2.1. Preliminaries: rotations and the rotation group

Before describing the model, we need to recall some facts about rotation matrices (see [Reference Degond18] for more detail). Throughout this paper, the dimension $n$ will be supposed greater or equal to $3$ . We denote by $\mathrm{M}_n$ the space of $n \times n$ matrices with real entries and by $\mathrm{SO}_n$ the subset of $\mathrm{M}_n$ consisting of rotation matrices:

\begin{equation*} \mathrm {SO}_n = \big \{ A \in \mathrm {M}_n \quad | \quad A A^T = A^T A = \mathrm {I} \quad \mathrm {and} \quad \mathrm {det}\, A = 1 \big \}, \end{equation*}

where $A^T$ is the transpose of $A$ , $\mathrm{I}$ is the identity matrix of $\mathrm{M}_n$ and $\mathrm{det}$ stands for the determinant. For $x$ , $y \in{\mathbb R}^n$ , we denote by $x \cdot y$ and $|x|$ the Euclidean inner product and norm. Likewise, we define a Euclidean inner product on $\mathrm{M}_n$ as follows:

(2.1) \begin{equation} M \cdot N = \frac{1}{2} \mathrm{Tr} (M^T N) = \frac{1}{2} \sum _{i,j} M_{ij}N_{ij}, \quad \forall M, \, N \in \mathrm{M}_n, \end{equation}

where $\mathrm{Tr}$ is the trace. We note the factor $\frac{1}{2}$ which differs from the conventional definition of the Frobenius inner product, but which is convenient when dealing with rotation matrices. We use the same symbol $\cdot$ for vector and matrix inner products as the context easily waives the ambiguity. The set $\mathrm{SO}_n$ is a compact Lie group, i.e. it is a group for matrix multiplication and an embedded manifold in $\mathrm{M}_n$ for which group multiplication and inversion are $C^\infty$ , and it is compact. Let $A \in \mathrm{SO}_n$ . We denote by $T_A$ the tangent space to $\mathrm{SO}_n$ at $A$ . The tangent space $T_{\mathrm{I}}$ at the identity is the Lie algebra $\mathfrak{so}_n$ of skew-symmetric matrices with real entries endowed with the Lie bracket $[X,Y] = XY-YX$ , $\forall X, \, Y \in \mathfrak{so}_n$ . Let $A \in \mathrm{SO}_n$ . Then,

(2.2) \begin{equation} T_A = A \, \mathfrak{so}_n = \mathfrak{so}_n \, A. \end{equation}

For $M \in \mathrm{M}_n$ , we denote by $P_{T_A} M$ its orthogonal projection onto $T_A$ . It is written:

\begin{equation*} P_{T_A} M = A \frac {A^T M - M^T A}{2} = \frac {M A^T - A M^T}{2} A. \end{equation*}

A Riemannian structure on $\mathrm{SO}_n$ is induced by the Euclidean structure of $\mathrm{M}_n$ following from (2.1). This Riemannian metric is given by defining the inner product of two elements $M$ , $N$ of $T_A$ by $M \cdot N$ . Given that there are $P$ and $Q$ in $\mathfrak{so}_n$ such that $M=AP$ , $N = AQ$ and that $A$ is orthogonal, we have $M \cdot N = P \cdot Q$ . As any Lie-group is orientable, this Riemannian structure gives rise to a Riemannian volume form and measures $\omega$ . This Riemannian measure is left-invariant by group translation [Reference Degond18, Lemma 2.1] and is thus equal to the normalised Haar measure on $\mathrm{SO}_n$ up to a multiplicative constant. We recall that on compact Lie groups, the Haar measure is also right invariant and invariant by group inversion. On Riemannian manifolds, the gradient $\nabla$ , divergence $\nabla \cdot$ and Laplacian $\Delta$ operators can be defined. Given a smooth map $f$ : $\mathrm{SO}_n \to{\mathbb R}$ , the gradient $\nabla f(A) \in T_A$ is defined by

(2.3) \begin{equation} \nabla f (A) \cdot X = df_A(X), \quad \forall X \in T_A, \end{equation}

where $df_A$ is the derivative of $f$ at $A$ and is a linear map $T_A \to{\mathbb R}$ , and $df_A(X)$ is the image of $X$ by $df_A$ . The divergence of a smooth vector field $\phi$ on $\mathrm{SO}_n$ (i.e. a map $\mathrm{SO}_n \to \mathrm{M}_n$ such that $\phi (A) \in T_A$ , $\forall A \in \mathrm{SO}_n$ ) is defined by duality by

\begin{equation*} \int _{\mathrm {SO}_n} \nabla \cdot \varphi \, f \, dA = - \int _{\mathrm {SO}_n} \varphi \cdot \nabla f \, dA, \quad \forall f \in C^{\infty }(\mathrm {SO}_n), \end{equation*}

where $C^{\infty }(\mathrm{SO}_n)$ denotes the space of smooth maps $\mathrm{SO}_n \to{\mathbb R}$ and where we have denoted the Haar measure by $dA$ . The Laplacian of a smooth map $f$ : $\mathrm{SO}_n \to{\mathbb R}$ is defined by $\Delta f = \nabla \cdot (\nabla f)$ .

It is not easy to find a convenient coordinate system on $\mathrm{SO}_n$ to express the divergence and Laplace operators, so we will rather use an alternate expression which uses the matrix exponential $\exp$ : $\mathfrak{so}_n \to \mathrm{SO}_n$ . Let $X \in \mathfrak{so}_n$ . Then, $\varrho (X)$ denotes the map $C^{\infty }(\mathrm{SO}_n) \to C^{\infty }(\mathrm{SO}_n)$ such that

(2.4) \begin{equation} \big ( \varrho (X)(f) \big )(A) = \frac{d}{dt} \big (f(A e^{tX}) \big )|_{t=0}, \quad \forall f \in C^{\infty }(\mathrm{SO}_n), \quad \forall A \in \mathrm{SO}_n. \end{equation}

We note that

(2.5) \begin{equation} \big ( \varrho (X)(f) \big )(A) = df_A(AX) = \nabla f(A) \cdot (AX). \end{equation}

Let $F_{ij}$ be the matrix with entries

(2.6) \begin{equation} (F_{ij})_{k\ell } = \delta _{ik} \delta _{j \ell } - \delta _{i \ell } \delta _{jk}. \end{equation}

We note that $(F_{ij})_{1 \leq i \lt j \leq n}$ forms an orthonormal basis of $\mathfrak{so}_n$ for the inner product (2.1). Then, we have [Reference Degond18, Lemma 2.2]

(2.7) \begin{equation} (\Delta f)(A) = \sum _{1 \leq i \lt j \leq n} \big ( \varrho (F_{ij})^2 f \big ) (A). \end{equation}

The expression remains valid if the basis $(F_{ij})_{1 \leq i \lt j \leq n}$ is replaced by another orthonormal basis of $\mathfrak{so}_n$ .

Finally, let $M \in \mathrm{M}_n^+$ where $\mathrm{M}_n^+$ is the subset of $\mathrm{M}_n$ consisting of matrices with positive determinant. There exists a unique pair $(A, S) \in \mathrm{SO}_n \times{\mathcal S}^+_n$ where ${\mathcal S}^+_n$ denotes the cone of symmetric positive-definite matrices, such that $M = A S$ . The pair $(A, S)$ is the polar decomposition of $M$ and we define a map $\mathcal P$ : $\mathrm{M}_n^+ \to \mathrm{SO}_n$ , $M \mapsto A$ . We note that ${\mathcal P}(M) = (M M^T)^{-1/2} M$ . We recall that a positive-definite matrix $S$ can be written $S = U D U^T$ where $U \in \mathrm{SO}_n$ and $D = \mathrm{diag}(d_1, \ldots, d_n)$ is the diagonal matrix with diagonal elements $d_1, \ldots, d_n$ with $d_i \gt 0$ , $\forall i=1, \ldots, n$ . Then, $S^{-1/2} = U D^{-1/2} U^T$ with $D^{-1/2} = \mathrm{diag}(d_1^{-1/2}, \ldots, d_n^{-1/2})$ .

2.2. The particle model

We consider a system of $N$ agents moving in an $n$ -dimensional space ${\mathbb R}^n$ . They have positions $(X_k(t))_{k=1}^N$ with $X_k(t) \in{\mathbb R}^n$ at time $t$ . All agents are identical rigid bodies. An agent’s attitude can be described by a moving direct orthonormal frame $(\omega _1^k(t), \ldots, \omega _n^k(t))$ referred to as the agent’s local body frame or body attitude. Let ${\mathbb R}^n$ be endowed with a reference direct orthonormal frame $({\textbf{e}}_1, \ldots,{\textbf{e}}_n)$ . We denote by $A_k(t)$ the unique rotation which maps $({\textbf{e}}_1, \ldots,{\textbf{e}}_n)$ to $(\omega _1^k(t), \ldots, \omega _n^k(t))$ , i.e. $\omega _j^k(t) = A_k(t){\textbf{e}}_j$ . Therefore, the $k$ -th agent’s body attitude can be described equivalently by the local basis $(\omega _1^k(t), \ldots, \omega _n^k(t))$ or by the rotation $A_k(t)$ .

The particle dynamics is as follows: particles move with speed $c_0$ in the direction of the first basis vector $\omega _1^k(t) = A_k(t){\textbf{e}}_1$ of the local body frame, hence leading to the equation

(2.8) \begin{equation} d X_k= c_0 A_k(t){\textbf{e}}_1 \, dt. \end{equation}

Body frames are subject to two processes. The first one tends to relax the particle’s body frame to a target frame which represents the average of the body frames of the neighbouring particles. The second one is diffusion noise. We first describe how the average of the body frames of the neighbouring particles is computed. Define

(2.9) \begin{equation} \tilde J_k(t) = \frac{1}{N \, R^n} \sum _{\ell =1}^N K \Big ( \frac{|X_k(t) - X_\ell (t)|}{R} \Big ) \, A_\ell (t), \end{equation}

where the sensing function $K$ : $[0,\infty ) \to [0,\infty )$ and the sensing radius $R \gt 0$ are given. Here we have assumed that the sensing function is radially symmetric for simplicity. Then, the body frame dynamics is as follows:

(2.10) \begin{equation} d A_k = \nu \, P_{T_{A_k}} \big ({\mathcal P}(\tilde J_k) \big ) \, dt + \sqrt{2D} \, P_{T_{A_k}} \circ dW_t^k, \end{equation}

where $\nu$ and $D$ are positive constants, $dW_t^k$ are independent Brownian motions on $\mathrm{M}_n$ and the symbol $\circ$ indicates that the stochastic differential equation is meant in the Stratonovich sense. Here, it is important to stress that these Brownian motions are defined using the metric induced by the inner product (2.1). In the first term, we note that the matrix $\tilde J_k$ is projected onto the rotation matrix issued from the polar decomposition by the map $\mathcal P$ . For consistency, we need to assume that $\tilde J_k(t)$ remains in $M_n^+$ , which will be true as long as the body frames of neighbouring particles are close to each other. The rotation $\Gamma _k ={\mathcal P}(\tilde J_k(t))$ can be seen as the average body frame of the neighbours to particle $k$ . For (2.10) to define a motion on $\mathrm{SO}_n$ , the right-hand side must be a tangent vector to $\mathrm{SO}_n$ at $A_k(t)$ . This is ensured by the projection operator $P_{T_{A_k}}$ which multiplies each term of the right-hand side of (2.10), and, for the second term, by the fact that the stochastic differential equation is taken in the Stratonovich sense. Indeed, according to [Reference Hsu43], the second term of (2.10) generates a Brownian motion on $\mathrm{SO}_n$ .

Finally, the particle system must be supplemented with initial conditions specifying the values of $(X_k,A_k)(0)$ . As discussed in ref. [Reference Degond, Frouvelle and Merino-Aceituno24], the first term of (2.10) relaxes the $k$ -th particle body frame to the neighbour’s average body frame $\Gamma _k$ and models the agents’ tendency to adopt the same body attitude as their neighbours. The second term of (2.10) is an idiosyncratic noise term that models either errors in the agent’s computation of the neighbour’s average body frame or the agent’s will to detach from the group and explore new environments.

2.3. The mean-field model

When $N \to \infty$ , the particle system can be approximated by a mean-field kinetic model. Denote by $f(x,A,t)$ the probability distribution of the particles, namely $f(x,A,t)$ is the probability density of the particles at $(x,A) \in{\mathbb R}^n \times \mathrm{SO}_n$ at time $t$ . Then, provided that $K$ satisfies

\begin{equation*} \int _{{\mathbb R}^n} K(|x|) \, dx = 1, \quad \int _{{\mathbb R}^n} K(|x|) \, |x|^2\, dx \lt \infty,\end{equation*}

$f$ is the solution of the following system:

(2.11) \begin{eqnarray} && \partial _t\, f + c_0 \, A{\textbf{e}}_1 \cdot \nabla _x f + \nu \nabla _A \cdot \big ( P_{T_A} ({\mathcal P} (\tilde J_f)) f \big ) - D \Delta _A f = 0, \end{eqnarray}

(2.12) \begin{eqnarray} && \tilde J_f(x,t) = \frac{1}{R^n} \int _{{\mathbb R}^n \times \mathrm{SO}_n} K \Big ( \frac{|x-y|}{R} \Big ) \, f(y, B, t) \, B \, dy \, dB, \end{eqnarray}

where again, in (2.12), the integral over $\mathrm{SO}_n$ is taken with respect to the normalised Haar measure. The operators $\nabla _A \cdot$ and $\Delta _A$ are respectively the divergence and Laplacian on $\mathrm{SO}_n$ as defined in Section 2.1. The index $A$ is there to distinguish them from analog operators acting on the spatial variable $x$ , which will be indicated (as in $\nabla _x f$ ) with an index $x$ . A small remark may be worth making: according to [Reference Hsu43], the stochastic process defined by the second term of (2.10) has infinitesimal generator $D \tilde \Delta$ where for any $f \in C^\infty (\mathrm{SO}_n)$ ,

\begin{equation*} \tilde \Delta f(A) = \sum _{i,j=1}^n (P_{T_A} E_{ij} \cdot \nabla _A)^2 f(A), \qquad \forall A \in \mathrm {SO}_n, \end{equation*}

and where $(E_{ij})_{i,j=1}^n$ is any orthogonal basis (for the inner product (2.1)) of $\mathrm{M}_n$ . For instance, we can take the matrices $E_{ij}$ with entries $(E_{ij})_{k \ell } = \sqrt{2} \delta _{ik} \delta _{j \ell }$ . It is shown in ref. [Reference Degond18, Lemma 2.2] that $\tilde \Delta$ coincides with the Laplacian $\Delta$ defined by (2.7). This gives a justification for the last term in (2.11). We note that (2.11) is a nonlinear Fokker-Planck equation, where the nonlinearity arises in the third term.

