A hypothetical small-angle pendulum estimate of a gravitational parameter: linearized propagation and a bounded nonlinearity remainder on declared data

Adopt as the declared physical content of a separate hypothetical constrained model the ideal planar point pendulum: one bob of inertial mass $m>0$, a fixed pivot at the origin of an inertial frame, and an ideal massless rigid rod of length $l>0$ constraining the bob to the vertical $e_1,e_3$ plane. Write its position as $x(t)=l(\sin \theta (t)e_1-\cos \theta (t)e_3)$ for frame time $t\in \mathbb{R}$, with a $C^2$ angle $\theta$ measured from the downward vertical; $\theta =0$ is the bottom position. Adopt the uniform acceleration field $-g{e}_3$, with $g>0$ a declared acceleration parameter of dimension $\mathrm{\ell }{\tau }^{-2}$, so the applied force is $-mg{e}_3$ using that same inertial mass. No separate gravitational mass or empirical mass-equality claim is introduced. The ideal rod reaction has zero tangential component. The adopted exact equation of this constrained model is $m{l}^2{\theta }^{\mathrm{\prime }\mathrm{\prime }}+mgl\sin \theta =0$, and small-angle idealization $T=2\pi \sqrt{l\mathrm{/}g}$ for the period $T$ of a small oscillation, whose linearization at $\theta =0$ is the quadratic mode of angular frequency $\omega =\sqrt{g\mathrm{/}l}$. The model and its small-angle idealization are declared data of the example, adopted from the cited treatments; the exact finite-amplitude period discrepancy is not derived here. This constrained system is not a force-branch operational observable model: for $N=1$, the rod and plane constraints give the configuration set $Q_{\mathrm{r}\mathrm{o}\mathrm{d}}=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_2=0,x_1^2+x_3^2=l^2\},$ a circle with empty interior, whereas Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error requires an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$. This example supplies only the constrained angle equation and a hypothetical data-reduction map. No measurement is performed and no measured value is used.

The proposed data-reduction map. Use numerical input coordinates $l,T$ for the pendulum length in metres and period in seconds, respectively. Algebraically solving the adopted small-angle relation $T=2\pi \sqrt{l\mathrm{/}g}$ for the parameter $g$ gives the reconstruction function $h:U⟶\mathbb{R},U:=\{(l,T)\in {\mathbb{R}}^2:l>0,T>0\},h(l,T):=\frac{4{\pi }^{2}l}{T^2},$ which is a data-reduction map, not the prediction map $\mathrm{\Phi }$ of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error. For a specified oscillation amplitude $0<\alpha <\pi$ released from rest, let $T_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}(l,g,\alpha )$ denote the period of that declared ideal motion when a finite period exists. The finite-amplitude discrepancy is $\delta :=\mathrm{\mid }{T}_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}(l,g,\alpha )-2\pi \sqrt{l\mathrm{/}g}\mathrm{\mid }\ge 0$, in units of $\tau$; this example neither estimates nor bounds $\delta$. Any actual apparatus calibration would also require a calibration datum as in Calibration of a mechanical apparatus against the primitive standards of the framework: reference inputs, deviations, rules and calibrated prediction maps, including its reference deviations and calibration rule; this item supplies no reference observations or realized setting. In the calculations below $l$ and $T$ are numerical coefficients in metres and seconds, $h$ is the numerical coefficient in metres per second squared, and Euclidean norms are applied to those fixed numerical coordinates, not to sums of physical quantities of different dimensions; the constants depend on this coordinate convention.

Declared data. Fix a nominal input $\mu =(l_0,T_0)\in U$ with $l_0,T_0>0$ and radii $0<{\rho }_l<l_0$, $0<{\rho }_T<T_0$, and put $U_B:=(l_0-{\rho }_l,l_0+{\rho }_l)\times (T_0-{\rho }_T,T_0+{\rho }_T)\subseteq U,$ an open convex box. Let $X=(l,T):\mathrm{\Omega }\to U_B$ be a square-integrable random vector with $E[X]=\mu ,\mathrm{Cov}(X)=\left(\begin{array}{cc}{\sigma }_l^2& {\sigma }_{lT}\\ {\sigma }_{lT}& {\sigma }_T^2\end{array}\right)\mathrm{.}$ All entries are declared model data: no reading of an apparatus is asserted. With $h$ as above:

