A cubic-potential chart system: the exact quadratic remainder, the explicit error constants and a checked finite-time bound

Let $m>0$, $k>0$ and $\lambda \in \mathbb{R}$ be constants, read with the dimensions $\mu$, $\mu {\tau }^{-2}$ and $\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), and consider the one-dimensional mechanical chart system of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart with constant metric field $M(q)=m$ and potential

$U(q)=\frac{1}{2}k{q}^2+\frac{\lambda }{6}{q}^3,$

For all field, norm and constant formulas below fix $s_1=1\mathrm{m}$, $t_{\ast }=1\mathrm{s}$ and $e_{\ast }=1\mathrm{J}$, and use the dimensionless numerical coordinates $q=Q\mathrm{/}{s}_1$, $t=t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{t}_{\ast }$, $v=(t_{\ast }\mathrm{/}{s}_1)\dot{Q}$. In these formulas $m,k,\lambda$ mean the numerical coefficients $m_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{s}_1^2\mathrm{/}(e_{\ast }{t}_{\ast }^2)$, $k_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{s}_1^2\mathrm{/}{e}_{\ast }$ and ${\lambda }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{s}_1^3\mathrm{/}{e}_{\ast }$, respectively. Consequently $r,\delta ,T,C,L,\mathrm{\parallel }A\mathrm{\parallel }$ are numerical normalized quantities, and $\max (1,(k+\mathrm{\mid }\lambda \mathrm{\mid }r\mathrm{/}2)\mathrm{/}m)$ compares dimensionless numbers. The physical horizon is $t_{\ast }T$, and the numerical error bound $b$ implies $\mathrm{\mid }Q-Q_{\mathrm{l}\mathrm{i}\mathrm{n}}\mathrm{\mid }\le s_{1}b$ and $\mathrm{\mid }\dot{Q}-{\dot{Q}}_{\mathrm{l}\mathrm{i}\mathrm{n}}\mathrm{\mid }\le (s_1\mathrm{/}{t}_{\ast })b$.

so that the mechanical field is $G(q,v)=(v,-(kq+\frac{\lambda }{2}{q}^2)\mathrm{/}m)$. Then:

1. $q_0=0$ is a mechanical equilibrium with $K:=D^{2}U(0)=k>0$; the small-oscillation model of clause (d) of The coupled quadratic oscillator model: mass metric, stiffness, generalized force, modal vocabulary and the small-oscillation model at a mechanical equilibrium is the one-dimensional quadratic model $m{\xi }^{\mathrm{\prime }\mathrm{\prime }}+k\xi =0$ with frequency $\omega =\sqrt{k\mathrm{/}m}$;
2. for the translated field $F(\delta q,\delta v)=(\delta v,-(k\delta q+\frac{\lambda }{2}\delta q^2)\mathrm{/}m)$ one has the exact remainder identity $F(\zeta )-A\zeta =(0,-\frac{\lambda }{2m}\delta q^2),A=\left(\begin{array}{cc}0& 1\\ -k\mathrm{/}m& 0\end{array}\right),\zeta =(\delta q,\delta v),$ so the constants of Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound can be taken as $C=\frac{\mid \lambda \mid }{2m},L=\max (1,\frac{k+\mid \lambda \mid r\mathrm{/}2}{m})$ on every ball $\parallel \zeta {\parallel }_2\le r$, and $\parallel A\parallel =\max (1,k\mathrm{/}m)$;
3. the numerical instance $m=1$ kg, $k=1$ kg/s${}^2$, $\lambda =1$ kg/(m$\cdot$s${}^2$), $r=1$, $\delta ={10}^{-2}$, $T=1$ in the normalized variables (a one-second horizon), for any initial normalized phase datum of norm $\delta$, satisfies $\delta e^{LT}={10}^{-2}{e}^{3\mathrm{/}2}\approx 0.04482<1$ and the resulting bound is $C{\delta }^{2}T{e}^{(\parallel A\parallel +2L)T}\approx 2.73\times {10}^{-3};$
4. for $\lambda =0$ the potential is exactly quadratic, $C=0$, and the nonlinear model coincides with its linearization: the bound of the theorem degenerates to the identity $z=y$.

• The cubic term is the first genuinely nonlinear term. Its remainder is exactly quadratic in the displacement, which is why the error-bound theorem's hypothesis holds globally here with the exact constant $C=\mid \lambda \mid \mathrm{/}(2m)$; for a general $U$ the constant comes from compactness and is not explicit, as in Finite-time error control for the small-oscillation approximation: the quadratic remainder of the mechanical field and the variation-of-constants bound.

• Decimal values of the instance. On the instance of clause 3 the smallness quantity is approximately $0.04482$ and the bound is approximately $2.73\times {10}^{-3}$; these decimals are evaluations of the exact expressions $e^{3\mathrm{/}2}\mathrm{/}100$ and $e^4\mathrm{/}20000$ and play no role in the verification.

• What the bound does not say. The estimate compares the nonlinear motion with the harmonic one only on the stated finite horizon and for the stated small datum. It gives no conclusion about the approximation error beyond that horizon. Likharev Section 5.2 discusses slowly accumulating perturbative effects under its own weak-nonlinearity assumptions, including a cubic force arising from a quartic potential. That discussion does not supply an all-time estimate or a deduction of secular behaviour for every motion of the cubic-potential model here; when $\lambda =0$ the equality in clause 4 holds at all times.