The proof of the convergence of the particle system (2.8), (2.10) to the kinetic model (2.11), (2.12) is still open. The difficulty is in the presence of the projection operator $\mathcal P$ in (2.10) which requires to control that the determinant of $\tilde J_k$ remains positive. In the Vicsek case where a similar singular behaviour is observed, a local-in-time convergence result is shown in ref. [Reference Briant, Diez and Merino-Aceituno11]. This result supports the conjecture that System (2.8), (2.10) converges to System (2.11), (2.12) in the limit $N \to \infty$ in small time. We will assume it.

2.4. Scaling and statement of the problem

Let $t_0$ be a time scale and define the spatial scale $x_0 = c_0 t_0$ . We introduce the following dimensionless parameters:

\begin{equation*} \tilde D = D t_0, \qquad \tilde R = \frac {R}{x_0}, \qquad \kappa = \frac {\nu }{D}. \end{equation*}

Then, we change variables and unknowns to dimensionless variables $x' = x/x_0$ , $t'=t/t_0$ and unknowns $f'(x',A,t') = x_0^n f(x,A,t)$ , $\tilde J'_{f'}(x',t') = x_0^n \tilde J_f(x,t)$ . Inserting these changes into (2.11), (2.12) leads to (omitting the primes for simplicity):

(2.13) \begin{eqnarray} && \partial _t f + A{\textbf{e}}_1 \cdot \nabla _x f + \tilde D \big [ \kappa \nabla _A \cdot \big ( P_{T_A} ({\mathcal P} (\tilde J_f)) f \big ) - \Delta _A f \big ] = 0, \end{eqnarray}

(2.14) \begin{eqnarray} && \tilde J_f(x,t) = \frac{1}{\tilde R^n} \int _{{\mathbb R}^n \times \mathrm{SO}_n} K \Big ( \frac{|x-y|}{\tilde R} \Big ) \, f(y, B, t) \, B\, dy \, dB, \end{eqnarray}

We introduce a small parameter $\varepsilon \ll 1$ and make the scaling assumption $ \frac{1}{\tilde D} = \tilde R = \varepsilon$ , while $\kappa$ is kept of order $1$ . By Taylor’s formula, we have $\tilde J_f = J_f +{\mathcal O}(\varepsilon ^2)$ where $J_f(x,t) = \int _{\mathrm{SO}_n} A \, f(x, A, t) \, dA$ . Since the map $\mathcal P$ is smooth on $\mathrm{M}_n^+$ , we get ${\mathcal P}(\tilde J_f) ={\mathcal P}(J_f) +{\mathcal O}(\varepsilon ^2)$ . Inserting these scaling assumptions and neglecting the ${\mathcal O}(\varepsilon ^2)$ terms in the above expansions (because they would have no influence on the result), we get the following perturbation problem:

(2.15) \begin{eqnarray} && \partial _t f^\varepsilon + A{\textbf{e}}_1 \cdot \nabla _x f^\varepsilon = \frac{1}{\varepsilon } \big [ - \kappa \nabla _A \cdot \big ( P_{T_A} ({\mathcal P} (J_{f^\varepsilon })) f^\varepsilon \big ) + \Delta _A f^\varepsilon \big ], \end{eqnarray}

(2.16) \begin{eqnarray} && J_f(x,t) = \int _{\mathrm{SO}_n} f(x, A, t) \, A\, dA. \end{eqnarray}

The goal of this paper is to provide the formal limit $\varepsilon \to 0$ of this problem. This problem is referred to as the hydrodynamic limit of the Fokker-Planck equation (2.15).

3.1. Statement of the results

We will need the following

Definition 3.1 (von Mises distribution). Let $\Gamma \in \mathrm{SO}_n$ and $\kappa \gt 0$ . The function $M_\Gamma$ : $\mathrm{SO}_n \to [0,\infty )$ such that

(3.1) \begin{equation} M_\Gamma (A) = \frac{1}{Z} \exp (\kappa \Gamma \cdot A), \quad Z = \int _{\mathrm{SO}_n} \exp (\kappa \mathrm{Tr} (A)/2) \, dA, \end{equation}

is called the von Mises distribution of orientation $\Gamma$ and concentration parameter $\kappa$ . It is the density of a probability measure on $\mathrm{SO}_n$ .

We note that $\int M_\Gamma (A) \, dA= \int \exp (\kappa \mathrm{Tr} (A)/2) \, dA$ does not depend on $\Gamma$ thanks to the translation invariance of the Haar measure.

The first main result of this paper is about the limit of the scaled kinetic System (2.15), (2.16) when $\varepsilon \to 0$ . We will need the following notations: for two vector fields $X=(X_i)_{i=1}^n$ and $Y=(Y_i)_{i=1}^n$ , we define the antisymmetric matrices $X \wedge Y = (X \wedge Y)_{ij}$ and $\nabla _x \wedge X = (\nabla _x \wedge X)_{ij}$ by

\begin{equation*} (X \wedge Y)_{ij} = X_i Y_j - X_j Y_i, \quad (\nabla _x \wedge X)_{ij} = \partial _{x_i} X_j - \partial _{x_j} X_i, \quad \forall i, \, j \in \{1, \ldots n\}. \end{equation*}

Then, we have

Theorem 3.2. We suppose that there is a smooth solution $f^\varepsilon$ to System (2.15), (2.16). We also suppose that $f^\varepsilon \to f^0$ as $\varepsilon \to 0$ as smoothly as needed. Then, there exist two functions $\rho$ : ${\mathbb R}^n \times [0,\infty ) \to [0,\infty )$ and $\Gamma$ : ${\mathbb R}^n \times [0,\infty ) \to \mathrm{SO}_n$ such that

(3.2) \begin{equation} f^0(x,A,t) = \rho (x,t) M_{\Gamma (x,t)}(A). \end{equation}

Furthermore, for appropriate real constants $c_1, \ldots, c_4$ , $\rho$ and $\Gamma$ satisfy the following system of equations:

(3.3) \begin{eqnarray} && \partial _t \rho + \nabla _x (\rho c_1 \Omega _1) = 0, \end{eqnarray}

(3.4) \begin{eqnarray} && \rho \big ( \partial _t \Gamma + c_2 (\Omega _1 \cdot \nabla _x) \Gamma \big ) ={\mathbb W} \Gamma, \end{eqnarray}

where

(3.5) \begin{equation} \Omega _k(x,t) = \Gamma (x,t){\textbf{e}}_k, \quad k \in \{1, \ldots, n\}, \end{equation}

and where

(3.6) \begin{equation}{\mathbb W} = - c_3 \nabla _x \rho \wedge \Omega _1 - c_4 \rho \big [ \big ( \Gamma (\nabla _x \cdot \Gamma ) \big ) \wedge \Omega _1 + \nabla _x \wedge \Omega _1 \big ]. \end{equation}

The notation $\nabla _x \cdot \Gamma$ stands for the divergence of the matrix $\Gamma$ , i.e. $(\nabla _x \cdot \Gamma )_i = \sum _{j=1}^n \partial _{x_i} \Gamma _{ij}$ and $\Gamma (\nabla _x \cdot \Gamma )$ is the vector arising from multiplying the vector $\nabla _x \cdot \Gamma$ on the left by the matrix $\Gamma$ .

System (3.3), (3.4) has been referred to in previous works [Reference Degond, Diez and Frouvelle19, Reference Degond, Frouvelle, Merino-Aceituno and Trescases27] as the Self-Organised Hydrodynamic model for body-attitude coordination. It consists of coupled first-order partial differential equation for the particle density $\rho$ and the average body-attitude $\Gamma$ and has been shown to be hyperbolic in ref. [Reference Degond, Frouvelle, Merino-Aceituno and Trescases27]. It models the system of interacting rigid bodies introduced in Section 2.2 as a fluid of which (3.3) is the continuity equation. The velocity of the fluid is $c_1 \Omega _1$ . Equation (3.4) is an evolution equation for the averaged body orientation of the particles within a fluid element, described by $\Gamma$ . The left-hand side of (3.4) describes pure transport at velocity $c_2 \Omega _1$ . In general, $c_2 \not = c_1$ , which means that such transport occurs at a velocity different from the fluid velocity. The right-hand side appears as the multiplication of $\Gamma$ itself on the left by the antisymmetric matrix $\mathbb W$ , which is a classical feature of rigid-body dynamics. The first term of (3.6) is the action of the pressure gradient which contributes to rotating $\Gamma$ so as to align $\Omega _1$ with $- \nabla _x \rho$ . The second term has a similar effect with the pressure gradient replaced by the vector $\Gamma (\nabla _x \cdot \Gamma )$ which encodes gradients of the mean body-attitude $\Gamma$ . Finally, the last term encodes self-rotations of the averaged body frame about the self-propulsion velocity $\Omega _1$ . The last two terms do not have counterparts in classical fluid hydrodynamics. We refer to [Reference Degond, Diez and Frouvelle19, Reference Degond, Diez and Na21, Reference Degond, Frouvelle, Merino-Aceituno and Trescases27] for a more detailed interpretation.

The second main result of this paper is to provide explicit formulas for the coefficients $c_1, \ldots, c_4$ . For this, we need to present additional concepts. Let $p = \lfloor \frac{n}{2} \rfloor$ the integer part of $\frac{n}{2}$ (i.e. $n=2p$ or $n = 2p+1$ ) and

(3.7) \begin{equation} \epsilon _n = \left \{ \begin{array}{ccc} 0 & \mathrm{ if } & n=2p \\[5pt] 1 & \mathrm{ if } & n=2p +1 \end{array} \right. . \end{equation}

Let ${\mathcal T} = [\!-\!\pi, \pi )^p$ . Let $\Theta = (\theta _1, \ldots, \theta _p) \in{\mathcal T}$ . We introduce

(3.8) \begin{eqnarray} u_{2p} (\Theta ) &=& \prod _{1 \leq j\lt k \leq p} \big ( \cos \theta _j - \cos \theta _k \big )^2, \quad \text{ for } p \geq 2, \end{eqnarray}

(3.9) \begin{eqnarray} u_{2p+1} (\Theta ) &=& \prod _{1 \leq j\lt k \leq p} \big ( \cos \theta _j - \cos \theta _k \big )^2 \, \prod _{j=1}^p \sin ^2 \frac{\theta _j}{2}, \quad \text{ for } p \geq 1, \end{eqnarray}

and

(3.10) \begin{equation} m(\Theta ) = \exp \Big ( \frac{\kappa }{2} \big ( 2 \sum _{k=1}^p \cos \theta _k + \epsilon _n \big ) \Big ) \, u_n(\Theta ). \end{equation}

Let $\nabla _\Theta$ (resp. $\nabla _\Theta \cdot$ ) denote the gradient (resp. divergence) operators with respect to $\Theta$ of scalar (resp. vector) fields on $\mathcal T$ . Then, we define $\alpha$ : ${\mathcal T} \to{\mathbb R}^p$ , with $\alpha = (\alpha _i)_{i=1}^p$ as a periodic solution of the following system:

(3.11) \begin{eqnarray} && - \nabla _\Theta \cdot \big ( m \nabla _\Theta \alpha _\ell \big ) + m \sum _{k \not = \ell } \Big ( \frac{\alpha _\ell - \alpha _k}{1 - \cos\!(\theta _\ell - \theta _k)} + \frac{\alpha _\ell + \alpha _k}{1 - \cos\!(\theta _\ell + \theta _k)} \Big ) \nonumber \\[5pt] &&\qquad + \epsilon _n m \frac{\alpha _\ell }{1 - \cos \theta _\ell } = m \, \sin \theta _\ell, \quad \forall \ell \in \{1, \ldots, p \}, \end{eqnarray}

A functional framework which guarantees that $\alpha$ exists is unique and satisfies an extra invariance property (commutation with the Weyl group) will be provided in Section 6.

Remark 3.2. In the case of $\mathrm{SO}_3$ , we have $p=1$ and a single unknown $\alpha _1(\theta _1)$ . From (3.11), we get that $\alpha _1$ satisfies

\begin{equation*} - \frac {\partial }{\partial \theta _1} \Big ( m \frac {\partial \alpha _1}{\partial \theta _1} \Big ) + \frac {m \, \alpha _1}{1 - \cos \theta _1} = m \sin \theta _1. \end{equation*}

We can compare this equation with [Reference Degond, Frouvelle and Merino-Aceituno24, Eq. (4.16)] and see that $\alpha _1$ coincides with the function $- \sin \theta \, \tilde \psi _0$ of [Reference Degond, Frouvelle and Merino-Aceituno24].

Thanks to these definitions, we have the

Theorem 3.3. The constants $c_1, \ldots, c_4$ involved in (3.3), (3.4), (3.6) are given by

(3.12) \begin{eqnarray} && c_1 = \frac{1}{n} \frac{\displaystyle \int \Big ( 2 \sum \cos \theta _k + \epsilon _n \Big ) \, m(\Theta ) \, d \Theta }{\displaystyle \int \, m(\Theta ) \, d \Theta }, \end{eqnarray}

(3.13) \begin{eqnarray} && c_2 = - \frac{1}{n^2-4} \times \nonumber \\[5pt] && \frac{\displaystyle \int \Big [ -n \big ( \sum \alpha _k \sin \theta _k \big ) \big ( 2 \sum \cos \theta _k + \epsilon _n \big ) + 4 \big ( \sum \alpha _k \sin \theta _k \cos \theta _k \big ) \Big ] m(\Theta ) d \Theta }{\displaystyle \int \big ( \sum \alpha _k \sin \theta _k \big ) m(\Theta ) d \Theta }, \end{eqnarray}

(3.14) \begin{eqnarray} && c_3 = \frac{1}{\kappa }, \end{eqnarray}

(3.15) \begin{eqnarray} && c_4 = - \frac{1}{n^2-4} \times \nonumber \\[5pt] && \frac{\displaystyle \int \Big [ - \big ( \sum \alpha _k \sin \theta _k \big ) \big ( 2 \sum \cos \theta _k + \epsilon _n \big ) + n \big ( \sum \alpha _k \sin \theta _k \cos \theta _k \big ) \Big ] m(\Theta ) d \Theta }{\displaystyle \int \big ( \sum \alpha _k \sin \theta _k \big ) m(\Theta ) d \Theta }, \end{eqnarray}

where the integrals are over $\Theta = (\theta _1, \ldots, \theta _p) \in{\mathcal T}$ and the sums over $k \in \{1, \ldots, p \}$ .

