Diagonal phase rotations and relative-phase coordinates

Let $V$ be a nonempty finite set and $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ be the phase torus. A phase is a real number modulo $2\pi$; write its addition and subtraction using representatives modulo $2\pi$. For a phase $c$, the diagonal rotation $R_c:T_V\to T_V$ is defined by $(R_c\theta {)}_x={\theta }_x+c$ for every $x\in V$. Two configurations are rotation equivalent, written $\theta \sim \varphi$, if $\varphi =R_c\theta$ for some phase $c$. Give the set $Q=T_V\mathrm{/}\sim$ the quotient topology and write $p:T_V\to Q$ for its projection.

Fix $v_0\in V$. The relative-phase map is $r(\theta )=({\theta }_x-{\theta }_{v_0}{)}_{x\in V\setminus \{v_0\}}$. Its induced map $\bar{r}:Q\to T_{V\setminus \{v_0\}}$ is a homeomorphism. If $V=\{v_0\}$, the target is the one-point empty product. A function of phase differences means a function whose value depends only on the family $({\theta }_x-{\theta }_y{)}_{x,y\in V}$; it has a unique induced function on $Q$.

The phasor average is $m(\theta )=\mathrm{\mid }V{\mathrm{\mid }}^{-1}{\sum }_{x\in V}{e}^{i{\theta }_x}$. If $m(\theta )\mathrm{\ne }0$, an argument is a phase $a$ with $e^{ia}=m(\theta )\mathrm{/}\mathrm{\mid }m(\theta )\mathrm{\mid }$; the rotation $R_{-a}$ makes the average positive real. At zero average no argument or preferred rotation is specified. A rotating coordinate frame with angular velocity $\omega \in \mathbb{R}$ replaces a time-dependent configuration $\theta (t)$ by $R_{-\omega t}\theta (t)$, where $-\omega t$ is read modulo $2\pi$.

Given: A nonempty finite $V$, a vertex $v_0\in V$, and the constructions in the definition.

Goal: Verify the rotation relation, relative-coordinate homeomorphism, descent, normalization, and rotating-frame description.

1.1 Addition modulo $2\pi$ is independent of the chosen representatives. It gives $R_0\theta =\theta$ and $R_c{R}_d\theta =R_{c+d}\theta$, with inverse $R_{-c}$; hence rotation equivalence is an equivalence relation. Each difference ${\theta }_x-{\theta }_y$ is unchanged by a common rotation, so $r$ is constant on equivalence classes. It therefore defines $\bar{r}$ uniquely by $\bar{r}(p(\theta ))=r(\theta )$. [F1, algebra]

2.1 Define $s:T_{V\setminus \{v_0\}}\to T_V$ by $s(z{)}_{v_0}=0$ and $s(z{)}_x=z_x$ otherwise. Then $r(s(z))=z$, while $s(r(\theta ))=R_{-{\theta }_{v_0}}\theta$. Consequently $p\circ s$ and $\bar{r}$ are inverse bijections. When $V$ has one element, the same formulas use the unique empty tuple and show that $Q$ has one point. [step 1.1, construct, algebra]

3.1 In quotient charts, each coordinate of $r$ is represented locally by subtraction of real coordinates followed by the circle quotient projection, so $r$ is continuous; $s$ is continuous by its constant and coordinate components. The quotient criterion makes $\bar{r}$ continuous, and $p\circ s$ is continuous. Thus the inverse bijections in step 2.1 are homeomorphisms. Any function depending only on differences is constant on each orbit and therefore has a unique set-theoretic factor through $p$; if that function is continuous, its factor is continuous by the same quotient criterion. [F1, F3, F4, step 1.1, step 2.1]

4.1 By the phasor addition law, $m(R_c\theta )=e^{ic}m(\theta )$. For nonzero $m$, the circle parametrization in [F1] supplies a phase $a$ with $e^{ia}=m\mathrm{/}\mathrm{\mid }m\mathrm{\mid }$; then $m(R_{-a}\theta )=\mathrm{\mid }m\mathrm{\mid }>0$. Such a phase is unique modulo $2\pi$. If $m=0$, every rotation leaves the average zero, so this normalization selects no rotation. Finally, $R_{-\omega t}$ subtracts the same phase from every coordinate and therefore leaves all relative phases unchanged. [F1, F2, step 1.1, algebra] ∎