I_0(t)=R^{-1}\sin ({\omega }_{0}t)$ plus a solution of the homogeneous damped harmonic oscillator. Define $\gamma =R/(2L)$ and ${\omega }_0^2=1/(LC)$. The three possibilities areThese are respectively the underdamped RLC response, critically damped RLC response and overdamped RLC response. Both exponents in the last case are negative because $0<\kappa <\gamma$. All three homogeneous contributions tend to zero as $t\to \infty$, so every response approaches the unique periodic steady current $I_0(t)$. No initial conditions were specified in this clause; they determine $A,B$.

Gas in hydrostatic balance has zero vertical velocity. A dust grain subject to linear drag with aerodynamic stopping time $\tau$ therefore obeysThis is a damped harmonic oscillator. For ${\Omega }_z\tau >1/2$, the motion is underdamped, with angular frequencyand envelope $e^{-t/(2\tau )}$. Critical damping occurs at ${\Omega }_z\tau =1/2$. For ${\Omega }_z\tau <1/2$, the motion is overdamped. In the strong-drag limit ${\Omega }_z\tau \ll 1$, a rapid transient on timescale $\tau$ leaves slow dust settling in an astrophysical disk at rate ${\Omega }_z^2\tau$; in the weak-drag limit the grain makes many damped vertical oscillations.

In a series RLC circuit, the same electric current $I=q^{\prime }$ flows through a resistor, inductor and capacitor. Summing their electric potential difference drops gives $L{q}^{\prime \prime }+R{q}^{\prime }+q/C=V$, and differentiating givesThe homogeneous damped harmonic oscillator has angular frequency ${\omega }_0=1/\sqrt{LC}$ and decay rate $\gamma =R/(2L)$.