We note that these formulas make $c_1, \ldots, c_4$ explicitely computable, at the expense of the resolution of System (3.11) and the computations of the integrals involved in the formulas above. In particular, it may be possible to compute numerical approximations of them for not-too-large values of $p$ . For large values of $p$ , analytical approximations will be required. Approximations of $\alpha$ may be obtained by considering the variational formulation (6.40) and restricting the unknown and test function spaces to appropriate (possibly finite-dimensional) subspaces.

The main objective of this paper is to prove Theorems 3.2 and 3.3. While the remainder of Section 3, as well as Section 4 rely on the same framework as [Reference Degond, Frouvelle and Merino-Aceituno24], the subsequent sections require completely new methodologies.

3.2. Equilibria

For $f$ : $\mathrm{SO}_n \to{\mathbb R}$ smooth enough, we define the collision operator

(3.16) \begin{equation} Q(f) = - \kappa \nabla _A \cdot \big ( P_{T_A} ({\mathcal P} (J_f)) f \big ) + \Delta _A f \quad \mathrm{with} \quad J_f = \int _{\mathrm{SO}_n} f \, A \, dA, \end{equation}

so that (2.15) can be recast into

(3.17) \begin{equation} \partial _t f^\varepsilon + A{\textbf{e}}_1 \cdot \nabla _x f^\varepsilon = \frac{1}{\varepsilon } Q(f^\varepsilon ). \end{equation}

It is clear that if $f^\varepsilon \to f^0$ as $\varepsilon \to 0$ strongly as well as all its first-order derivatives with respect to $x$ and second-order derivatives with respect to $A$ , then, we must have $Q(f^0) = 0$ . Solutions of this equation are called equilibria.

We have the following lemma whose proof can be found in ref. [Reference Degond, Diez and Frouvelle19, Appendix 7] and is not reproduced here:

Lemma 3.4. We have

\begin{equation*} \int _{\mathrm {SO}_n} A \, M_\Gamma (A) \, dA = c_1 \Gamma, \quad c_1 = \Big \langle \frac { \mathrm {Tr}(A)}{n} \Big \rangle _{\exp (\kappa \mathrm {Tr} (A)/2)}, \end{equation*}

where for two functions $f$ , $g$ : $\mathrm{SO}_n \to{\mathbb R}$ , with $g \geq 0$ and $g \not \equiv 0$ , we note

\begin{equation*} \langle f(A) \rangle _{g(A)} = \frac {\int _{\mathrm {SO}_n} f(A) \, g(A) \, dA}{\int _{\mathrm {SO}_n} g(A) \, dA}. \end{equation*}

The function $c_1$ : ${\mathbb R} \to{\mathbb R}$ , $\kappa \mapsto c_1(\kappa )$ is a nonnegative, nondecreasing function which satisfies $c_1(0) = 0$ , and $\lim _{\kappa \to \infty } c_1(\kappa ) = 1$ . It is given by (3.12).

From now on, we will use $\Gamma _f ={\mathcal P}(J_f)$ . We have different expressions of $Q$ expressed in the following

Lemma 3.5. We have

(3.18) \begin{eqnarray} Q(f) &=& - \kappa \nabla _A \cdot \big ( P_{T_A} \Gamma _f f \big ) + \Delta _A f \end{eqnarray}

(3.19) \begin{eqnarray} &=& \nabla _A \cdot \Big [ M_{\Gamma _f} \nabla _A \Big ( \frac{f}{M_{\Gamma _f}} \Big ) \Big ] \end{eqnarray}

(3.20) \begin{eqnarray} &=& \nabla _A \cdot \Big [ f \nabla _A \big (\!-\! \kappa \Gamma _f \cdot A + \log f \big ) \Big ]. \end{eqnarray}

Proof. Formula (3.18) is nothing but (3.16) with $\Gamma _f$ in place of ${\mathcal P}(J_f)$ . To get the other two expressions, we note that

(3.21) \begin{equation} \nabla _A (\Gamma \cdot A) = P_{T_A} \Gamma, \quad \forall A, \, \Gamma \in \mathrm{SO}_n. \end{equation}

Then, (3.20) follows immediately and for (3.19) we have

\begin{eqnarray*} \nabla _A \cdot \Big [ M_{\Gamma _f} \nabla _A \Big ( \frac{f}{M_{\Gamma _f}} \Big ) \Big ] &=& \Delta _A f - \nabla _A \cdot \big [f \nabla _A \big ( \log (M_{\Gamma _f}) \big ) \big ] \\[5pt] &=& \Delta _A f - \kappa \nabla _A \cdot \big [f P_{T_{A}} \Gamma _f \big ] = Q(f), \end{eqnarray*}

which ends the proof.

The following gives the equilibria of $Q$ :

Lemma 3.6. Let $f$ : $\mathrm{SO}_n \to{\mathbb R}$ be smooth enough such that $f \geq 0$ , $\rho _f \gt 0$ and $\mathrm{det} J_f \gt 0$ . Then, we have

\begin{equation*} Q(f) = 0 \Longleftrightarrow \exists (\rho,\Gamma ) \in (0,\infty ) \times \mathrm {SO}_n \text { such that } f = \rho M_{\Gamma }. \end{equation*}

Proof. Suppose $f$ fulfils the assumptions of the lemma and is such that $Q(f)=0$ . Using (3.19) and Stokes’ theorem, this implies:

\begin{equation*} 0 = \int _{\mathrm {SO}_n} Q(f) \frac {f}{M_{\Gamma _f}} \, dA = - \int _{\mathrm {SO}_n} \Big | \nabla _A \Big ( \frac {f}{M_{\Gamma _f}} \Big ) \Big |^2 \, M_{\Gamma _f} \, dA. \end{equation*}

Hence, $f/M_{\Gamma _f}$ is constant. So, there exists $\rho \in (0,\infty )$ such that $f = \rho M_{\Gamma _f}$ which shows that $f$ is of the form $\rho M_{\Gamma }$ for some $(\rho,\Gamma ) \in (0,\infty ) \times \mathrm{SO}_n$ .

Conversely, let $f = \rho M_\Gamma$ for some $(\rho,\Gamma ) \in (0,\infty ) \times \mathrm{SO}_n$ . If we show that $\Gamma = \Gamma _f$ , then, by (3.19) we deduce that $Q(f) = 0$ . By Lemma 3.4, we have $J_{\rho M_\Gamma } = \rho c_1 \Gamma$ with $\rho c_1 \gt 0$ . Thus, $\Gamma _f ={\mathcal P}(\rho c_1 \Gamma ) = \Gamma$ , which ends the proof. Note that knowing that $c_1 \gt 0$ is crucial in that step of the proof.

Now, we are looking for the equations satisfied by $\rho$ and $\Gamma$ .

3.3. The continuity equation

Proof. By Stokes’s theorem, for all second-order differentiable function $f$ : $\mathrm{SO}_n \to{\mathbb R}$ , we have $ \int _{\mathrm{SO}_n} Q(f) \, dA = 0$ . Therefore, integrating (3.17) with respect to $A$ , we obtain,

(3.22) \begin{equation} \int _{\mathrm{SO}_n} (\partial _t + A{\textbf{e}}_1 \cdot \nabla _x) f^\varepsilon \, dA = 0. \end{equation}

For any distribution function $f$ , we define

\begin{equation*} \rho _f(x,t) = \int _{\mathrm {SO}_n} f(x,A,t) \, dA. \end{equation*}

Thus, with (2.16), Eq. (3.22) can be recast into

(3.23) \begin{equation} \partial _t \rho _{f^\varepsilon } + \nabla _x \cdot (J_{f^\varepsilon }{\textbf{e}}_1) = 0. \end{equation}

Now, given that the convergence of $f^\varepsilon \to f^0$ as $\varepsilon \to 0$ is supposed strong enough, and thanks to Lemma 3.4, we have

\begin{equation*} \rho _{f^\varepsilon } \to \rho _{f^0} = \rho _{\rho M_\Gamma } = \rho, \qquad J_{f^\varepsilon } \to J_{f^0} = J_{\rho M_\Gamma } = \rho c_1 \Gamma. \end{equation*}

Then, passing to the limit $\varepsilon \to 0$ in (3.23) leads to (3.3).

4.1. Definition and first characterisation

Now, we need an equation for $\Gamma$ . We see that the proof of Prop. 3.8 can be reproduced if we can find functions $\psi$ : $\mathrm{SO}_n \to{\mathbb R}$ such that for all second-order differentiable function $f$ : $\mathrm{SO}_n \to{\mathbb R}$ , we have

(4.1) \begin{equation} \int _{\mathrm{SO}_n} Q(f) \, \psi \, dA = 0. \end{equation}

Such a function is called a collision invariant. In the previous proof, the collision invariant $\psi = 1$ was used. Unfortunately, it can be verified that the only collision invariants of $Q$ are the constants. Thus, the previous proof cannot be reproduced to find an equation for $\Gamma$ . In order to find more equations, we have to relax the condition that (4.1) must be satisfied for all functions $f$ . This leads to the concept of generalised collision invariant (GCI). We first introduce a few more definitions.

Given $\Gamma \in \mathrm{SO}_n$ , we define the following linear Fokker-Planck operator, defined for second-order differentiable functions $f$ : $\mathrm{SO}_n \to{\mathbb R}$ :

\begin{equation*} {\mathcal Q}(f,\Gamma ) = \nabla \cdot \Big [ M_\Gamma \nabla \Big ( \frac {f}{M_\Gamma } \Big ) \Big ]. \end{equation*}

For simplicity, in the remainder of the present section as well as in Sections 5 and 6, we will drop the subscript $A$ to the $\nabla$ , $\nabla \cdot$ and $\Delta$ operators as all derivatives will be understood with respect to $A$ .

We note that

(4.2) \begin{equation} Q(f) ={\mathcal Q}(f,\Gamma _f). \end{equation}

Definition 4.1. Given $\Gamma \in \mathrm{SO}_n$ , a GCI associated with $\Gamma$ is a function $\psi$ : $\mathrm{SO}_n \to{\mathbb R}$ such that

(4.3) \begin{equation} \int _{\mathrm{SO}_n}{\mathcal Q}(f,\Gamma ) \, \psi \, dA = 0 \quad \text{for all} \quad f: \, \mathrm{SO}_n \to{\mathbb R} \quad \text{such that} \quad P_{T_\Gamma } J_f = 0. \end{equation}

The set ${\mathcal G}_\Gamma$ of GCI associated with $\Gamma$ is a vector space.

From this definition, we have the following lemma which gives a justification for why this concept is useful for the hydrodynamic limit.

Lemma 4.2. We have

(4.4) \begin{equation} \psi \in{\mathcal G}_{\Gamma _f} \quad \Longrightarrow \quad \int _{\mathrm{SO}_n} Q(f) \, \psi \, dA = 0. \end{equation}

Proof. By (4.2) and (4.3), it is enough to show that $P_{T_{\Gamma _f}} J_f = 0$ . But $\Gamma _f ={\mathcal P}(J_f)$ , so there exists a symmetric positive-definite matrix $S$ such that $J_f = \Gamma _f S$ . So,

\begin{equation*} P_{T_{\Gamma _f}} J_f = \Gamma _f \frac {\Gamma _f^T J_f - J_f^T \Gamma _f}{2} = \Gamma _f \frac {S-S^T}{2} = 0. \end{equation*}

The following lemma provides the equation solved by the GCI.

Lemma 4.3. The function $\psi$ : $\mathrm{SO}_n \to{\mathbb R}$ belongs to ${\mathcal G}_\Gamma$ if and only if $\exists P \in T_\Gamma$ such that

(4.5) \begin{equation} \nabla \cdot \big ( M_\Gamma \nabla \psi \big ) = P \cdot A \, M_\Gamma. \end{equation}

Proof. On the one hand, we can write

\begin{equation*} \int _{\mathrm {SO}_n} {\mathcal Q}(f,\Gamma ) \, \psi \, dA = \int _{\mathrm {SO}_n} f \, {\mathcal Q}^*(\psi,\Gamma ) \, dA, \end{equation*}

where ${\mathcal Q}^*(\cdot,\Gamma )$ is the formal $L^2$ -adjoint to ${\mathcal Q}(\cdot,\Gamma )$ and is given by

\begin{equation*} {\mathcal Q}^*(\psi,\Gamma ) = M_\Gamma ^{-1} \nabla \cdot \big ( M_\Gamma \nabla \psi \big ). \end{equation*}

On the other hand, we have

\begin{eqnarray*} P_{T_\Gamma } J_f =0 & \Longleftrightarrow & P_{T_\Gamma } \int _{\mathrm{SO}_n} f \, A \, dA = 0 \\[5pt] & \Longleftrightarrow & \int _{\mathrm{SO}_n} f \, A \cdot P \, dA = 0, \quad \forall P \in T_\Gamma \\[5pt] & \Longleftrightarrow & f \in \{ A \mapsto A \cdot P \, \, | \, \, P \in T_\Gamma \}^\bot \end{eqnarray*}

where the orthogonality in the last statement is meant in the $L^2$ sense. So, by (4.3), $\psi \in{\mathcal G}_\Gamma$ if and only if

\begin{equation*} \{ A \mapsto A \cdot P \, \, | \, \, P \in T_\Gamma \}^\bot \subset \{ {\mathcal Q}^*(\psi,\Gamma ) \}^\bot, \end{equation*}

or by taking orthogonals again, if and only if

(4.6) \begin{equation} \mathrm{Span} \{{\mathcal Q}^*(\psi,\Gamma ) \} \subset \{ A \mapsto A \cdot P \, \, | \, \, P \in T_\Gamma \}, \end{equation}

because both sets in (4.6) are finite-dimensional, hence closed. Statement (4.6) is equivalent to the statement that there exists $P \in T_\Gamma$ such that (4.5) holds true.

