An oscillator observed in a uniformly rotating chart: the conserved chart Hamiltonian is not the inertial mechanical energy

Work in the one-label force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, time window $I_0=\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$, and adopt the potential

$U(x):=\frac{k}{2}\mid x{\mid }^2,k>0\text{ with }[k]=\mu {\tau }^{-2},$

so that $F(t,x,v)=-\mathrm{\nabla }U(x)=-kx$ and the model is a global potential model $d\equiv 0$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets whose motions satisfy $m{x}^{\mathrm{\prime }\mathrm{\prime }}=-kx$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model). Fix a rotation rate $\omega \mathrm{\ne }0$ of dimension ${\tau }^{-1}$ and let $\mathit{\omega }:=\omega e_3$. Define

$R(t):=\left(\begin{array}{ccc}\cos (\omega t)& -\sin (\omega t)& 0\\ \sin (\omega t)& \cos (\omega t)& 0\\ 0& 0& 1\end{array}\right)\mathrm{.}$

This matrix is the rotation about $e_3$ by angle $\omega t$; the identity ${\sin }^2(\omega t)+{\cos }^2(\omega t)=1$ gives $R(t)\in SO(3)$, and direct differentiation gives $R^{\mathrm{\prime }}(t)z=\mathit{\omega }\times R(t)z$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$, $SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$, The cross product in ${\mathbb{R}}^3$). Consider the time-dependent chart

$X(t,q):=R(t)q,\mathrm{\Omega }:=\mathbb{R}\times {\mathbb{R}}^3,$

which is of class $C^{\mathrm{\infty }}$ with $X_q(t,q)=R(t)$ invertible, and the substituted Lagrangian of Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation:

$\tilde{L}(t,q,v)=\frac{m}{2}\mid v+\mathit{\omega }\times q{\mid }^2-\frac{k}{2}\mid q{\mid }^2\mathrm{.}$

Verify:

1. Legendre data and chart Hamiltonian. $\tilde{L}$ is of mechanical type with $G=m{I}_3$, $A(q)=m\mathit{\omega }\times q$ and $V(q)=\frac{k}{2}\mid q{\mid }^2-\frac{m}{2}\mid \mathit{\omega }\times q{\mid }^2$, hence hyperregular with chart momentum $p=m(v+\mathit{\omega }\times q)$ and Hamiltonian

$H(t,q,p)=\frac{\mid p{\mid }^2}{2m}-p\cdot (\mathit{\omega }\times q)+\frac{k}{2}\mid q{\mid }^2\mathrm{.}$

2. Difference from the inertial mechanical energy. Along every $C^2$ curve $q:J\to {\mathbb{R}}^3$ on an interval $J\subseteq \mathbb{R}$ with nonempty interior, the inertial mechanical energy $E_{\mathrm{i}\mathrm{n}}=\frac{m}{2}\mid x^{\mathrm{\prime }}{\mid }^2+\frac{k}{2}\mid x{\mid }^2$ with $x=X(t,q)$ and the chart energy function $e=q^{\mathrm{\prime }}\cdot {\tilde{L}}_v-\tilde{L}$ satisfy the difference identity of The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions with $⟨X_t,\mathsf{M}{x}^{\mathrm{\prime }}⟩=p\cdot (\mathit{\omega }\times q)$, i.e. $E_{\mathrm{i}\mathrm{n}}=H+p\cdot (\mathit{\omega }\times q)$ along solutions.
3. Explicit circular motion. If $k=m{\omega }^2$, then $x(t):=rR(t)e_1$ with $r>0$ of dimension $\mathrm{\ell }$ is a motion of the model on $\mathbb{R}$, its chart description is the constant curve $q(t)=r{e}_1$ with $v\equiv 0$ and $p\equiv m\mathit{\omega }\times (r{e}_1)$, and along it $H\equiv 0$ while $E_{\mathrm{i}\mathrm{n}}\equiv m{\omega }^2{r}^2$; the substituted Lagrangian has ${\tilde{L}}_t\equiv 0$, so $H$ is a conserved quantity of the chart equations that is not the inertial mechanical energy.
4. Numerical instance. For $m=1\mathrm{k}\mathrm{g}$, $\omega =2{\mathrm{s}}^{-1}$, $r=1\mathrm{m}$ and $k=4\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}$ one has at $t=0$ the data $x=(1,0,0)\mathrm{m}$, $x^{\mathrm{\prime }}=(0,2,0)\mathrm{m}{\mathrm{s}}^{-1}$, $q=(1,0,0)$, $v=0$ and $p=(0,2,0)\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, with $H=0\mathrm{J}$ and $E_{\mathrm{i}\mathrm{n}}=4\mathrm{J}$.

All values are model values of the declared idealization; no measurement is reported.

• Comparison with the source picture. The structure matches the rotating-ring computation of the Likharev source, where the Hamiltonian and the mechanical energy differ by an $\omega$-dependent term while the Hamiltonian is the conserved one; here the difference is $p\cdot (\mathit{\omega }\times q)$, the exact general form of clause 2 of The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions in this chart. The chart Hamiltonian is a conserved quantity of the chart equations; it is not a measurement of the inertial mechanical energy, and the example attaches no empirical meaning to its value.

• Consistency of the two conserved quantities. For the circular motion both $H=0$ and $E_{\mathrm{i}\mathrm{n}}=m{\omega }^2{r}^2$ are constant, as they must be, and they differ; the difference $p\cdot (\mathit{\omega }\times q)$ is constant along this particular motion but is not a constant function on the chart's state space, so no global additive constant reconciles the two functions.