Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. Take the rotation parameter $s$ to be measured in radians, treated as dimensionless in SI, and take $a$ to be dimensionless; thus $\mid a\mid s$ is dimensionless and the charge $a\cdot J$ has angular-momentum units (action). Fix a nonzero vector $a\in {\mathbb{R}}^3$ and put, for $s\in \mathbb{R}$ and $z\in {\mathbb{R}}^3$,

$R_a(s)z:=z+\frac{\sin (\mid a\mid s)}{\mid a\mid }a\times z+\frac{1-\cos (\mid a\mid s)}{\mid a{\mid }^2}a\times (a\times z),$

where $\times$ is the cross product of ${\mathbb{R}}^3$ (The cross product in ${\mathbb{R}}^3$). For a configuration $x=(x_1,\dots ,x_N)$ write $R_a(s)x:=(R_a(s)x_1,\dots ,R_a(s)x_N)$, and let ${\xi }^a:Q\to {\mathbb{R}}^{3N}$ be the vector field with ${\xi }_i^a(x):=a\times x_i$. Write $J(t):={\sum }_i{x}_i(t)\times m_i{x}_i^{\mathrm{\prime }}(t)$ for the total angular momentum about the inertial origin (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame). Then:

1. (infinitesimal and finite invariance). The following are equivalent: (i) ${\sum }_i⟨a\times x_i,{\mathrm{\nabla }}_{x_i}U(t,x)⟩=0$ for every $(t,x)\in I_0\times Q$; (ii) $U(t,R_a(s)x)=U(t,x)$ whenever $\{R_a(r)x:r\text{ between }0\text{ and }s\}$ lies in $Q$. More generally put $D_x:=\{s\in \mathbb{R}:R_a(s)x\in Q\}$, an open set containing $0$, with interval components its maximal open intervals. Under (i), $s\mapsto U(t,R_a(s)x)$ is constant on each interval component. Invariance for every $s\in D_x$, including components separated by an excluded arc, holds if and only if all these constants equal $U(t,x)$. This agreement is automatic if $D_x$ is connected, in particular if the whole rotation orbit is contained in $Q$.

2. (the symmetry datum). Under these equivalent hypotheses the strict datum $\tau \equiv 0$, $\xi \equiv {\xi }^a$, $B\equiv 0$ satisfies the model's Noether identity on $O$ in the sense of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions.

3. (conservation of the angular-momentum component). If in addition $d\equiv 0$, then for every motion $x:J\to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t\mapsto a\cdot J(t)={\sum }_i⟨a,x_i(t)\times m_i{x}_i^{\mathrm{\prime }}(t)⟩$ is constant on $J$.

4. (sharpness). If $d\equiv 0$ and there is a state $(t_0,x_0)\in I_0\times Q$ with $g:={\sum }_i⟨a\times x_{0,i},{\mathrm{\nabla }}_{x_i}U(t_0,x_0)⟩\mathrm{\ne }0$, then the maximal motion $x_{\max }:J_{\max }\to Q$ through the initial datum $(t_0,x_0,0)$ satisfies $\frac{d}{dt}(a\cdot J(t)){\mathrm{\mid }}_{t_0}=-g\mathrm{\ne }0$; in particular $a\cdot J$ is not constant on $J_{\max }$.

5. (vector form). If ${\sum }_i{x}_i\times {\mathrm{\nabla }}_{x_i}U(t,x)=0$ for every $(t,x)\in I_0\times Q$, then with $d\equiv 0$ the total angular momentum vector $J$ is constant along every motion.

The hypothesis is a condition on the declared potential — equivalently, by clause 1, invariance of $U$ along admissible arcs of the simultaneous rotations $R_a(s)$ of all labels, with full endpoint invariance requiring the component agreement specified there; the conservation clauses are conditional deductions of the global potential branch, and no symmetry of any particular force law and no empirical content is asserted.

• Separated arcs. On two disjoint balls centred at opposite points of a circle about the axis, the potential equal to $0$ on one ball and $1$ on the other has zero rotational derivative everywhere, but a rotation between the two centres changes its value. Differential conservation holds on arbitrary open $Q$; comparing separated arcs requires the compatibility in clause 1.

• Torque reading. Under the potential presentation the total torque of the potential forces about the origin is $-{\sum }_i{x}_i\times {\mathrm{\nabla }}_{x_i}U$, so hypothesis (i) for all $a$ is the vanishing of that torque; clause 5 is then the familiar statement that a model whose potential exerts no torque about the origin conserves its total angular momentum. The balance theorem Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case reaches the conservation statement from centrality of the pair forces and vanishing external torque, a different hypothesis; the two routes are complementary and neither is a consequence of the other.

• Angular momentum about other origins. The charge constructed here is the angular momentum about the inertial origin, which is where the rotation family $R_a$ acts. For rotations about another fixed point $o$ the same computation with $x_i-o$ in place of $x_i$ produces the component of the angular momentum about $o$; that variant is not needed on this page and is not asserted.

• Empirical status. The statement identifies a conserved quantity of the model under an explicit rotational invariance hypothesis of the declared potential; it is a conditional deduction inside the model, and no claim is made that any actual force law is rotation invariant, nor is any converse from an observed conservation to a symmetry of a model.