4.2. Existence and invariance properties

We now state an existence result for the GCI. First, we introduce the following spaces: $L^2(\mathrm{SO}_n)$ stands for the space of square integrable functions $f$ : $\mathrm{SO}_n \to{\mathbb R}$ endowed with the usual $L^2$ -norm $\|f\|^2_{L^2} = \int _{\mathrm{SO}_n} |f(A)|^2 \, dA$ . Then, we define $H^1(\mathrm{SO}_n) = \{ f \in L^2(\mathrm{SO}_n) \, \, | \, \, \nabla f \in L^2(\mathrm{SO}_n)\}$ (where $\nabla f$ is meant in the distributional sense), endowed with the usual $H^1$ -norm $\|f\|^2_{H^1} = \|f\|^2_{L^2} + \|\nabla f\|^2_{L^2}$ . Finally, $H^1_0(\mathrm{SO}_n)$ are the functions of $H^1(\mathrm{SO}_n)$ with zero mean, i.e. $f \in H^1_0(\mathrm{SO}_n) \Longleftrightarrow f \in H^1(\mathrm{SO}_n)$ and

(4.7) \begin{equation} \int _{\mathrm{SO}_n} f \, dA = 0. \end{equation}

We will solve (4.5) in the variational sense. We note that for $P \in T_\Gamma$ , we have

\begin{equation*} \int _{\mathrm {SO}_n} A \cdot P \, M_\Gamma (A) \, dA = c_1 \Gamma \cdot P = 0, \end{equation*}

i.e. the right-hand side of (4.5) satisfies (4.7). Hence, if $\psi$ is a smooth solution of (4.5) satisfying (4.7), it satisfies

(4.8) \begin{equation} \int _{\mathrm{SO}_n} M_\Gamma \, \nabla \psi \, \nabla \chi \, dA = - \int _{\mathrm{SO}_n} M_\Gamma \, A \cdot P \, \chi \, dA, \end{equation}

for all functions $\chi$ satisfying (4.7). This suggests to look for solutions of (4.8) in $H^1_0(\mathrm{SO}_n)$ . Indeed, we have

Proof. This is a classical application of Lax-Milgram’s theorem. We only need to verify that the bilinear form

\begin{equation*} a(\psi,\chi ) = \int _{\mathrm {SO}_n} M_\Gamma \, \nabla \psi \, \nabla \chi \, dA, \end{equation*}

is coercive on $H^1_0(\mathrm{SO}_n)$ . Since $\mathrm{SO}_n$ is compact, there exists $C\gt 0$ such that $M_\Gamma \geq C$ . So,

\begin{equation*} a(\psi,\psi ) \geq C \int _{\mathrm {SO}_n} |\nabla \psi |^2\, dA. \end{equation*}

This is the quadratic form associated with the Laplace operator $- \Delta$ on $\mathrm{SO}_n$ . But the lowest eigenvalue of $- \Delta$ is $0$ and its associated eigenspace are the constant functions. Then, there is a spectral gap and the next eigenvalue $\lambda _2$ is positive. Hence, we have

(4.9) \begin{equation} \int _{\mathrm{SO}_n} |\nabla \psi |^2\, dA \geq \lambda _2 \int _{\mathrm{SO}_n} |\psi |^2\, dA, \quad \forall \psi \in H^1_0(\mathrm{SO}_n), \end{equation}

(see e.g. [Reference Degond18, Section 4.3] for more detail). This implies the coercivity of $a$ on $H^1_0(\mathrm{SO}_n)$ and ends the proof.

For any $P \in T_\Gamma$ , there exists $X \in \mathfrak{so}_n$ such that $P = \Gamma X$ . We denote by $\psi _X^\Gamma$ the unique solution of (4.8) in $H^1_0(\mathrm{SO}_n)$ associated with $P = \Gamma X$ . Then, we have the

Corollary 4.5. The space ${\mathcal G}_\Gamma$ is given by

(4.10) \begin{equation}{\mathcal G}_\Gamma = \mathrm{Span} \big ( \{1\} \cup \{\psi _X^\Gamma \, \, | \, \, X \in \mathfrak{so}_n \} \big ), \end{equation}

and we have

(4.11) \begin{equation} \mathrm{dim} \,{\mathcal G}_\Gamma = \mathrm{dim} \, \mathfrak{so}_n + 1 = \frac{n(n-1)}{2}+1. \end{equation}

Proof. If $\psi \in{\mathcal G}_\Gamma$ , then, $\psi - \bar \psi \in{\mathcal G}_\Gamma \cap H^1_0(\mathrm{SO}_n)$ where $\bar \psi = \int _{\mathrm{SO}_n} \psi \, dA$ . Then, $\exists X \in \mathfrak{so}_n$ such that $\psi - \bar \psi = \psi _X^\Gamma$ , which leads to (4.10). Now, the map $\mathfrak{so}_n \to H^1_0(\mathrm{SO}_n)$ , $X \mapsto \psi _X^\Gamma$ is linear and injective. Indeed, suppose $\psi _X^\Gamma = 0$ . Then, inserting it into (4.8), we get that

\begin{equation*} \int _{\mathrm {SO}_n} M_\Gamma (A) \, A \cdot (\Gamma X) \, \chi (A) \, dA = 0, \quad \forall \chi \in H^1_0(\mathrm {SO}_n), \end{equation*}

and by density, this is still true for all $\chi \in L^2(\mathrm{SO}_n)$ . This implies that

\begin{equation*} M_\Gamma (A) \, A \cdot (\Gamma X) = 0, \quad \forall A \in \mathrm {SO}_n, \end{equation*}

and since $M_\Gamma \gt 0$ , that $A \cdot (\Gamma X) = (\Gamma ^T A) \cdot X = 0$ , for all $A \in \mathrm{SO}_n$ . Now the multiplication by $\Gamma ^T$ on the left is a bijection of $\mathrm{SO}_n$ , so we get that $X$ satisfies

\begin{equation*} A \cdot X = 0, \quad \forall A \in \mathrm {SO}_n. \end{equation*}

Then, taking $A = e^{tY}$ with $Y \in \mathfrak{so}_n$ and differentiating with respect to $t$ , we obtain

\begin{equation*} Y \cdot X = 0, \quad \forall Y \in \mathfrak {so}_n, \end{equation*}

which shows that $X=0$ . Hence, ${\mathcal G}_\Gamma$ is finite-dimensional and (4.11) follows.

From now on, we will repeatedly use the following lemma.

Lemma 4.6. (i) Let $g \in \mathrm{SO}_n$ and let $\ell _g$ , $r_g$ and $\xi _g$ be the left and right translations and conjugation maps of $\mathrm{SO}_n$ , respectively:

(4.12) \begin{equation} \ell _g(A) = gA, \quad r_g(A) = Ag, \quad \xi _g = \ell _g \circ r_{g^{-1}} = \ell _g \circ r_{g^T}, \quad \forall A \in \mathrm{SO}_n. \end{equation}

Let $f$ : $\mathrm{SO}_n \to{\mathbb R}$ be smooth. Then, we have for any $X \in \mathfrak{so}_n$ :

(4.13) \begin{eqnarray} \nabla (f \circ \ell _g) (A) \cdot AX &=& \nabla f (gA) \cdot gAX, \end{eqnarray}

(4.14) \begin{eqnarray} \nabla (f \circ \xi _g) (A) \cdot AX &=& \nabla f (gAg^T) \cdot gAXg^T. \end{eqnarray}

(ii) If $f$ and $\varphi$ : $\mathrm{SO}_n \to{\mathbb R}$ are smooth, then,

(4.15) \begin{eqnarray} \nabla (f \circ \ell _g) (A) \cdot \nabla (\varphi \circ \ell _g) (A) &=& \nabla f(gA) \cdot \nabla \varphi (gA), \end{eqnarray}

(4.16) \begin{eqnarray} \nabla (f \circ \xi _g) (A) \cdot \nabla (\varphi \circ \xi _g) (A) &=& \nabla f(gAg^T) \cdot \nabla \varphi (gAg^T), \end{eqnarray}

Proof. (i) We show (4.13). (4.14) is shown in a similar way. By (2.5), we have

\begin{equation*} \nabla (f \circ \ell _g) (A) \cdot AX = \frac {d}{dt} \big ( (f \circ \ell _g)(A e^{tX}) \big )|_{t=0} = \frac {d}{dt} \big ( f(g A e^{tX}) \big )|_{t=0} = \nabla f(g A) \cdot g A X. \end{equation*}

(ii) Again, we show (4.15), the proof of (4.16) being similar. Applying (4.13) twice, we have

\begin{eqnarray*} \nabla f(gA) \cdot \nabla \varphi (gA) &=& \nabla (f \circ \ell _g) (A) \cdot \big (g^T \nabla \varphi (gA) \big ) = \nabla \varphi (gA) \cdot \big ( g \nabla (f \circ \ell _g) (A) \big ) \\[5pt] &=& \nabla (\varphi \circ \ell _g) (A) \cdot \nabla (f \circ \ell _g) (A). \end{eqnarray*}

Proposition 4.7 (translation invariance). We have

(4.17) \begin{equation} \psi _X^{\mathrm{I}}(A) = \psi _X^\Gamma (\Gamma A), \quad \forall A, \, \Gamma \in \mathrm{SO}_n, \quad \forall X \in \mathfrak{so}_n. \end{equation}

Proof. $\psi = \psi _X^\Gamma$ is the unique solution in $H^1_0(\mathrm{SO}_n)$ of the following variational formulation:

\begin{eqnarray*} && \int _{\mathrm{SO}_n} \exp \big (\frac{\kappa }{2} \mathrm{Tr} (\Gamma ^T A) \big ) \, \nabla \psi (A) \cdot \nabla \chi (A) \, dA \\[5pt] &&\quad = - \frac{1}{2} \int _{\mathrm{SO}_n} \exp \big (\frac{\kappa }{2} \mathrm{Tr} (\Gamma ^T A) \big ) \, \mathrm{Tr} (A^T \Gamma X) \, \chi (A) \, dA, \quad \forall \chi \in H^1_0(\mathrm{SO}_n). \end{eqnarray*}

By the change of variables $A' = \Gamma ^T A$ , the translation invariance of the Haar measure and (4.15), we get, dropping the primes for simplicity:

(4.18) \begin{eqnarray} && \int _{\mathrm{SO}_n} \exp (\frac{\kappa }{2} \mathrm{Tr} A ) \, \nabla (\psi \circ \ell _\Gamma ) (A) \cdot \nabla (\chi \circ \ell _\Gamma ) (A) \, dA \nonumber \\[5pt] &&\quad = - \frac{1}{2} \int _{\mathrm{SO}_n} \exp (\frac{\kappa }{2} \mathrm{Tr} A ) \, \mathrm{Tr} (A^T X) \, \chi \circ \ell _\Gamma (A) \, dA, \quad \forall \chi \in H^1_0(\mathrm{SO}_n), \end{eqnarray}

We remark that the mapping $H^1_0(\mathrm{SO}_n) \to H^1_0(\mathrm{SO}_n)$ , $\chi \to \chi \circ \ell _\Gamma$ is a linear isomorphism and an isometry (the proof is analogous to the proof of Prop. 6.4 below and is omitted). Thus, we can replace $\chi \circ \ell _\Gamma$ in (4.18) by any test function $\tilde \chi \in H^1_0(\mathrm{SO}_n)$ , which leads to a variational formulation for $\psi \circ \ell _\Gamma$ which is identical with that of $\psi _X^{\mathrm{I}}$ . By the uniqueness of the solution of the variational formulation, this leads to (4.17) and finishes the proof.

Proposition 4.8 (Conjugation invariance). We have

(4.19) \begin{equation} \psi _X^{\mathrm{I}}(gAg^T) = \psi _{g^TXg}^{\mathrm{I}}(A), \quad \forall A, \, g \in \mathrm{SO}_n. \end{equation}

From this point onwards, the search for GCI differs significantly from [Reference Degond, Frouvelle and Merino-Aceituno24] where the assumption of dimension $n=3$ was crucial. We will need further concepts about the rotation groups which are summarised in the next section.

6.1. Introduction of $\boldsymbol\alpha = (\boldsymbol\alpha _\boldsymbol{i})_{\boldsymbol{i}=\textbf{1}}^\boldsymbol{p}$ and first properties

This section is devoted to the introduction of the function $\alpha = (\alpha _i)_{i=1}^p$ , ${\mathcal T} \to{\mathbb R}^p$ which will be eventually shown to solve System (3.11).

For simplicity, we denote $\psi _X^{\mathrm{I}}$ simply by $\psi _X$ and $M_{\mathrm{I}}$ by $M$ . Let $A, \, \Gamma \in \mathrm{SO}_n$ be fixed. The map $\mathfrak{so}_n \to{\mathbb R}$ , $X \mapsto \psi _X^\Gamma (A)$ is a linear form. Hence, there exists a map $\mu ^\Gamma$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ such that

(6.1) \begin{equation} \psi _X^\Gamma (A) = \mu ^\Gamma (A) \cdot X, \quad \forall A \in \mathrm{SO}_n, \quad \forall X \in \mathfrak{so}_n. \end{equation}

We abbreviate $\mu ^{\mathrm{I}}$ into $\mu$ .

