The undamped oscillator: Dirichlet non-uniqueness at $T=n\pi \mathrm{/}\omega$ despite unique initial-value solutions

Let $m>0$ and $k>0$, put $\omega :=(k\mathrm{/}m{)}^{1\mathrm{/}2}>0$ (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$), and let $T>0$. Consider the undamped homogeneous oscillator equation

$m{x}^{\mathrm{\prime }\mathrm{\prime }}(t)+kx(t)=0$

of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with time domain $I=\mathbb{R}$ and zero forcing, and its solutions on intervals with nonempty interior. The Dirichlet curves below are restrictions to $[0,T]$ of its global solutions. Then:

1. for every real $A$ the function $x_A(t):=A\sin (\omega t)$ is a $C^2$ (indeed $C^{\mathrm{\infty }}$) solution on $\mathbb{R}$ with $x_A(0)=0$ and $x_A(T)=A\sin (\omega T)$; moreover $x_A(T)=0$ if and only if $A=0$ or $\omega T\in \pi \mathbb{Z}$ (The zero sets of sine and cosine and the least positive common period 2 pi);
2. if $T=n\pi \mathrm{/}\omega$ for some integer $n\ge 1$, then the set of $C^2$ solutions of the Dirichlet problem on $[0,T]$,

$m{x}^{\mathrm{\prime }\mathrm{\prime }}+kx=0,x(0)=x(T)=0,$

is exactly $\{x_A:A\in \mathbb{R}\}$; this set is a one-dimensional real vector space, so the boundary-value problem has infinitely many distinct solutions with the same endpoint data $(0,0)$, one for each initial velocity $A\omega$; 3. by contrast, for every $(x_0,v_0)\in {\mathbb{R}}^2$ the initial-value problem $m{x}^{\mathrm{\prime }\mathrm{\prime }}+kx=0$, $x(0)=x_0$, $x^{\mathrm{\prime }}(0)=v_0$, has exactly one solution on $\mathbb{R}$, namely $x(t)=x_0\cos (\omega t)+(v_0\mathrm{/}\omega )\sin (\omega t)$ (The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data, clauses 3 and 4 with $c=0$); uniqueness for the initial-value problem therefore does not imply uniqueness for the boundary-value problem; 4. for the chart with $O=\mathbb{R}\times \mathbb{R}\times \mathbb{R}$ and $\mathcal{L}(q,v)=\frac{1}{2}(v^2-{\omega }^2{q}^2)$ over $[0,T]\times \mathbb{R}$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations), each $x_A$ is a stationary curve of the action; and if $T=n\pi \mathrm{/}\omega$, the stationary curves on $[0,T]$ satisfying the endpoint conditions $x(0)=x(T)=0$ are exactly the curves $x_A$, so the fixed-endpoint problem with zero data has infinitely many stationary curves (Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression).

• At $T=\pi \mathrm{/}\omega$ the second-variation value in direction ${\eta }_1=\sin (\omega t)$ 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 zero and the Dirichlet solution space is one-dimensional. More generally, its nonzero solutions occur exactly when $T=n\pi \mathrm{/}\omega$ for an integer $n\ge 1$, as clauses 1-2 show. The computations here are mathematical statements about the model; no physical forcing or measured period is asserted.

• The one-parameter family $\{x_A\}$ is not an artefact of dropping a hypothesis: it persists as the full solution set of the Dirichlet problem, and it is a legitimate underdetermined boundary-value problem, in contrast with the exactly solvable initial-value problem.