Scope and limits of the verification items on this page

Every item on this page is mathematics: the oscillator, the two-mass spring system, the pendulum field and the mass distributions are mathematical objects with explicit parameters, and no item asserts a physical law, a unit convention or a measured value. The following limits belong to the claims and must be carried by any consumer.

• The second-variation sign results are finite-time statements about the action. The definition The fixed-endpoint second variation of a time-dependent action at a curve and the formula of The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula concern the fixed-endpoint family $q+\epsilon \eta$ of a $C^2$ Lagrangian chart on a compact interval and produce one real number per pair $(q,\eta )$. The saddle computation of Second variation of the oscillator action on a finite interval: both signs for $T>\pi \mathrm{/}\omega$ and the value at $T=\pi \mathrm{/}\omega$ is a statement about the action functional of the quadratic Lagrangian on $[0,T]$; it does not assert that any physical motion extremises a physical action, and it does not contradict any minimum principle, because stationarity and minimisation are distinct conditions (the scope note of Tong, Section 2.1). For $T>\pi \mathrm{/}\omega$ both signs occur; the quantitative statement is exactly the one displayed, with the variations ${\eta }_n=\sin (n\pi t\mathrm{/}T)$ and the exact identity $S(\epsilon {\eta }_n)=\frac{{\epsilon }^2}{2}{\delta }^{2}S(q_0)[{\eta }_n,{\eta }_n]$.

• Endpoint degeneracy at $T=\pi \mathrm{/}\omega$ is not a failure of the model. At this value the displayed second-variation value in direction ${\eta }_1$ is zero and the Dirichlet problem of The undamped oscillator: Dirichlet non-uniqueness at $T=n\pi \mathrm{/}\omega$ despite unique initial-value solutions has the one-dimensional solution space $\{A\sin (\omega t)\}$, while initial-value problems remain uniquely solvable. More generally, the Dirichlet problem is non-unique exactly at $T=n\pi \mathrm{/}\omega$ for positive integers $n$. This non-uniqueness is an underdetermination of a boundary-value problem, not a contradiction between the differential equation and its solutions. At $T=\pi \mathrm{/}\omega$, the item also computes positive second-variation values in directions ${\eta }_n$ for $n\ge 2$; it makes no claim about all remaining directions or about overall extremality at this duration.

• The parallel-axis theorem is the centred positive-mass case. Parallel-axis theorem: the inertia tensor about a displaced point of a centred mass distribution assumes $\mu (\nu )=0$, that is, coordinates whose origin is the centre of mass, together with finite second moments, which for a bounded finite Borel measure are proved in Finiteness and a uniform bound for the entries of the inertia tensor about an arbitrary point and in the finiteness lemma of Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix. The theorem covers point, line, surface and volume measures, but only the centred identity $I_{\nu }(a)=I(\nu )+m(\parallel a{\parallel }_2^2{I}_3-a{a}^{\mathsf{T}})$; for a non-centred distribution the additional first-moment terms displayed in the proof of that theorem are present, and no physical assertion about any body is made.

• The coupled-oscillator proposition is an exact statement about a linear system. Two equal masses with two outer springs and a coupling spring: eigenmodes, decoupling and the explicit normal frequencies identifies the eigenmodes of the constant matrix $K=\left(\begin{array}{cc}k+c& -c\\ -c& k+c\end{array}\right)$ and proves the decoupling bijection; it claims exactness for all times for the linear system with the stated parameter ranges. The degenerate case $k=0$ produces affine motion of the in-phase mode and is reported as such; no oscillatory frequency is attached to a vanishing eigenvalue.

• The pendulum bound is finite-time, conditional and of order ${\delta }^2$. The pendulum field near the downward equilibrium: bounded force derivative, cubic remainder and the finite-time small-angle error bound is a conditional consequence of the explicit field $F(\theta ,v)=(v,-{\omega }^2\sin \theta )$, the smallness condition $\delta e^{LT}<r$ and the cubic remainder $\mid \sin \theta -\theta \mid \le \mid \theta {\mid }^3\mathrm{/}6$. It is not an assertion that the nonlinear pendulum is exactly harmonic, not a global or long-time statement, and not a report about any measured pendulum; the exactly harmonic system is the linearised equation $y_1^{\mathrm{\prime }\mathrm{\prime }}=-{\omega }^2{y}_1$ alone.