We define $H^1_0(\mathrm{SO}_n, \mathfrak{so}_n)$ as the space of functions $\chi$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ such that each component of $\chi$ in an orthonormal basis of $\mathfrak{so}_n$ is a function of $H^1_0(\mathrm{SO}_n)$ (with similar notations for $L^2(\mathrm{SO}_n, \mathfrak{so}_n)$ and $H^1(\mathrm{SO}_n, \mathfrak{so}_n)$ ). Obviously, the definition does not depend on the choice of the orthonormal basis. Now, let $\chi \in H^1(\mathrm{SO}_n, \mathfrak{so}_n)$ . Then $\nabla \chi (A)$ (which is defined almost everywhere) can be seen as an element of $\mathfrak{so}_n \otimes T_A$ by the relation

\begin{equation*} \nabla \chi (A) \cdot (X \otimes AY) = \nabla (\chi \cdot X) (A) \cdot (AY), \quad \forall X, \, Y \in \mathfrak {so}_n, \end{equation*}

where we have used (2.2) to express an element of $T_A$ as $AY$ for $Y \in \mathfrak{so}_n$ and where we define the inner product on $\mathfrak{so}_n \otimes T_A$ by

\begin{equation*} \big ( X \otimes AY \cdot X' \otimes AY' \big ) = (X \cdot X') \, (A Y \cdot AY') = (X \cdot X') \, (Y \cdot Y'), \end{equation*}

for all $X, \, Y \, X', \, Y' \, \in \mathfrak{so}_n$ . With this identification, if $(\Phi _i)_{i=1}^{\mathcal N}$ and $(\Psi _i)_{i=1}^{\mathcal N}$ (with ${\mathcal N} = \frac{n(n-1)}{2}$ ) are two orthonormal bases of $\mathfrak{so}_n$ , then $(\Phi _i \otimes A \Psi _j)_{i,j = 1}^{\mathcal N}$ is an orthonormal basis of $\mathfrak{so}_n \otimes T_A$ and we can write

(6.2) \begin{equation} \nabla \chi (A) = \sum _{i,j=1}^{\mathcal N} \big ( \nabla (\chi \cdot \Phi _i)(A) \cdot (A \Psi _j) \big ) \, \Phi _i \otimes A \Psi _j. \end{equation}

Consequently, if $\mu \in H^1(\mathrm{SO}_n, \mathfrak{so}_n)$ is another function, we have, thanks to Parseval’s formula and (2.5):

(6.3) \begin{eqnarray} \nabla \mu (A) \cdot \nabla \chi (A) &=& \sum _{i,j=1}^{\mathcal N} \big ( \nabla (\mu \cdot \Phi _i) (A) \cdot A \Psi _j \big ) \, \big ( \nabla (\chi \cdot \Phi _i) (A) \cdot A \Psi _j \big ) \nonumber \\[5pt] &=& \sum _{i,j=1}^{\mathcal N} \Big ( \big ( \varrho (\Psi _j) (\mu \cdot \Phi _i) \big ) (A) \Big ) \, \Big ( \big ( \varrho (\Psi _j) (\chi \cdot \Phi _i) \big ) (A) \Big ). \end{eqnarray}

In general, we will use (6.3) with identical bases $(\Phi _i)_{i=1}^{\mathcal N} = (\Psi _i)_{i=1}^{\mathcal N}$ , but this is not necessary. The construction itself shows that these formulae are independent of the choice of the orthonormal bases of $\mathfrak{so}_n$ .

Now, we have the following properties:

Proposition 6.1 (Properties of $\mu$ ). (i) The function $\mu$ is the unique variational solution in $H^1_0(\mathrm{SO}_n, \mathfrak{so}_n)$ of the equation

(6.4) \begin{equation} M^{-1} \nabla \cdot \big ( M \, \nabla \mu \big ) (A) = \frac{A - A^T}{2}, \quad \forall A \in \mathrm{SO}_n, \end{equation}

where the differential operator at the left-hand side is applied componentwise. The variational formulation of (6.4) is given by

(6.5) \begin{equation} \left \{ \begin{array}{l} \displaystyle \mu \in H^1_0(\mathrm{SO}_n, \mathfrak{so}_n), \\[5pt] \displaystyle \int _{\mathrm{SO}_n} \nabla \mu \cdot \nabla \chi \, M \, dA = - \int _{\mathrm{SO}_n} \frac{\displaystyle A-A^T}{\displaystyle 2} \cdot \chi \, M \, dA, \quad \forall \chi \in H^1_0(\mathrm{SO}_n, \mathfrak{so}_n), \end{array} \right. \end{equation}

with the interpretation (6.3) of the left-hand side of (6.5).

(ii) We have (conjugation invariance):

(6.6) \begin{equation} \mu (gAg^T) = g \mu (A) g^T, \quad \forall A, \, g \in \mathrm{SO}_n. \end{equation}

(iii) We have (translation invariance):

(6.7) \begin{equation} \mu ^\Gamma (A) = \mu (\Gamma ^T A), \quad \forall \Gamma, \, A \in \mathrm{SO}_n. \end{equation}

Proof. (i) Since $X$ is antisymmetric, we have $A \cdot X = \frac{A-A^T}{2} \cdot X$ . Hence, (4.5) (with $\Gamma = \mathrm{I}$ ) can be written

\begin{equation*} \Big ( M^{-1} \nabla \cdot \big ( M \nabla \mu \big ) - \frac {A-A^T}{2} \Big ) \cdot X = 0, \quad \forall X \in \mathfrak {so}_n. \end{equation*}

Since the matrix to the left of the inner product is antisymmetric and the identity is true for all antisymmetric matrices $X$ , we find (6.4). We can easily reproduce the same arguments on the variational formulation (4.8) (with $\Gamma = \mathrm{I}$ ), which leads to the variational formulation (6.5) for $\mu$ .

(ii) (4.19) reads

\begin{equation*} \mu (gAg^T) \cdot X = \mu (A) \cdot (g^T X g) = \big ( g \mu (A) g^T \big ) \cdot X, \end{equation*}

hence (6.6).

(iii) (4.17) reads

\begin{equation*} \mu (A) \cdot X = \mu ^\Gamma (\Gamma A) \cdot X, \end{equation*}

hence (6.7).

The generic form of a function satisfying (6.6) is given in the next proposition.

Proposition 6.2. (i) Let $\chi$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ be a smooth map satisfying

(6.8) \begin{equation} \chi (gAg^T) = g \chi (A) g^T, \quad \forall A, \, g \in \mathrm{SO}_n. \end{equation}

Define $p$ such that $n=2p$ or $n=2p+1$ .

(i) There exists a $p$ -tuple $\tau = (\tau _i)_{i=1}^p$ of periodic functions $\tau _i$ : ${\mathcal T} \to{\mathbb R}$ such that

(6.9) \begin{equation} \chi (A_\Theta ) = \sum _{k=1}^p \tau _k(\Theta ) F_{2k-1 \, 2k}, \quad \forall \Theta \in{\mathcal T}. \end{equation}

Furthermore, $\tau$ commutes with the Weyl group, i.e.

(6.10) \begin{equation} \tau \circ W = W \circ \tau, \quad \forall W \in \mathfrak{W}. \end{equation}

(ii) $\chi$ has expression

(6.11) \begin{equation} \chi (A) = \sum _{k=1}^p \tau _k(\Theta ) \, gF_{2k-1 \, 2k}g^T, \end{equation}

where $g \in \mathrm{SO}_n$ and $\Theta \in{\mathcal T}$ are such that $A = g A_\Theta g^T$ .

Proof. (i) Let $A$ , $g \in{\mathbb T}$ . Then, since $\mathbb T$ is abelian, $gAg^T=A$ and (6.8) reduces to

(6.12) \begin{equation} g \chi (A) g^T = \chi (A), \quad \forall A, \, g \in{\mathbb T}. \end{equation}

Fixing $A$ , letting $g = e^{tX}$ with $X \in \mathfrak{h}$ and differentiating (6.12) with respect to $t$ , we get

\begin{equation*} [X, \chi (A)] = 0, \quad \forall X \in \mathfrak {h}. \end{equation*}

This means that $\mathfrak{h} \oplus (\chi (A) \,{\mathbb R})$ is an abelian subalgebra of $\mathfrak{so}_n$ . But $\mathfrak{h}$ is a maximal subalgebra of $\mathfrak{so}_n$ . So, it implies that $\chi (A) \in \mathfrak{h}$ . Hence, there exist functions $\tau _i$ : ${\mathcal T} \to{\mathbb R}$ , $\Theta \mapsto \tau _i(\Theta )$ for $i=1, \ldots, p$ such that (6.9) holds.

Now, we show (6.10) on a set of generating elements of $\mathfrak{W}$ . For this, we use (6.8) with such generating elements $g$ , reminding that $\mathfrak{W} \approx N({\mathbb T})/{\mathbb T}$ as described in Section 5. We distinguish the two parity cases of $n$ .

Case $n=2p$ even. In this case, the Weyl group is generated by $(C_{ij})_{1 \leq i \lt j \leq p}$ and $(D_{ij})_{1 \leq i \lt j \leq p}$ (see Section 5). First, we take $g = C_{ij}$ as defined in Section 5. Then, conjugation by $C_{ij}$ exchanges $F_{2i-1 \, 2i}$ and $F_{2j-1 \, 2j}$ and leaves $F_{2k-1 \, 2k}$ for $k \not = i, \, j$ invariant. On the other hand conjugation by $C_{ij}$ changes $A_\Theta$ into $A_{C_{ij} \Theta }$ where we recall that, by abuse of notation, we also denote by $C_{ij}$ the transformation of $\Theta$ generated by conjugation by $C_{ij}$ . Thus, from (6.8) and (6.9) we get

\begin{equation*} \chi (A_{C_{ij} \Theta }) = \sum _{k \not = i, \, j} \tau _k (\Theta ) F_{2k-1 \, 2k} + \tau _i(\Theta ) F_{2j-1 \, 2j} + \tau _j(\Theta ) F_{2i-1 \, 2i}. \end{equation*}

On the other hand, direct application of (6.9) leads to

\begin{equation*} \chi (A_{C_{ij} \Theta }) = \sum _{k=1}^p \tau _k (C_{ij} \Theta ) F_{2k-1 \, 2k}. \end{equation*}

Equating these two expressions leads to

\begin{eqnarray*} \tau _k \big ( C_{ij} (\Theta )\big ) &=& \tau _k (\Theta ), \quad \forall k \not = i, \, j. \\[5pt] \tau _i \big ( C_{ij} (\Theta )\big ) &=& \tau _j (\Theta ), \qquad \tau _j \big ( C_{ij} (\Theta )\big ) = \tau _i (\Theta ), \end{eqnarray*}

Hence, we get

(6.13) \begin{equation} \tau \big ( C_{ij}(\Theta ) \big ) = C_{ij} \big ( \tau (\Theta ) \big ). \end{equation}

Next, we take $g = D_{ij}$ . Conjugation by $D_{ij}$ changes $F_{2i-1 \, 2i}$ into $-F_{2i-1 \, 2i}$ and $F_{2j-1 \, 2j}$ into $-F_{2j-1 \, 2j}$ and leaves $F_{2k-1 \, 2k}$ for $k \not = i, \, j$ invariant. Besides, conjugation by $D_{ij}$ changes $A_\Theta$ into $A_{D_{ij} \Theta }$ . Thus, using the same reasoning as previously, we get

\begin{eqnarray*} \tau _k \big ( D_{ij} \Theta \big ) &=& \tau _k ( \Theta ), \quad \forall k \not = i, \, j, \\[5pt] \tau _i \big ( D_{ij} \Theta \big ) &=& - \tau _i ( \Theta ), \qquad \tau _j \big ( D_{ij} \Theta \big ) = - \tau _j ( \Theta ). \end{eqnarray*}

Hence, we find

\begin{equation*} \tau \big ( D_{ij}(\Theta ) \big ) = D_{ij} \big ( \tau (\Theta ) \big ). \end{equation*}

Case $n=2p+1$ odd. Here, the Weyl group is generated by $(C_{ij})_{1 \leq i \lt j \leq p}$ and $(D_i)_{1 =1}^p$ . Taking $g=C_{ij}$ as in the previous case, we get (6.13) again. Now, taking $g=D_i$ , conjugation by $D_i$ changes $F_{2i-1 \, 2i}$ into $-F_{2i-1 \, 2i}$ and leaves $F_{2k-1 \, 2k}$ for $k \not = i$ invariant. Besides, conjugation by $g$ changes $A_\Theta$ into $A_{D_i \Theta }$ . Thus, we get

\begin{eqnarray*} \tau _k \big ( D_i \Theta \big ) &=& \tau _k (\Theta ), \quad \forall k \not = i, \\[5pt] \tau _i \big ( D_i \Theta \big ) &=& - \tau _i ( \Theta ). \end{eqnarray*}

Thus, we finally get

\begin{equation*} \tau \big ( D_i(\Theta ) \big ) = D_i \big ( \tau (\Theta ) \big ), \end{equation*}

which ends the proof.

(ii) The fact that $\tau$ commutes with the Weyl group guarantees that formula (6.11) is well-defined, i.e. if $(g,\Theta )$ and $(g',\Theta ')$ are two pairs in $\mathrm{SO}_n \times{\mathcal T}$ such that $A = g A_\Theta g^T = g' A_{\Theta '}{g'}^T$ , then, the two expressions (6.11) deduced from each pair are the same. Applying (6.8) to (6.9) shows that (6.11) is necessary. It can be directly verified that (6.11) satisfies (6.8) showing that it is also sufficient.

The following corollary is a direct consequence of the previous discussion:

Corollary 6.3. Let $\mu$ be the solution of the variational formulation (6.4). Then, there exists $\alpha = (\alpha _i)_{i=1}^p$ : ${\mathcal T} \to{\mathbb R}^p$ such that

(6.14) \begin{equation} \mu (A_\Theta ) = \sum _{k=1}^p \alpha _k(\Theta ) F_{2k-1 \, 2k}, \quad \forall \Theta \in{\mathcal T}, \end{equation}

and $\alpha$ commutes with the Weyl group,

\begin{equation*} \alpha \circ W = W \circ \alpha, \quad \forall W \in \mathfrak {W}. \end{equation*}

Now, we wish to derive a system of PDEs for $\alpha$ . There are two ways to achieve this aim.

• - The first one consists of deriving the system in strong form by directly computing the differential operators involved in (6.4) at a point $A_\Theta$ of the maximal torus $\mathbb T$ . This method applies the strategy exposed in ref. [Reference Faraut30, Section 8.3] and used in ref. [Reference Degond18] to derive expressions of the radial Laplacian on rotation groups. However, this method does not give information on the well-posedness of the resulting system. We develop this method in Appendix A for the interested reader and as a cross-validation of the following results.

• - The second method, which is developed below, consists of deriving the system in weak form, using the variational formulation (6.5). We will show that we can restrict the space of test functions $\chi$ to those satisfying the invariance relation (6.8). This will allow us to derive a variational formulation for the system satisfied by $\alpha$ which will lead us to its well-posedness and eventually to the strong form of the equations.

