Example: 1D spring (/Harmonic oscillator)

• - Force \(F(t)=-k\,(p(t)-l^0)\;\;\), \(k\) spring stiffness, \(l^0\) rest length
• - Second order differential equation: \(m\,p''(t)+k\,p(t)=k\,l^0\)

• - First order system \(\displaystyle \underbrace{\left(\begin{array}{c}p \\ v\end{array}\right)'(t)}_{u'(t)}=\underbrace{\left(\begin{array}{c}v(t) \\ -k/m\left(p(t)-l^0\right)\end{array}\right)}_{\mathcal{F}(u,t)}\)
• - Linear system \(\displaystyle \underbrace{\left(\begin{array}{c}p \\ v\end{array}\right)'(t)}_{u'(t)}=\underbrace{\left(\begin{array}{cc} 0 & 1 \\ -k/m & 0\end{array}\right)}_{\mathrm{A}}\,\underbrace{\left(\begin{array}{c}p \\ v\end{array}\right)(t)}_{u(t)} + \underbrace{\left(\begin{array}{c}0 \\ k/m\;l^0\end{array}\right)}_{b}\)

• - Exact solution known: \(\displaystyle p(t)=A\,\sin(\omega\,t+\varphi)+l^0\)
• • \(\omega=\sqrt{k/m}\), \(\displaystyle A^2=(p^0-l^0)^2+\left(\frac{v_0}{\omega}\right)^2\), \(\displaystyle \tan(\varphi)=\frac{p^0-l^0}{v^0/\omega}\)