Reeb flows, pseudo-holomorphic curves and transverse foliations

Abstract

Pseudo-holomorphic curves in symplectizations, as introduced by Hofer in 1993, and then developed by Hofer, Wysocki, and Zehnder, have brought new insights to Hamiltonian dynamics, providing new approaches to some classical questions in Celestial Mechanics. This short survey presents some recent developments in Reeb dynamics based on the theory of pseudo-holomorphic curves in symplectizations, focusing on transverse foliations near critical energy surfaces.

Appendices

Appendix A: The action functional

Let M be a smooth 3-manifold equipped with a contact form $\lambda$. Denote by $\varphi _t,t\in {\mathbb {R}},$ the Reeb flow of $\lambda$.

The action functional associated with $\lambda$ is defined on the space of smooth curves $\gamma : {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M$ as

$$\begin{aligned} {\mathcal {A}}(\gamma ) = \int _{{\mathbb {R}}/ {\mathbb {Z}}} \gamma ^*\lambda , \quad \forall \gamma \in C^\infty ({\mathbb {R}}/ {\mathbb {Z}},M). \end{aligned}$$

Let $\eta \in \Gamma (\gamma ^*TM)$ be a vector field along $\gamma$ and let $u:(-\epsilon ,\epsilon )\times {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M$ be a variation of $\gamma$ so that $u(0,t)=\gamma (t)$ for all $t\in {\mathbb {R}}/ {\mathbb {Z}}$ and $u_s(0,\cdot )=\eta$. We compute the first variation of ${\mathcal {A}}$ at $\gamma$ in the direction of $\eta$

$$\begin{aligned} \begin{aligned} d{\mathcal {A}}(\gamma ) \cdot \eta&= \int _{0 \times {\mathbb {R}}/ {\mathbb {Z}}} {\mathcal {L}}_{\partial _s} (u^*\lambda )\\&= \int _{0 \times {\mathbb {R}}/ {\mathbb {Z}}} d i_{\partial _s} (u^*\lambda )+\int _{0 \times {\mathbb {R}}/ {\mathbb {Z}}} i_{\partial _s} (u^*d\lambda )\\&= \int _0^1 d\lambda |_{\gamma (t)}(\eta (t), {{\dot{\gamma }}}(t))dt. \end{aligned} \end{aligned}$$

(6)

Since $d\lambda |_{\xi =\ker \lambda }$ is nondegenerate, this implies that $\gamma$ is a critical point of ${\mathcal {A}}$ if and only if ${{\dot{\gamma }}} \subset \ker d\lambda |_\gamma .$ In particular, if $\dot{\gamma }$ never vanishes, then $\gamma$ is a critical point of ${\mathcal {A}}$ if and only if $\gamma$ is a reparametrization of a closed Reeb orbit.

Now let $P=(x,T)$ be a closed Reeb orbit of $\lambda$. Then $x_T = x(T\cdot ):{\mathbb {R}}/ {\mathbb {Z}}\rightarrow M$ is a critical point of ${\mathcal {A}}$. Let $\eta \in \Gamma (x_T^*\xi )$ and let $u:(-\epsilon ,\epsilon ) \times {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M$ be a variation of $x_T$ so that $u_s(0,\cdot )=\eta$. We can assume that $u_s(s,\cdot )\subset \xi , \forall s$. Using a finite covering of a tubular neighborhood of $x_T({\mathbb {R}})$, we may assume that $x_T$ is an embedding. Let $\Xi$ be a vector field extending $\eta$ in a neighborhood of $x({\mathbb {R}})$.