6.2. Reduction to a conjugation-invariant variational formulation

We first define the following spaces:

\begin{eqnarray*} L^2_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n) &=& \big \{ \chi \in L^2(\mathrm{SO}_n, \mathfrak{so}_n) \text{ such that } \\[5pt] && \qquad \chi \text{ satisfies (6.8) a.e. } A \in \mathrm{SO}_n, \, \, \forall g \in \mathrm{SO}_n \big \}, \\[5pt] H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n) &=& \big \{ \chi \in H^1(\mathrm{SO}_n, \mathfrak{so}_n) \text{ such that } \\[5pt] && \qquad \chi \text{ satisfies (6.8) a.e. } A \in \mathrm{SO}_n \, \, \forall g \in \mathrm{SO}_n \big \}, \end{eqnarray*}

where $\mathrm{a.e.}$ stands for ‘for almost every’. Concerning these spaces, we have the

Proof. (i) Let $g \in \mathrm{SO}_n$ . We introduce the conjugation map $\Xi _g$ mapping any function $\chi$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ to another function $\Xi _g \chi$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ such that

\begin{equation*} \Xi _g \chi (A) = g^T \chi (gAg^T) g, \quad \forall A \in \mathrm {SO}_n.\end{equation*}

We prove that $\Xi _g$ is an isometry of $L^2(\mathrm{SO}_n, \mathfrak{so}_n)$ and of $H^1(\mathrm{SO}_n, \mathfrak{so}_n)$ for any $g \in \mathrm{SO}_n$ . The result follows as

\begin{equation*} L^2_{\mathrm {inv}}(\mathrm {SO}_n, \mathfrak {so}_n) = \bigcap _{g \in \mathrm {SO}_n} \mathrm {ker}_{L^2} (\Xi _g - \mathrm {I}), \quad H^1_{\mathrm {inv}}(\mathrm {SO}_n, \mathfrak {so}_n) = \bigcap _{g \in \mathrm {SO}_n} \mathrm {ker}_{H^1} (\Xi _g - \mathrm {I}). \end{equation*}

Thanks to the cyclicity of the trace and the translation invariance of the Haar measure, we have

\begin{eqnarray*} \int _{\mathrm{SO}_n} |\Xi _g \chi (A)|^2 \, dA &=& \int _{\mathrm{SO}_n} \big ( g^T \chi (gAg^T) g \big ) \cdot \big ( g^T \chi (gAg^T) g \big ) \, dA \\[5pt] &=& \int _{\mathrm{SO}_n} |\chi (gAg^T)|^2 \, dA = \int _{\mathrm{SO}_n} |\chi (A)|^2 \, dA, \end{eqnarray*}

where, for $X \in \mathfrak{so}_n$ , we have denoted by $|X| = (X \cdot X)^{1/2}$ the Euclidean norm on $\mathfrak{so}_n$ . This shows that $\Xi _g$ is an isometry of $L^2(\mathrm{SO}_n, \mathfrak{so}_n)$ .

Now, with (6.3) and (6.8), we have

\begin{eqnarray*} |\nabla (\Xi _g \chi ) (A)|^2 &=& \frac{1}{2} \sum _{i,j=1}^n \nabla (\Xi _g \chi )_{ij} \cdot \nabla (\Xi _g \chi )_{ij} \\[5pt] &=& \frac{1}{2} \sum _{i,j,k,\ell,k^{\prime},\ell ^{\prime}=1}^n g_{ki} \, g_{\ell j} \, g_{k^{\prime} i} \, g_{\ell ^{\prime} j} \, \nabla (\chi _{k \ell } \circ \xi _g)(A) \cdot \nabla (\chi _{k^{\prime} \ell ^{\prime}} \circ \xi _g)(A) \\[5pt] &=& \frac{1}{2} \sum _{k,\ell =1}^n |\nabla \chi _{k \ell } (gAg^T)|^2 = |\nabla \chi (gAg^T)|^2, \end{eqnarray*}

where we used that $\sum _{i=1}^n g_{ki} g_{k^{\prime}i} = \delta _{k k^{\prime}}$ and similarly for the sum over $j$ , as well as (4.16). Now, using the translation invariance of the Haar measure, we get

\begin{equation*} \int _{\mathrm {SO}_n} |\nabla (\Xi _g \chi ) (A)|^2 \, dA = \int _{\mathrm {SO}_n} |\nabla \chi (A)|^2 \, dA, \end{equation*}

which shows that $\Xi _g$ is an isometry of $H^1(\mathrm{SO}_n, \mathfrak{so}_n)$ .

(ii) We use the fact that, for any integrable function $f$ : $\mathrm{SO}_n \to \mathfrak{so}_n$ , we have

(6.15) \begin{equation} \int _{\mathrm{SO}_n} f(A) \, dA = \int _{\mathrm{SO}_n} \Big ( \int _{\mathrm{SO}_n} f(gAg^T) \, dg \Big ) dA. \end{equation}

Let $\chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . With (6.8), we have

\begin{equation*} \int _{\mathrm {SO}_n} \chi (gAg^T) \, dg = \int _{\mathrm {SO}_n} g \chi (A) g^T\, dg. \end{equation*}

We define the linear map $T$ : $\mathfrak{so}_n \to{\mathbb R}$ , $ X \mapsto \int _{\mathrm{SO}_n} (g \chi (A) g^T) \, dg \cdot X$ . By translation invariance of the Haar measure, we have $T(hXh^T) = T(X)$ , for all $h \in \mathrm{SO}_n$ . Thus, $T$ intertwines the representation $\mathfrak{so}_n$ of $\mathrm{SO}_n$ (i.e. the adjoint representation $\mathrm{Ad}$ ) and its trivial representation $\mathbb R$ . $\mathrm{Ad}$ is irreducible except for dimension $n=4$ where it decomposes into two irreducible representations. Neither of these representations is isomorphic to the trivial representation. Then, by Schur’s Lemma, $T=0$ . This shows that $\int _{\mathrm{SO}_n} \chi (gAg^T) \, dg =0$ and by application of (6.15), that $\int _{\mathrm{SO}_n} \chi (A) \, dA =0$ .

We now show that the space $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ may replace $H^1_0(\mathrm{SO}_n, \mathfrak{so}_n)$ in the variational formulation giving $\mu$ . More specifically, we have the

Proposition 6.5. (i) $\mu$ is the unique solution of the variational formulation (6.5) if and only if it is the unique solution of the variational formulation

(6.16) \begin{equation} \left \{ \begin{array}{l} \displaystyle \mu \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n), \\[5pt] \displaystyle \int _{\mathrm{SO}_n} \nabla \mu \cdot \nabla \chi \, M \, dA = - \int _{\mathrm{SO}_n} \frac{A-A^T}{2} \cdot \chi \, M \, dA, \quad \forall \chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n). \end{array} \right. \end{equation}

(ii) The variational formulation (6.16) can be equivalently written

(6.17) \begin{equation} \left \{ \begin{array}{l} \displaystyle \mu \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n), \\[5pt] \displaystyle \int _{{\mathcal T}} (\nabla \mu \cdot \nabla \chi ) (A_\Theta ) \, M(A_\Theta ) \, u_n(\Theta ) \, d\Theta \\[5pt] \displaystyle \quad = - \int _{{\mathcal T}} \frac{A_\Theta -A_\Theta ^T}{2} \cdot \chi (A_\Theta ) \, M(A_\Theta ) \, u_n(\Theta ) \, d\Theta, \quad \forall \chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n). \end{array} \right. \end{equation}

Proof. (i) We remark that (6.16) has a unique solution. Indeed, (4.9) can be extended componentwise to all $\psi \in H^1_0(\mathrm{SO}_n, \mathfrak{so}_n)$ , and in particular, to all $\psi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Thus, the bilinear form at the left-hand side of (6.16) is coercive on $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Existence and uniqueness follow from Lax-Milgram’s theorem.

Let $\mu$ be the solution of (6.5). Since $\mu$ satisfies (6.6), it belongs to $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Furthermore, restricting (6.5) to test functions in $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ , it satisfies (6.16).

Conversely, suppose that $\mu$ is the unique solution of (6.16). We will use (6.15). Let $\chi \in H^1_0(\mathrm{SO}_n, \mathfrak{so}_n)$ . Thanks to (4.16) and to the fact that $M$ is a class function, we have

\begin{eqnarray*} && I(A) \;=\!:\; \int _{\mathrm{SO}_n} M(gAg^T) \, \nabla \mu (gAg^T) \cdot \nabla \chi (gAg^T) \, dg \\[5pt] && = \frac{1}{2} \sum _{i,j=1}^n \int _{\mathrm{SO}_n} M(A) \, \nabla \mu _{ij} (gAg^T) \cdot \nabla \chi _{ij}(gAg^T) \, dg \\[5pt] && = \frac{1}{2} \sum _{i,j=1}^n \int _{\mathrm{SO}_n} M(A) \, \nabla (\mu _{ij} \circ \xi _g) (A) \cdot \nabla (\chi _{ij} \circ \xi _g)(A) \, dg. \end{eqnarray*}

Now, by (6.6), we have $ (\mu _{ij} \circ \xi _g) (A) = \sum _{k,\ell = 1}^n g_{ik} \, g_{j \ell } \, \mu _{k \ell }(A)$ . We deduce that

\begin{eqnarray*} I(A) &=& \frac{1}{2} \sum _{k,\ell = 1}^n M(A) \, \nabla \mu _{k \ell }(A) \cdot \nabla \Big ( \sum _{i,j=1}^n \int _{\mathrm{SO}_n} g_{ik} \, g_{j \ell } \, (\chi _{ij} \circ \xi _g) \, dg \Big )(A)\\[5pt] &=& \frac{1}{2} \sum _{k,\ell = 1}^n M(A) \, \nabla \mu _{k \ell }(A) \cdot \nabla \bar \chi _{k \ell }(A) = M(A) \nabla \mu (A) \cdot \nabla \bar \chi (A), \end{eqnarray*}

where $\bar \chi$ is defined by

(6.18) \begin{equation} \bar \chi (A) = \int _{\mathrm{SO}_n} g^T \chi (gAg^T) g \, dg, \quad \forall A \in \mathrm{SO}_n. \end{equation}

Similarly, we have

\begin{eqnarray*} && J(A) \;=\!:\; \int _{\mathrm{SO}_n} M(gAg^T) \, \Big ( g \frac{A-A^T}{2} g^T \Big ) \cdot \chi (gAg^T) \, dg \\[5pt] && = M(A) \, \frac{A-A^T}{2} \cdot \Big ( \int _{\mathrm{SO}_n} g^T \chi (gAg^T) g \, dg \Big ) = M(A) \, \frac{A-A^T}{2} \cdot \bar \chi (A). \end{eqnarray*}

Applying (6.15), we get that

(6.19) \begin{eqnarray} && \int _{\mathrm{SO}_n} \Big ( \nabla \mu \cdot \nabla \chi + \frac{A-A^T}{2} \cdot \chi \Big ) M \, dA = \int _{\mathrm{SO}_n} (I(A) + J(A)) \, dA \nonumber \\[5pt] &&\qquad = \int _{\mathrm{SO}_n} \Big ( \nabla \mu \cdot \nabla \bar \chi + \frac{A-A^T}{2} \cdot \bar \chi \Big ) M \,dA. \end{eqnarray}

Now, we temporarily assume that

(6.20) \begin{equation} \bar \chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n), \end{equation}

Then, because $\mu$ is the unique solution of (6.16), the right-hand side of (6.19) is equal to zero. This shows that $\mu$ is the unique solution of (6.5).

Now, we show (6.20). That $\bar \chi$ satisfies the invariance relation (6.8) is obvious. We now show that $ \| \bar \chi \|_{H^1} \leq \| \chi \|_{H^1}$ .

We first show that $ \| \bar \chi \|_{L^2} \leq \| \chi \|_{L^2}$ . By Cauchy-Schwarz inequality, Fubini’s theorem and the translation invariance of the Haar measure, we have

\begin{eqnarray*} \int _{\mathrm{SO}_n} |\bar \chi (A)|^2 \, dA &=& \int _{\mathrm{SO}_n} \Big | \int _{\mathrm{SO}_n} g^T \chi (gAg^T) g \, dg \Big |^2 \, dA \leq \int _{\mathrm{SO}_n} \Big ( \int _{\mathrm{SO}_n} | g^T \chi (gAg^T) g |^2 \, dg \Big ) dA \\[5pt] &=& \int _{\mathrm{SO}_n} \Big ( \int _{\mathrm{SO}_n} | \chi (gAg^T) |^2 \, dA \Big ) dg = \int _{\mathrm{SO}_n} | \chi (A) |^2 \, dA. \end{eqnarray*}

We now show that $ \| \nabla \bar \chi \|_{L^2} \leq \| \nabla \chi \|_{L^2}$ . Differentiating (6.18) with respect to $A$ and using (4.14), we get

\begin{eqnarray*} && |\nabla \bar \chi (A)|^2 = \frac{1}{2} \sum _{k, \ell = 1}^n |\nabla \bar \chi _{k \ell }(A)|^2 = \frac{1}{2} \sum _{k,\ell =1}^n \Big | \int _{\mathrm{SO}_n} \Big (\sum _{i,j=1}^n g_{ik} \, g_{j \ell } \, g^T \nabla \chi _{ij}(gAg^T) g \Big ) \, dg \Big |^2. \end{eqnarray*}

Applying Cauchy-Schwarz formula, this leads to

\begin{eqnarray*} && |\nabla \bar \chi (A)|^2 \leq \frac{1}{2} \sum _{k,\ell =1}^n \int _{\mathrm{SO}_n} \Big | \sum _{i,j=1}^n g_{ik} \, g_{j \ell } \, g^T \nabla \chi _{ij}(gAg^T) g \Big |^2 \, dg \\[5pt] && = \frac{1}{2} \sum _{i,j,i^{\prime},j^{\prime},k,\ell =1}^n \int _{\mathrm{SO}_n} g_{ik} \, g_{j \ell } \, g_{i^{\prime}k} \, g_{j^{\prime} \ell } \, \big ( g^T \nabla \chi _{i^{\prime} j^{\prime}}(gAg^T) g \big ) \cdot \big ( g^T \nabla \chi _{ij} (gAg^T) g \big ) \, dg \\[5pt] && = \frac{1}{2} \sum _{i,j=1}^n \int _{\mathrm{SO}_n} \nabla \chi _{i j}(gAg^T) \cdot \nabla \chi _{ij} (gAg^T) \, dg, \end{eqnarray*}

where we have applied that $\sum _{k=1}^n g_{ik} g_{i^{\prime}k} = \delta _{i i^{\prime}}$ and similarly for the sum over $\ell$ . Thus,

\begin{eqnarray*} && \int _{\mathrm{SO}_n} |\nabla \bar \chi (A)|^2 \, dA \leq \frac{1}{2} \sum _{i,j=1}^n \int _{\mathrm{SO}_n} \Big ( \int _{\mathrm{SO}_n} \nabla \chi _{i j}(gAg^T) \cdot \nabla \chi _{ij} (gAg^T) \, dA \Big ) \, dg \\[5pt] && = \frac{1}{2} \sum _{i,j=1}^n \int _{\mathrm{SO}_n} \nabla \chi _{i j}(A) \cdot \nabla \chi _{ij} (A) \, dA = \int _{\mathrm{SO}_n} | \nabla \chi (A)|^2 \, dA, \end{eqnarray*}

which shows the result and ends the proof of (i)

(ii) Let $\chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Then, the functions $A \mapsto M(A) \nabla \mu (A) \cdot \nabla \chi (A)$ and $A \mapsto M \, \frac{A-A^T}{2} \cdot \chi (A)$ are class functions (the proof relies on similar computations as those just made above and is omitted). Then, (6.17) is simply a consequence of Weyl’s integration formula (5.7). This ends the proof.

