A two-label pair-potential model: translation, rotation, boost and energy charges verified on the model class and on an explicit rigid-rotation motion

Declare a two-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=2$, masses $m_1,m_2>0$, total mass $M:=m_1+m_2$ and reduced mass $\mu :=m_1{m}_2\mathrm{/}M$, time window $I_0=\mathbb{R}$, configuration domain $Q=\{(x_1,x_2)\in ({\mathbb{R}}^3{)}^2:x_1\mathrm{\ne }{x}_2\}$ (open), and let $\psi :(0,\mathrm{\infty })\to \mathbb{R}$ be of class $C^2$; its argument is the squared separation of dimension ${\mathrm{\ell }}^2$, and its values have energy dimension. Adopt the potential

$U(x_1,x_2):=\psi (s(x)),s(x):=\mid x_1-x_2{\mid }^2,$

of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, so that 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. Verify:

For the translation family use a dimensionless direction and a length-valued parameter; for rotations use a dimensionless generator and a parameter in radians; for time translations use a parameter in seconds; for boosts use a dimensionless direction and a velocity-valued parameter. In the explicit potential below, $c_0$ has energy units $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-2}$ and $\kappa$ has units $\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}$ (energy per squared length).

1. gradients and pair reading. For every $x\in Q$,

${\mathrm{\nabla }}_{x_1}U(x)=2{\psi }^{\mathrm{\prime }}(s(x))(x_1-x_2),{\mathrm{\nabla }}_{x_2}U(x)=-2{\psi }^{\mathrm{\prime }}(s(x))(x_1-x_2);$

so the model invokes the external-plus-pair presentation of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch with $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}=0$ and a single pair force $F_{12}=-F_{21}=-2{\psi }^{\mathrm{\prime }}(s)(x_1-x_2)$ that is antisymmetric and central. 2. translation. ${\sum }_i{\mathrm{\nabla }}_{x_i}U=0$ identically, so the differential condition of Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness holds for every $a\in {\mathbb{R}}^3$; consequently $P$ and the boost charge $a\cdot (tP-MX(t))$ of The Galilean boost charge of the model: exact condition, uniform centre-of-mass motion, the induced motions and the boundary of the conservation law are constant along every motion and the centre of mass moves uniformly. 3. rotation. For every $a\in {\mathbb{R}}^3$, ${\sum }_i⟨a\times x_i,{\mathrm{\nabla }}_{x_i}U⟩=0$ identically, so the condition of Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin holds for every axis; consequently the total angular momentum $J={\sum }_i{x}_i\times m_i{x}_i^{\mathrm{\prime }}$ is constant along every motion. 4. energy. $U$ has no explicit time dependence, so the mechanical energy $E=K+U$ is constant along every motion by Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential. 5. explicit rigid-rotation motion (Hooke-type pair). Specialise to $\psi (s)=c_0+\kappa s$ with constants $c_0\in \mathbb{R}$ and $\kappa >0$, and put $\mathrm{\Omega }:=\sqrt{2\kappa \mathrm{/}\mu }$, $X_0\in {\mathbb{R}}^3$, $e(t):=(\cos \mathrm{\Omega }t,\sin \mathrm{\Omega }t,0)$, $r_1:=(\mu \mathrm{/}{m}_1)\rho$ and $r_2:=-(\mu \mathrm{/}{m}_2)\rho$ for a constant $\rho >0$; $X_0$ and $\rho$ have length dimension and $\mathrm{\Omega }$ has dimension ${\tau }^{-1}$. Then $x_i(t):=X_0+r_{i}e(t)$ defines a motion of the model on $\mathbb{R}$, and along it $P\equiv 0$, $J=\mathrm{\Omega }\mu {\rho }^2(0,0,1)$ and $E=c_0+2\kappa {\rho }^2$ are constant. 6. numerical instance. For $m_1=m_2=1\mathrm{k}\mathrm{g}$, $\kappa =1\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}$, $\rho =1\mathrm{m}$, $c_0=0\mathrm{J}$, $X_0=0$: $\mu =\frac{1}{2}$, $\mathrm{\Omega }=2{\mathrm{s}}^{-1}$, $r_1=\frac{1}{2}\mathrm{m}$, $r_2=-\frac{1}{2}\mathrm{m}$ and at $t=0$ one has $x_1=(\frac{1}{2},0,0)$, $x_2=(-\frac{1}{2},0,0)$, $x_1^{\mathrm{\prime }}=(0,1,0)$, $x_2^{\mathrm{\prime }}=(0,-1,0)$ in the stated units, $P=0$, $J=(0,0,1)\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $E=2\mathrm{J}$; also ${\mathrm{\nabla }}_{x_1}U(0)=2\kappa (x_1-x_2)=(2,0,0)$ in newtons and ${\mathrm{\nabla }}_{x_2}U(0)=(-2,0,0)$.

All values are model values of the declared idealization; no measurement is reported and no property of a physical pair interaction is asserted.

• Both routes agree here. For this model the pair presentation is antisymmetric and central, so the balance theorems Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces and Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case also apply and yield the same conservation statements under their own hypotheses. The Noether route verified here uses the potential presentation; the two are independent deductions, and the example checks the hypotheses of both at once.

• Representative of the potential. The additive constant $c_0$ enters the energy statement but not the forces; the numerical instance fixes $c_0=0$ as a declared normalization, in the sense of clause (c) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. No physical meaning is attached to the value $E=0$.