= Negative-energy direction gives exponential growth in a self-adjoint wave equation
{title2=$\|q(t)\|\geq\|q_0\|e^{\alpha t}$}
Let $\ddot q=Fq$ on a <Hilbert space>, with time-independent <self-adjoint operator> $F$, and conserved quadratic energy $E=\|\dot q\|^2/2-\langle q,Fq\rangle/2$. If $\langle q_0,Fq_0\rangle>0$, set $\dot q(0)=\alpha q_0$ with $\alpha^2=\langle q_0,Fq_0\rangle/\|q_0\|^2$ and $\alpha>0$. This gives $E=0$. For $N=\|q\|^2$, $N'=2\operatorname{Re}\langle q,\dot q\rangle$ and $N''=4\|\dot q\|^2$, so
$$
(\ln N)''=\frac{4[N\|\dot q\|^2-(\operatorname{Re}\langle q,\dot q\rangle)^2]}{N^2}\geq0
$$
by the <Cauchy-Schwarz inequality>. Since $(\ln N)'(0)=2\alpha$, $N(t)\geq N(0)e^{2\alpha t}$. Positivity of $N$ follows from $N'(0)>0$ and $N''\geq0$. Thus a negative direction of the restoring energy $W=-\langle q,Fq\rangle/2$ yields an exponentially growing solution without a normal-mode expansion or a variational spectral theorem. If $W$ is positive definite, conserved $E$ instead controls the energy norm; a bound in the original <Hilbert space> norm additionally follows from a coercive lower bound on $W$.
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