6.3. Derivation and well-posedness of system (3.11) for $\alpha$

Now, we investigate how the condition $\chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ translates onto $\tau$ and define the following spaces:

• • $C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ is the set of periodic, $C^\infty$ functions ${\mathcal T} \to{\mathbb R}^p$ which commute with the Weyl group, i.e. such that (6.10) is satisfied in $\mathcal T$ , for all $W \in \mathfrak{W}$ ,

• • $\mathcal H$ is the closure of $C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ for the norm

  \begin{equation*} \| \tau \|_{\mathcal H}^2 \;=\!:\; \sum _{i=1}^p \int _{{\mathcal T}} |\tau _i(\Theta )|^2 \, u_n(\Theta ) \, d \Theta, \end{equation*}

• • $\mathcal V$ is the closure of $C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ for the norm

  \begin{eqnarray*} && \| \tau \|_{\mathcal V}^2 \;=\!:\; \| \tau \|_{\mathcal H}^2 + \int _{{\mathcal T}} \Big \{ \sum _{i,j=1}^p \Big | \frac{\partial \tau _i}{\partial \theta _j}(\Theta ) \Big |^2 + \sum _{1 \leq i \lt j \leq p} \Big ( \frac{|(\tau _i - \tau _j)(\Theta )|^2}{1 - \cos\!(\theta _i - \theta _j)} \\[5pt] &&\qquad + \frac{|(\tau _i + \tau _j)(\Theta )|^2}{1 - \cos\!(\theta _i + \theta _j)} \Big ) + \epsilon _n \sum _{i=1}^p \frac{|\tau _i(\Theta )|^2}{1 - \cos \theta _i} \Big \} \, u_n(\Theta ) \, d \Theta. \end{eqnarray*}

Remark 6.2. We note that for $\tau \in C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ , we have $\| \tau \|_{\mathcal V} \lt \infty$ , which shows that the definition of $\mathcal V$ makes sense. Indeed since $\tau$ commutes with $\mathfrak{W}$ , we have

\begin{eqnarray*} && (\tau _i - \tau _j) (\ldots, \theta _i, \ldots, \theta _j \ldots ) = \tau _i (\ldots, \theta _i, \ldots, \theta _j \ldots ) - \tau _i (\ldots, \theta _j, \ldots, \theta _i \ldots ) \\[5pt] && = (\theta _j - \theta _i) \Big ( \big ( \frac{\partial }{\partial \theta _j} - \frac{\partial }{\partial \theta _i} \big ) \tau _i \Big ) (\ldots, \theta _i, \ldots, \theta _i \ldots ) +{\mathcal O} \big ((\theta _j - \theta _i)^2\big ) ={\mathcal O} \big (|\theta _j - \theta _i| \big ), \end{eqnarray*}

as $\theta _j - \theta _i \to 0$ , while

\begin{equation*} 1 - \cos\!(\theta _i - \theta _j) = 2 \sin ^2 \Big ( \frac {\theta _i - \theta _j}{2} \Big ) = \frac {1}{2} |\theta _j - \theta _i|^2 + {\mathcal O} \big (|\theta _j - \theta _i|^4 \big ). \end{equation*}

Thus,

\begin{equation*} \frac {|(\tau _i - \tau _j)(\Theta )|^2}{ 1 - \cos\!(\theta _i - \theta _j)} \lt \infty. \end{equation*}

The same computation holds when $\theta _j + \theta _i \to 0$ and, in the odd-dimensional case, when $\theta _i \to 0$ .

Proof. Let $\chi \in C^{\infty }_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ be an element of the space of smooth functions satisfying the invariance relation (6.8). Then, it is clear that $\tau$ associated with $\chi$ through (6.9) belongs to $C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ . For such $\chi$ and $\tau$ , we temporarily assume that

(6.21) \begin{equation} \| \chi \|_{H^1}^2 = \frac{\gamma _n}{(2 \pi )^p} \| \tau \|^2_{\mathcal V}. \end{equation}

We also assume that

(6.22) \begin{equation} C^{\infty }_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n) \, \, \text{ is dense in } \, \, H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n). \end{equation}

Let now $\chi$ be an element of $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ and $(\chi ^q)_{q \in{\mathbb N}}$ be a sequence of elements of $C^{\infty }_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ which converges to $\chi$ in $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Then, the associated sequence $(\tau ^q)_{q \in{\mathbb N}}$ is such that $\tau ^q \in C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ and by (6.21) $(\tau ^q)_{q \in{\mathbb N}}$ is a Cauchy sequence in $\mathcal V$ . Thus, there is an element $\tau \in{\mathcal V}$ such that $\tau ^q \to \tau$ in $\mathcal V$ . Now, up to the extraction of subsequences, we have $\chi ^q \to \chi$ , a.e. in $\mathrm{SO}_n$ and $\tau ^q \to \tau$ , a.e. in $\mathcal T$ . Hence,

\begin{equation*} \chi ^q (A_\Theta ) = \sum _{i=1}^p \tau ^q_i (\Theta ) F_{2i-1 \, 2i} \to \sum _{i=1}^p \tau _i(\Theta ) F_{2i-1 \, 2i} = \chi (A_\Theta ) \quad \mathrm {a.e.} \quad \Theta \in {\mathcal T}, \end{equation*}

and, since $\chi$ satisfies (6.8), $\chi (A)$ is given by (6.11). Therefore, if $\chi$ belongs to $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ , there exists $\tau \in{\mathcal V}$ such that $\chi$ is given by (6.11).

Therefore, Prop. 6.6 will be proved if we prove the converse property, namely

(6.23) \begin{equation} \mathrm{For \, any \,\,} \tau \in{\mathcal V}, \mathrm{\, \, then \,\, } \chi \mathrm{\, \, given \, by \, (6.9) \, belongs \, to} \,\, \mathrm{H}^1_{\mathrm{inv}}(\mathrm{SO}_{\mathrm{n}}, \mathfrak{so}_{\mathrm{n}}), \end{equation}

as well as Properties (6.21) and (6.22).

Proof of (6.21). Let $\chi \in C^{\infty }_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Since the function $A \mapsto |\chi (A)|^2$ is a class function, we have, thanks to Weyl’s integration formula (5.7):

(6.24) \begin{equation} \| \chi \|_{L^2}^2 = \frac{\gamma _n}{(2 \pi )^p} \int _{{\mathcal T}} |\chi (A_\Theta )|^2 \, u_n(\Theta ) \, d \Theta = \frac{\gamma _n}{(2 \pi )^p} \int _{{\mathcal T}} \sum _{i=1}^p |\tau _i(\Theta )|^2 \, u_n(\Theta ) \, d \Theta, \end{equation}

which shows that $\| \chi \|^2_{L^2} = \frac{\gamma _n}{(2 \pi )^p} \| \tau \|^2_{\mathcal H}$ .

Now, we show that

(6.25) \begin{eqnarray} \| \nabla \chi \|_{L^2}^2 &=& \frac{\gamma _n}{(2 \pi )^p} \int _{{\mathcal T}} \Big \{ \sum _{i,j=1}^p \Big |\frac{d \tau _i}{d \theta _j} (\Theta ) \Big |^2 + \sum _{1 \leq i \lt j \leq p} \Big ( \frac{|(\tau _i - \tau _j)(\Theta )|^2}{1 - \cos\!(\theta _i - \theta _j)} + \frac{|(\tau _i + \tau _j)(\Theta )|^2}{1 - \cos\!(\theta _i + \theta _j)} \Big ) \nonumber \\[5pt] && \qquad + \epsilon _n \sum _{i=1}^p \frac{\tau _i^2(\Theta )}{1 - \cos \theta _i} \, \Big \} u_n(\Theta ) \, d \Theta. \end{eqnarray}

The function $A \mapsto |\nabla \chi (A)|^2$ is a class function (the proof relies on similar computations to those made in the proof of Prop. 6.5 and is omitted), and we can use Weyl’s integration formula to evaluate $\| \nabla \chi \|_{L^2}^2$ . For this, we need to compute $|\nabla \chi (A_\Theta )|^2$ . This will be done by means of (6.3), which requires to find a convenient basis of $\mathfrak{so}_n$ and to compute the action of the derivation operator $\varrho$ on each term. The latter will be achieved thanks to (5.8). Given that $\chi$ satisfies (6.8), we get, for all $A \in \mathrm{SO}_n$ and $X \in \mathfrak{so}_n$ :

\begin{equation*} \frac {d}{dt} \big ( \chi ( e^{tX} A e^{-tX}) \big ) \big |_{t=0} = \frac {d}{dt} \big ( e^{tX} \chi (A) e^{-tX} \big ) \big |_{t=0} = [X, \chi (A)]. \end{equation*}

When inserted into (5.8), this leads to

(6.26) \begin{equation} \Big (\varrho \big (\mathrm{Ad}(A^{-1}) X - X \big ) \chi \Big )(A) = [X, \chi (A)], \quad \forall A \in \mathrm{SO}_n, \quad \forall X \in \mathfrak{so}_n. \end{equation}

We will also use the formula

(6.27) \begin{eqnarray} && \big ( \varrho (F_{2k-1 \, 2k}) (\chi ) \big )(A_\Theta ) = \frac{d}{dt} \chi (A_\Theta e^{t F_{2k-1 \, 2k}}) \big |_{t=0} = \frac{d}{dt} \chi (A_{(\theta _1, \ldots, \theta _k-t, \ldots, \theta _p)}) \big |_{t=0} \nonumber \\[5pt] && = \sum _{\ell =1}^p \frac{d}{dt} \tau _\ell (\theta _1, \ldots, \theta _k-t, \ldots, \theta _p) \big |_{t=0} \, F_{2\ell -1 \, 2 \ell } = - \sum _{\ell =1}^p \frac{\partial \tau _\ell }{\partial \theta _k}(\Theta ) \, F_{2\ell -1 \, 2 \ell }, \end{eqnarray}

which is a consequence of (6.9).

It will be convenient to first treat the special examples of dimension $n=3$ and $n=4$ , before generalising them to dimensions $n=2p$ and $n=2p+1$ .

Case of $\mathrm{SO}_3$ . This is an even-dimensional case $n=2p+1$ with $p=1$ . We have $\Theta = \theta _1$ . Then, $A_\Theta = R_{\theta _1}$ (where $R_\theta$ is given by (5.1)) and we have $\chi (A_\Theta ) = \tau _1(\theta _1) F_{12}$ . The triple $(F_{12}, G^+, G^-)$ with $G^\pm = \frac{1}{\sqrt{2}} (F_{13} \pm F_{23})$ is an orthonormal basis of $\mathfrak{so}_3$ which we will use to compute (6.3). Thanks to (6.27), we first have

\begin{equation*} \big ( \varrho (F_{12}) \chi \big )(A_\Theta ) = - \frac {d \tau _1}{d \theta _1} (\theta _1) F_{12}. \end{equation*}

Now, we apply (6.26) with $A=A_\Theta$ and $X=G^+$ or $G^-$ . Since $A_\Theta ^{-1} = A_{- \Theta }$ , easy computations (see also [Reference Degond18]) lead to

\begin{eqnarray*} \mathrm{Ad}(A_{-\Theta }) G^+ - G^+ &=& (\cos \theta _1 - 1) G^+ + \sin \theta _1 G^-, \\[5pt] \mathrm{Ad}(A_{-\Theta }) G^- - G^- &=& - \sin \theta _1 G^+ + (\cos \theta _1 - 1) G^-, \end{eqnarray*}

and, using (5.9),

\begin{equation*} [G^+, \chi (A_\Theta )] = - \tau _1(\theta _1) G^-, \qquad [G^-, \chi (A_\Theta )] = \tau _1(\theta _1) G^+. \end{equation*}

Thus,

\begin{eqnarray*} \Big ( \big ( (\cos \theta _1 - 1) \varrho (G^+) + \sin \theta _1 \varrho (G^-) \big ) \chi \Big ) (A_\Theta ) &=& - \tau _1(\theta _1) G^-, \\[5pt] \Big ( \big (\!-\! \sin \theta _1 \varrho (G^+) + (\cos \theta _1 - 1) \varrho (G^-) \big ) \chi \Big ) (A_\Theta ) &=& \tau _1(\theta _1) G^+, \end{eqnarray*}

which leads to

(6.28) \begin{eqnarray} \big ( \varrho (G^+) \chi \big ) (A_\Theta ) &=& - \frac{\tau _1(\theta _1)}{2 (1 - \cos \theta _1)} \big ( \sin \theta _1 G^+ + (\cos \theta _1 - 1) G^- \big ), \end{eqnarray}

(6.29) \begin{eqnarray} \big ( \varrho (G^-) \chi \big ) (A_\Theta ) &=& \frac{\tau _1(\theta _1)}{2 (1 - \cos \theta _1)} \big ( (\cos \theta _1 - 1) G^+ - \sin \theta _1 G^- \big ). \end{eqnarray}

From (6.3), it follows that

\begin{equation*} |\nabla \chi (A_\Theta ) |^2 = \Big |\frac {d \tau _1}{d \theta _1} (\theta _1) \Big |^2 + \frac {\tau _1^2(\theta _1)}{1 - \cos \theta _1}, \end{equation*}

which leads to (6.25) for $n=3$ .