The second variation of ${\mathcal {A}}$ at $x_T$ in the direction of $\eta$ is

$$\begin{aligned} \begin{aligned} d^2{\mathcal {A}}(x_T) \cdot (\eta ,\eta )&= \int _{0 \times {\mathbb {R}}/ {\mathbb {Z}}} {\mathcal {L}}_{\partial _s}({\mathcal {L}}_{\partial _s} (u^*\lambda ))\\&= \int _{0 \times {\mathbb {R}}/ {\mathbb {Z}}} i_{\partial _s}d (i_{\partial _s} (u^*d\lambda )))\\&= \int _{ {\mathbb {R}}/ {\mathbb {Z}}} x_T^* (i_\Xi d(i_\Xi d\lambda ))\\&= \int _0^1 d(i_\eta d\lambda )|_{x_T(t)}(\eta (t), \dot{x}_T(t))dt\\&= \int _0^1 d\lambda |_{x_T(t)} ( -({\mathcal {L}}_{\dot{x}_T} \eta )(t),\eta (t)) \, dt.\\ \end{aligned} \end{aligned}$$

(7)

We have used the usual relation $d\alpha (X,Y)=X \cdot \alpha (Y) - Y \cdot \alpha (X) - \alpha ([X,Y])$, where X, Y are vector fields and $\alpha$ is a 1-form.

Observe that the last expression obtained in (7) depends only on the linearized flow along P

$$\begin{aligned} \begin{aligned} ({\mathcal {L}}_{\dot{x}_T} \eta )(t)&: = \frac{d}{ds}\big |_{s=0} \left\{ D\varphi ^{-1}_{sT}(x_T(t+s)) \cdot \eta (t+s) \right\} . \end{aligned} \end{aligned}$$

A similar analysis of the action functional can be found in [32].

Appendix B: The asymptotic operator

Let $P=(x,T)$ be a closed orbit of the Reeb flow of $\lambda$ and let $J:\xi \rightarrow \xi$ be a $d\lambda$-compatible complex structure. The asymptotic operator associated with P and J is the linear operator

$$\begin{aligned} \begin{aligned} A_{P,J}(\eta ) := -J|_{x_T} \cdot {\mathcal {L}}_{\dot{x}_T}\eta \in L^2(x_T^* \xi ), \quad \forall \eta \in W^{1,2}(x_T^*\xi ), \\ \end{aligned} \end{aligned}$$

where ${\mathcal {L}}_{\dot{x}_T} \eta$ is defined in the previous section.

It is an exercise to check that $A_{P,J}$ admits 0 as an eigenvalue if and only if P is degenerate.

Since J preserves $d\lambda$ we see from (7) that

$$\begin{aligned} d^2{\mathcal {A}}(x_T) \cdot (\eta ,\eta ) = \int _0^1 d\lambda |_{x_T(t)} (A_{P,J} \cdot \eta (t), J_t\cdot \eta (t))\, dt, \end{aligned}$$

where $J_t = J|_{x_T(t)}$.

If $\nu \in {\mathbb {R}}$ is an eigenvalue of $A_{P,J}$ and $\eta _\nu$ is a non-trivial $\nu$-eigenfunction, then

$$\begin{aligned} \begin{aligned} d^2{\mathcal {A}}(x_T) \cdot (\eta _\nu ,\eta _\nu )&= \int _0^1 d\lambda _{x_T(t)}(A_{P,J} \cdot \eta _\nu (t), J_t\cdot \eta _\nu (t))\, dt,\\&= \nu \int _0^1 g_{J_t}(\eta _\nu (t),\eta _{\nu }(t))\, dt, \end{aligned} \end{aligned}$$

where $g_J(\cdot , \cdot ) = d\lambda (\cdot , J \cdot )$ is the positive-definite inner product on $\xi$ induced by $d\lambda$ and J. Hence $d^2{\mathcal {A}}(x_T) \cdot (\eta _\nu ,\eta _\nu )$ vanishes if and only if $\nu =0$ and, otherwise, its sign coincides with the sign of $\nu$.

