Scope and limits of the oscillator, linearization, averaging and adiabatic items on this page

Every item on this page is mathematics: parameters, forcing functions, initial data and error constants are mathematical objects, and none of the statements asserts a physical law, a unit convention or an empirical fact. The following limits are part of the claims and must be carried by any consumer.

• Linear oscillator results are exact for the stated model only. The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data and The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0 classify all solutions of the constant-coefficient model with the stated parameter ranges ($k>0$, resp. $k\le 0$) and are exact equivalences, including the free zero mode when $k=0$ and the exponential growth when $k<0$. Strictly damped homogeneous oscillator with $k>0$: every solution tends to zero needs $c>0$ and $k>0$; for $c=0$ no decay is claimed. The forced oscillator with $k\ge 0$: unique global solution and the convolution (Green's function) formula gives the exact convolution representation for continuous forcing and $k\ge 0$; the kernel formula is a formula for that model, not a physical superposition principle. Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $\mathrm{\Omega }=\sqrt{k\mathrm{/}m}$ separates the steady response from the homogeneous transient; the resonant clause supplies an exact solution of the undamped resonant model, unbounded when $f_0\mathrm{\ne }0$ and identically zero when $f_0=0$, and The amplitude of the sinusoidally forced damped oscillator: location and value of its maximum locates the unique maximum of the response amplitude for $c>0$ and $f_0\mathrm{\ne }0$ (for $c=0$ and $f_0\mathrm{\ne }0$ the non-resonant amplitude is unbounded near ${\omega }_0$; for $f_0=0$ it is zero wherever its formula is defined). Forced quadratic system $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=\phi (t)$: decoupled forced modes and the convolution solution is exact for constant coefficient matrices $M,K$ and continuous forcing; it asserts nothing about nonlinear systems or configuration-dependent $M$.

• Finite-time linearization is not a stability statement. Finite-time linearization error bound: quadratic remainder, exponential growth bound and the variation-of-constants estimate is conditional on the explicit smallness condition $\delta e^{LT}<r$ and on a quadratic remainder bound on the ball, and concludes only $\mathrm{\mid }z(t)-y(t)\mathrm{\mid }=O({\delta }^2)$ on the finite horizon $[0,T]$. It neither asserts long-time agreement of the nonlinear and linear evolutions nor Lyapunov stability of the origin. For a mechanical system with a configuration-dependent kinetic metric, the remainder hypothesis must be verified for the full first-order field (derivatives of $M$ and $M^{-1}$ included), not for the potential alone.

• Averaging compares two different models. Averaging for fast periodic oscillations: order-$\epsilon$ agreement on the $O(1\mathrm{/}\epsilon )$ time scale compares the solutions of $x^{\mathrm{\prime }}=\epsilon f(t,x)$ and $y^{\mathrm{\prime }}=\epsilon \bar{f}(y)$ and gives agreement of order $\epsilon$ on the horizon $[0,A\mathrm{/}\epsilon ]$ under the trapping condition $\epsilon K<\rho$. The averaged equation is not equivalent to the oscillatory one; a longer horizon requires the trapping and smallness hypotheses to be checked again, and the theorem says nothing about stability or about invariant sets of the exact flow.

• Adiabatic bounds need the chart, the frequency gap and the trapping. Adiabatic action drift bound: order-$\epsilon$ change of the action on the $O(1\mathrm{/}\epsilon )$ time scale is conditional on a $C^2$ parameter-dependent action-angle datum with $\mathrm{\mid }{\mathrm{\partial }}_{I}h\mathrm{\mid }\ge {\omega }_0>0$ on the stated compact sets and on the branch solution staying in them. The existence of such a family for a given mechanical model, uniform in the parameter, is a separate hypothesis; resonances ($\omega =0$), separatrix crossings and action values leaving the chart are excluded, and the conclusion is a bound of order $\epsilon$ on a finite rescaled-time horizon, not exact conservation of the action.

• Downstream use. Physical pages that read these results must adopt their own postulates and interpretations for masses, forces, damping, driving and adiabatic parameter variation, and must re-state the empirical qualifications of any measured input; none of the mathematics above supplies such interpretations.