1. $h$ is $C^{\mathrm{\infty }}$ on $U$, with $\frac{\mathrm{\partial }h}{\mathrm{\partial }l}(l,T)=\frac{4{\pi }^2}{T^2}=\frac{h(l,T)}{l},\frac{\mathrm{\partial }h}{\mathrm{\partial }T}(l,T)=-\frac{8{\pi }^{2}l}{T^3}=-\frac{2h(l,T)}{T},$ so $Dh(\mu )$ has the two components $h(\mu )\mathrm{/}{l}_0$ and $-2h(\mu )\mathrm{/}{T}_0$;
2. the Hessian on $U$ is $H_h(l,T)=\left(\begin{array}{cc}0& -{\displaystyle \frac{8{\pi }^2}{T^3}}\\ -{\displaystyle \frac{8{\pi }^2}{T^3}}& {\displaystyle \frac{24{\pi }^{2}l}{T^4}}\end{array}\right),$ and on $U_B$ its operator norm is at most the Frobenius bound $B:=\left(2\right(\frac{8{\pi }^2}{(T_0-{\rho }_T{)}^3}{)}^2+(\frac{24{\pi }^2(l_0+{\rho }_l)}{(T_0-{\rho }_T{)}^4}{)}^2{)}^{1\mathrm{/}2};$
3. the covariance of the linearization $T_{\mu }(x)=h(\mu )+Dh(\mu )\cdot (x-\mu )$ is the $1\times 1$ matrix $\mathrm{Cov}(T_{\mu }(X))=\frac{16{\pi }^4}{T_0^4}{\sigma }_l^2-\frac{64{\pi }^4{l}_0}{T_0^5}{\sigma }_{lT}+\frac{64{\pi }^4{l}_0^2}{T_0^6}{\sigma }_T^2,$ which for uncorrelated inputs (${\sigma }_{lT}=0$) reads $\mathrm{Cov}(T_{\mu }(X))=\frac{16{\pi }^4}{T_0^4}({\sigma }_l^2+\frac{4l_0^2}{T_0^2}{\sigma }_T^2)=h(\mu {)}^2(\frac{{\sigma }_l^2}{l_0^2}+\frac{4{\sigma }_T^2}{T_0^2});$
4. for the nonlinearity remainder $R:=h(X)-h(\mu )-Dh(\mu )\cdot (X-\mu )$ one has the pointwise bound $\mid R\mid \le \frac{B}{2}\mid X-\mu {\mid }^2$, hence $\mid E[R]\mid \le E\mid R\mid \le \frac{B}{2}E\mid X-\mu {\mid }^2$, and if $E\mid X-\mu {\mid }^4<+\mathrm{\infty }$ then $E[R^2]\le \frac{B^2}{4}E\mid X-\mu {\mid }^4$.

Since $E[X]=\mu$ gives $E[Dh(\mu )(X-\mu )]=0$, the mean of the reading $\widehat{g}:=h(X)$ satisfies $\mid E[\widehat{g}]-h(\mu )\mid \le \frac{B}{2}E\mid X-\mu {\mid }^2$, while the exact variance of the linearization is the matrix of clause 3. The actual variance has the explicit correction of Mean, mean-square and covariance bounds for second-order linearization error: with $s^2=\mathrm{Var}(T_{\mu }(X))$ and $M_4=E\mathrm{\mid }X-\mu {\mathrm{\mid }}^4$, $\mathrm{\mid }\mathrm{Var}(\widehat{g})-s^2\mathrm{\mid }\le B\sqrt{s^2{M}_4}+\frac{B^2}{4}{M}_4\mathrm{.}$ The box ensures $M_4\le ({\rho }_l^2+{\rho }_T^2{)}^2<\mathrm{\infty }$; the model discrepancy $\delta$ and the reference deviations of the calibration contribute further unquantified terms. The protocol is a proposal: no value of $l$, $T$, $g$, ${\sigma }_l$, ${\sigma }_T$, ${\sigma }_{lT}$, $\delta$ or $B$ is a measurement here.