In order to better describe the spectrum of the operator $A_{P,J}$ we need some normalization. Choose a unitary trivialization $\Psi : x_T^* \xi \rightarrow {\mathbb {R}}/ {\mathbb {Z}}\times {\mathbb {R}}^2$. This means that

$$\begin{aligned} \Psi ^*(dx \wedge dy) = d\lambda |_\xi \quad {\text{ and }} \quad \Psi _* J = J_0:=\left( \begin{array}{cc}0 &{} -1 \\ 1 &{} 0 \end{array}\right) . \end{aligned}$$

Here (x, y) are coordinates in ${\mathbb {R}}^2$.

We claim that the operator

$$\begin{aligned} {\mathcal {L}}_S:= \Psi \circ A_{P,J} \circ \Psi ^{-1} \end{aligned}$$

has the form

$$\begin{aligned} {\mathcal {L}}_S= -J_0\frac{d}{dt} - S, \end{aligned}$$

(8)

for some smooth loop $t \mapsto S(t)$ of $2\times 2$ symmetric matrices.

Indeed, recall that the Reeb flow of $\lambda$ preserves $d\lambda |_\xi$ and thus we find a smooth path of $2\times 2$ symplectic matrices $\Phi (t)$, satisfying $\Phi (0)=I$ and

$$\begin{aligned} \Phi (1+t) = \Phi (t) \Phi (1), \quad \forall t\in {\mathbb {R}}/ {\mathbb {Z}}. \end{aligned}$$

(9)

It represents the linearized flow of $\frac{1}{T}\lambda$ restricted to $(\xi ,d\lambda |_\xi )$ along $x_T$ in coordinates $\Psi$. In particular, a solution ${\bar{\zeta }}(t)\in {\mathbb {R}}^2 \simeq \xi |_{x_T(t)}$ to the linearized flow with initial condition $\bar{\zeta }(t+s)=\zeta (t+s) \in {\mathbb {R}}^2 \simeq \xi _{x_T(t+s)}$ satisfies

$$\begin{aligned} {\bar{\zeta }}(t) = \Phi (t)\Phi (t+s)^{-1}\zeta (t+s), \quad \forall s,t. \end{aligned}$$

Note that ${\bar{\zeta }}(t)$ depends on s.

Denoting

$$\begin{aligned} \zeta (t) = \Psi \circ \eta (t)\in {\mathbb {R}}^2 \quad \forall t, \end{aligned}$$

let us compute ${\mathcal {L}}_{\dot{x}_T} \eta$ in coordinates $\Psi$

$$\begin{aligned} \begin{aligned} \Psi \circ {\mathcal {L}}_{\dot{x}_T}\eta (t)&= \lim _{s\rightarrow 0} \frac{{\bar{\zeta }}(t) - \zeta (t)}{s}\\&= \lim _{s\rightarrow 0} \frac{\Phi (t)\Phi (t+s)^{-1}\zeta (t+s) - \zeta (t)}{s}\\&= \lim _{s\rightarrow 0} \left\{ \Phi (t)\Phi (t+s)^{-1}\frac{\zeta (t+s)-\zeta (t)}{s}+\Phi (t)\frac{\Phi (t+s)^{-1}-\Phi (t)^{-1}}{s}\zeta (t)\right\} \\&= {{\dot{\zeta }}}(t) +\Phi (t)\dot{[\Phi (t)^{-1}]}\zeta (t)\\&= {{\dot{\zeta }}}(t) - {{\dot{\Phi }}}(t) \Phi (t)^{-1}\zeta (t) \end{aligned} \end{aligned}$$

(10)

Now let

$$\begin{aligned} S(t):= -J_0 {{\dot{\Phi }}}(t) \Phi (t)^{-1}. \end{aligned}$$

(11)

Since $\Phi (t)$ is symplectic we have

$$\begin{aligned} \Phi (t)^T J_0 \Phi (t) = J_0 \Rightarrow {{\dot{\Phi }}}(t)^T J_0 \Phi (t) + \Phi (t)^T J_0 {{\dot{\Phi }}}(t) = 0, \quad \forall t. \end{aligned}$$