Case of $\mathrm{SO}_4$ : this is an odd-dimensional case $n=2p$ with $p=2$ . We have $\Theta = (\theta _1, \theta _2)$ and

\begin{equation*} \chi (A_\Theta ) = \tau _1(\theta _1,\theta _2) F_{12} + \tau _2(\theta _1,\theta _2) F_{34}.\end{equation*}

The system $(F_{12}, F_{34}, H^+, H^-, K^+, K^-)$ with $H^\pm = \frac{1}{\sqrt{2}} (F_{13} \pm F_{24})$ and $K^\pm = \frac{1}{\sqrt{2}} (F_{14} \pm F_{23})$ is an orthonormal basis of $\mathfrak{so}_4$ , which will be used to express (6.3). Then, we have

(6.30) \begin{eqnarray} \big ( \varrho (F_{12}) \chi \big )(A_\Theta ) &=& - \frac{d \tau _1}{d \theta _1} (\Theta ) F_{12} - \frac{d \tau _2}{d \theta _1} (\Theta ) F_{34}, \end{eqnarray}

(6.31) \begin{eqnarray} \big ( \varrho (F_{34}) \chi \big )(A_\Theta ) &=& - \frac{d \tau _1}{d \theta _2} (\Theta ) F_{12} - \frac{d \tau _2}{d \theta _2} (\Theta ) F_{34}. \end{eqnarray}

Now, we compute:

\begin{eqnarray*} \mathrm{Ad}(A_{-\Theta }) H^+ - H^+ &=& (c_1 c_2 + s_1 s_2 - 1) H^+ - (c_1 s_2 - s_1 c_2) K^-, \\[5pt] \mathrm{Ad}(A_{-\Theta }) H^- - H^- &=& - (c_1 s_2 + s_1 c_2) K^+ + (c_1 c_2 - s_1 s_2 - 1)H^-, \\[5pt] \mathrm{Ad}(A_{-\Theta }) K^+ - K^+ &=& (c_1 c_2 - s_1 s_2 - 1) K^+ + (c_1 s_2 + s_1 c_2) H^-, \\[5pt] \mathrm{Ad}(A_{-\Theta }) K^- - K^- &=& (c_1 s_2 - s_1 c_2) H^+ + (c_1 c_2 + s_1 s_2 - 1)K^-, \end{eqnarray*}

with $c_i = \cos \theta _i$ and $s_i = \sin \theta _i$ , $i=1, \, 2$ . We also compute, using (5.9):

\begin{eqnarray*} [H^+, \chi (A_\Theta )] &=& (\!-\! \tau _1 + \tau _2) K^-, \quad [H^-, \chi (A_\Theta )] = (\tau _1 + \tau _2) K^+, \\[5pt] \mbox{} [K^+, \chi (A_\Theta )] &=& - (\tau _1 + \tau _2) H^-, \quad [K^-, \chi (A_\Theta )] = (\tau _1 - \tau _2) H^+. \end{eqnarray*}

where we omit the dependence of $\tau _i$ on $\Theta$ for simplicity. Applying (6.26), we get two independent linear systems of equations for $((\varrho (H^+)\chi )(A_\Theta ), (\varrho (K^-)\chi )(A_\Theta ))$ on one hand and $((\varrho (K^+)\chi )(A_\Theta ), (\varrho (H^-)\chi )(A_\Theta ))$ on the other hand, which can both easily be resolved into

(6.32) \begin{eqnarray} \big ( \varrho (H^+)\chi \big )(A_\Theta ) &=& \frac{1}{2} (\tau _1-\tau _2) \Big ( K^- - \frac{\sin (\theta _1-\theta _2)}{1-\cos\!(\theta _1-\theta _2)} H^+ \Big ), \end{eqnarray}

(6.33) \begin{eqnarray} \big ( \varrho (K^-)\chi \big )(A_\Theta ) &=& \frac{1}{2} (\tau _1-\tau _2) \Big (\!-\! \frac{\sin (\theta _1-\theta _2)}{1-\cos\!(\theta _1-\theta _2)} K^- - H^+ \Big ), \end{eqnarray}

(6.34) \begin{eqnarray} \big ( \varrho (H^-)\chi \big )(A_\Theta ) &=& \frac{1}{2} (\tau _1+\tau _2) \Big (\!-\! K^+ - \frac{\sin (\theta _1+\theta _2)}{1-\cos\!(\theta _1+\theta _2)} H^- \Big ), \end{eqnarray}

(6.35) \begin{eqnarray} \big ( \varrho (K^+)\chi \big )(A_\Theta ) &=& \frac{1}{2} (\tau _1+\tau _2) \Big (\!-\! \frac{\sin (\theta _1+\theta _2)}{1-\cos\!(\theta _1+\theta _2)} K^+ + H^- \Big ). \end{eqnarray}

Taking the squared norms in $\mathfrak{so}_4$ of (6.30) to (6.35), we get

\begin{equation*}|\nabla \chi (A_\Theta ) |^2 = \sum _{i,j=1}^2 \Big |\frac {d \tau _i}{d \theta _j} \Big |^2 + \frac {|\tau _1 - \tau _2|^2}{1 - \cos\!(\theta _1 - \theta _2)} + \frac {|\tau _1 + \tau _2|^2}{1 - \cos\!(\theta _1 + \theta _2)}, \end{equation*}

which leads to (6.25) for $n=4$ .

Case of $\mathrm{SO}_{2p}$ : Define $H_{jk}^\pm = \frac{1}{\sqrt{2}} (F_{2j-1 \, 2k-1} \pm F_{2j \, 2k})$ and $K_{jk}^\pm = \frac{1}{\sqrt{2}} (F_{2j-1 \, 2k} \pm F_{2j \, 2k-1})$ for $1 \leq j \lt k \leq p$ . Then, the system $\big ( (F_{2j-1 \, 2j})_{j=1, \ldots, p}, (H_{jk}^+, H_{jk}^-, K_{jk}^+, K_{jk}^-)_{1 \leq j \lt k \leq p} \big )$ is the orthonormal basis of $\mathfrak{so}_{2p}{\mathbb R}$ which will be used to evaluate (6.3). Then, we remark that the computations of $\mathrm{Ad}(A_{-\Theta }) H_{jk}^\pm$ and $\mathrm{Ad}(A_{- \Theta }) K_{jk}^\pm$ only involve the $4 \times 4$ matrix subblock corresponding to line and column indices belonging to $\{ 2j-1, 2j \} \cup \{ 2k-1, 2k \}$ . Thus, restricted to these $4 \times 4$ matrices, the computations are identical to those done in the case of $\mathrm{SO}_{4}{\mathbb R}$ . This directly leads to (6.25) for $n=2p$ .

Case of $\mathrm{SO}_{2p+1}$ : this case is similar, adding to the previous basis the elements $G_j^{\pm } = \frac{1}{\sqrt{2}} (F_{2j-1 \, 2p+1} \pm F_{2j \, 2p+1})$ . These additional terms contribute to terms like in the $\mathrm{SO}_3$ case, which leads to (6.25) for $n=2p+1$ .

This finishes the proof of (6.25).

Proof of (6.23). Consider $\tau \in{\mathcal V}$ and a sequence $(\tau ^q)_{q \in{\mathbb N}}$ of elements of $C_{\mathrm{per}}^{\infty,\mathfrak{W}}({\mathcal T},{\mathbb R}^p)$ which converges to $\tau$ in $\mathcal V$ . Let $\chi ^q$ be associated with $\tau ^q$ through (6.11). From (6.9), we see that the function $\Theta \to \chi ^q(A_\Theta )$ belongs to $C^{\infty }({\mathcal T}, \mathfrak{so}_n)$ . However, we cannot deduce directly from it that $\chi ^q$ belongs to $C^{\infty }_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Indeed, for a given $A \in \mathrm{SO}_n$ , the pair $(g,\Theta ) \in \mathrm{SO}_n \times{\mathcal T}$ such that $A = g A_\Theta g^T$ is not unique, and consequently, the map $\mathrm{SO}_n \times{\mathcal T} \to \mathrm{SO}_n$ , $(g, \Theta ) \mapsto g A_\Theta g^T$ is not invertible. Therefore, we cannot deduce that $\chi ^q$ is smooth from the smoothness of the map $\mathrm{SO}_n \times{\mathcal T} \to \mathfrak{so}_n$ , $(g, \Theta ) \mapsto g \chi ^q(A_\Theta ) g^T$ . The result is actually true but the proof requires a bit of topology (see Remark 6.3 below). Here, we give a proof of (6.23) that avoids proving that $\chi ^q$ is smooth and which is thus simpler.

Temporarily dropping the superscript $q$ from $\chi ^q$ , we first show that $\chi$ is a differentiable function $\mathrm{SO}_n \to \mathfrak{so}_n$ at any point $A \in{\mathbb T}$ (which means that the derivatives in all directions of the tangent space $T_A = A \mathfrak{so}_n$ of $\mathrm{SO}_n$ at $A \in{\mathbb T}$ are defined, not only those derivatives in the direction of an element of the tangent space $T_A{\mathbb T} = A \mathfrak{h}$ of $\mathbb T$ at $A$ ). Indeed, this is a consequence of the proof of (6.21) above. This proof shows that for an orthonormal basis $(\Phi _i)_{i=1}^{\mathcal N}$ (with ${\mathcal N} = \mathrm{dim} \, \mathfrak{so}_n$ ) whose precise definition depends on $n$ (see proof of (6.21)), then $(\varrho (\Phi _i) (\chi ))(A)$ exists for all $A \in{\mathbb T}$ and all $i \in \{1, \ldots{\mathcal N}\}$ , which is exactly saying that the derivatives of $\chi$ in all the direction of $T_A$ exist for all $A \in{\mathbb T}$ . If we refer to the $n=3$ case for instance, $(\varrho (F_{12}) (\chi ))(A_\Theta )$ exists because $(d \tau _1/d \theta )(\theta _1)$ exists since $\tau$ is $C^\infty$ . Then, (6.28) and (6.29) tell us that $(\varrho (G^+) (\chi ))(A_\Theta )$ and $(\varrho (G^-) (\chi ))(A_\Theta )$ exist, except maybe when $\cos \theta _1 = 1$ . But, because $\tau$ commutes with the Weyl group, Eqs. (6.28) and (6.29) lead to finite values of $(\varrho (G^\pm ) (\chi ))(A_\Theta )$ when $\cos \theta _1 = 1$ thanks to a Taylor expansion similar to that made in Remark 6.2. It is straightforward to see that dimensions $n \geq 4$ can be treated in a similar fashion.

From this, we deduce that $\nabla \chi (A)$ exists for all $A \in \mathrm{SO}_n$ . Indeed, by construction, $\chi$ satisfies (6.8). So, we deduce that

(6.36) \begin{equation} \nabla (\chi \circ \xi _g) (A) = \nabla (g \chi g^T) (A), \quad \forall A, \, g \in \mathrm{SO}_n, \end{equation}

where $\xi _g$ is defined by (4.12). Applying (6.2) with the basis $(\Phi _i)_{i=1}^{\mathcal N} = (\Psi _i)_{i=1}^{\mathcal N} = (F_{ij})_{1 \leq i \lt j \leq n}$ , we get, thanks to (4.14),

(6.37) \begin{equation} \nabla (\chi \circ \xi _g) (A) = \sum _{k \lt \ell } \sum _{k^{\prime} \lt \ell ^{\prime}} \big ( \nabla \chi _{k \ell } (g A g^T) \cdot g A F_{k^{\prime} \ell ^{\prime}} g^T \big ) \, F_{k \ell } \otimes A F_{k^{\prime} \ell ^{\prime}}, \end{equation}

while

(6.38) \begin{equation} \nabla (g \chi g^T)(A) = \sum _{k \lt \ell } \sum _{k^{\prime} \lt \ell ^{\prime}} \big ( \nabla \chi _{k \ell }(A) \cdot A F_{k^{\prime} \ell ^{\prime}} \big ) \, (g F_{k \ell } g^T) \otimes A F_{k^{\prime} \ell ^{\prime}}, \end{equation}

where $\sum _{k \lt \ell }$ means the sum over all pairs $(k,\ell )$ such that $1 \leq k \lt \ell \leq n$ and similarly for $\sum _{k^{\prime} \lt \ell ^{\prime}}$ . Thus, using the fact that the basis $(F_{k \ell } \otimes A F_{k^{\prime} \ell ^{\prime}})_{k\lt \ell, k^{\prime}\lt \ell ^{\prime}}$ is an orthonormal basis of $\mathfrak{so}_n \otimes T_A$ and that the two expressions (6.37) and (6.38) are equal thanks to (6.36), we get

(6.39) \begin{equation} \nabla \chi _{k \ell }(gAg^T) \cdot g A F_{k' \ell '} g^T = \sum _{k_1 \lt \ell _1} \big ( \nabla \chi _{k_1 \ell _1} (A) \cdot A F_{k' \ell '} \big ) \big ( g F_{k_1 \ell _1} g^T \cdot F_{k \ell } \big ). \end{equation}

We apply this formula with $A = A_\Theta$ . Because we know that $\nabla \chi (A_\Theta )$ exists for all $\Theta \in{\mathcal T}$ , the right-hand side of (6.39) is well-defined. Thus, the left-hand side of (6.39) tells us that $\nabla \chi (A)$ is defined (by its components on the basis $(F_{k \ell } \otimes A F_{k' \ell '})_{k\lt \ell, k'\lt \ell '}$ ) for all $A$ such that there exists $(g,\Theta ) \in \mathrm{SO}_n \times{\mathcal T}$ with $A = g A_\Theta g^T$ . But of course, such $A$ range over the whole group $\mathrm{SO}_n$ , which shows that $\nabla \chi$ exists everywhere.

Now, we note that formula (6.21) applies to $\chi$ . Indeed, because $\chi$ satisfies (6.8), $|\chi |^2$ and $|\nabla \chi |^2$ are class functions, so they only depend on their values on $\mathbb T$ , which are given by the same formulas in terms of $\tau$ as those found in the proof of (6.21). Hence, the value of $\| \chi \|_{H^1}^2$ is still given by (6.21). In particular, it is finite and so, we have that $\chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ .

Now, putting the superscript $q$ back, we have $\chi ^q \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ and up to an unimportant constant, $\| \chi ^q \|_{H^1} = \| \tau ^q \|_{\mathcal V}$ . Since $\tau ^q \to \tau$ in $\mathcal V$ , $\chi ^q$ is a Cauchy sequence in $H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ . Thus, it converges to an element $\chi \in H^1_{\mathrm{inv}}(\mathrm{SO}_n, \mathfrak{so}_n)$ and by the same reasoning as when we proved the direct implication, we deduce that $\chi$ and $\tau$ are related with each other by (6.11), which ends the proof.