Hence

$$\begin{aligned} {{\dot{\Phi }}}(t)^TJ_0 = \Phi (t)^T S(t). \end{aligned}$$

(12)

Using the definition of S we compute

$$\begin{aligned} S(t)^T = (\Phi (t)^T)^{-1}{{\dot{\Phi }}}(t)^T J_0 \Rightarrow \dot{\Phi }(t)^T J_0 = \Phi (t)^T S(t)^T. \end{aligned}$$

It follows from (12) that $S(t)^T=S(t), \forall t$. The reader can easily check that (9) implies $S(t+1)=S(t) \forall t.$ Finally, (8) follows from (10), (11) and the identity

$$\begin{aligned} {\mathcal {L}}_S = -J_0 \cdot \Psi \circ {\mathcal {L}}_{\dot{x}_T} \circ \Psi ^{-1}. \end{aligned}$$

We check the symmetry of $A_{P,J}$ using coordinates $\Psi$. The $L^2$-product on $\Gamma (x_T^*\xi )$ induced by $d\lambda$ and J takes the form

$$\begin{aligned} \langle \zeta _1,\zeta _2\rangle = \int _0^1 \langle \zeta _1(t),\zeta _2(t) \rangle \, dt, \quad \forall \zeta _1,\zeta _2\in L^2({\mathbb {R}}/ {\mathbb {Z}},{\mathbb {R}}^2). \end{aligned}$$

Integrating by parts, we obtain

$$\begin{aligned} \begin{aligned} \langle \zeta _1,{\mathcal {L}}_S\cdot \zeta _2\rangle&= \int _0^1 \langle \zeta _1(t),-J_0{{\dot{\zeta }}}_2(t) - S(t)\zeta _2(t)\rangle \, dt\\&= \int _0^1 \langle -J_0{{\dot{\zeta }}}_1(t) - S(t)\zeta _1(t),\zeta _2(t)\rangle \, dt\\&= \langle {\mathcal {L}}_S \cdot \zeta _1,\zeta _2\rangle . \end{aligned} \end{aligned}$$

The eigenvalues of ${\mathcal {L}}_S$ are real. A non-trivial $\nu$-eigenfunction $\zeta _\nu$ of ${\mathcal {L}}_S$ satisfies the smooth linear ODE

$$\begin{aligned} {{\dot{\zeta }}}_\nu (t)= J_0 (S(t)+\nu I) \zeta _\nu (t) , \quad \forall t, \end{aligned}$$

and thus $\zeta _\nu$ is smooth and never vanishes. In particular, $\zeta _\nu$ has a well-defined winding number

$$\begin{aligned} \mathrm{wind}(\nu ) = \frac{\Theta (1) - \Theta (0)}{2\pi } \end{aligned}$$

where $\zeta _\nu (t) = (r(t)\cos (\Theta (t)),r(t) \sin (\Theta (t))) \forall t,$ for continuous functions $r>0,\Theta$. It does not depend on the $\nu$-eigenfunction.

If the linearized first return map $D\varphi _T(x_T(0)):\xi |_{x_T(0)} \rightarrow \xi _{x_T(0)}$ is the identity map then it is always possible to choose $J\in {\mathcal {J}}_+(\xi )$ and a trivialization $\Psi$ so that $S \equiv 0$. In this case, the eigenvalues of ${\mathcal {L}}_0=-J_0\frac{d}{dt}$ are

$$\begin{aligned} \nu _k = 2\pi k, k \in {\mathbb {Z}}. \end{aligned}$$

Each $\nu _k$ admits a 2-dimensional eigenspace

$$\begin{aligned} \{t\mapsto A(\cos (2 \pi kt),\sin (2\pi kt)) + B(\sin (2\pi kt),-\cos (2\pi kt)), t\in {\mathbb {R}}/ {\mathbb {Z}},A,B\in {\mathbb {R}}\}, \end{aligned}$$

whose eigenfunctions have winding number k.

The next theorem asserts that the general case is similar.

Theorem B.1

([19]) Let $t\mapsto S(t)$ be a smooth loop of $2\times 2$ symmetric matrices and let ${\mathcal {L}}_S$ be defined as in (8). Then

• (i) The spectrum $\sigma ({\mathcal {L}}_S)$ consists of real eigenvalues which accumulate precisely at $\pm \infty$.

• (ii) The winding number $\mathrm{wind}(\nu )\in {\mathbb {Z}},\nu \in \sigma ({\mathcal {L}}_S),$ is independent of the $\nu$-eigenfunction.

• (iii) The map

  $$\begin{aligned} \nu \mapsto \mathrm{wind}(\nu )\in {\mathbb {Z}}, \end{aligned}$$

  is a surjective increasing map. For every $k\in {\mathbb {Z}}$, there exist precisely two eigenvalues $\nu _k^1,\nu _k^2$, counting multiplicities, so that

  $$\begin{aligned} \mathrm{wind}(\nu _k^1)=\mathrm{wind}(\nu _k^2)=k. \end{aligned}$$

• (iv) If $\nu _1\ne \nu _2$ satisfy $\mathrm{wind}(\nu _1)=\mathrm{wind}(\nu _2)$, then any two $\nu _1,\nu _2$-eigenfunctions are pointwise linearly independent.

Appendix C: The generalized Conley–Zehnder index

Let $P=(x,T)$ be a closed Reeb orbit of $\lambda$ and let $\tau$ be a unitary trivialization of $x_T^*\xi$. Let ${\mathcal {L}}_S$ be the operator defined in (8). Let

$$\begin{aligned} \begin{aligned} \nu ^\tau _-(P):= \max \{ \nu : \nu \in \sigma ({\mathcal {L}}_S)\cap (-\infty ,0)\},\\ \nu ^\tau _+(P):= \min \{ \nu : \nu \in \sigma ({\mathcal {L}}_S)\cap [0,+\infty )\}, \end{aligned} \end{aligned}$$

and let

$$\begin{aligned} \mathrm{wind}_+^\tau (P):= \mathrm{wind}(\nu ^\tau _+(P)) \quad {\text{ and }} \quad \mathrm{wind}^\tau _-(P):= \mathrm{wind}(\nu ^\tau _-(P)) . \end{aligned}$$

These winding numbers satisfy $0\le \mathrm{wind}^\tau _+(P)-\mathrm{wind}^\tau _-(P)\le 1$ and they do not depend on J.

Definition C.1

( [20]) The generalized Conley–Zehnder index of $P=(x,T)$ with respect to $\tau$ is defined as

$$\begin{aligned} \mathrm{CZ}^\tau (P):=\mathrm{wind}_+^\tau (P)+\mathrm{wind}_-^\tau (P). \end{aligned}$$

This definition immediately implies that if $\nu \in \sigma (A_{P,J})$, then

$$\begin{aligned} \begin{aligned} \nu < 0 \ \Rightarrow \mathrm{wind}^\tau (\nu ) \le \frac{\mathrm{CZ}^\tau (P)}{2},\\ \nu \ge 0 \ \Rightarrow \mathrm{wind}^\tau (\nu ) \ge \frac{\mathrm{CZ}^\tau (P)}{2}. \end{aligned} \end{aligned}$$

A more geometric definition of $\mathrm{CZ}^\tau$ is as follows. Consider $\Phi _t,t\in [0,1]$, the family of $2\times 2$ symplectic matrices representing the linearized Reeb flow on $\xi$ along P in coordinates induced by $\tau$ as in the previous section.

For each initial condition $0\ne {\bar{\zeta }}$ let $\Theta _{\bar{\zeta }}(t)$ be a continuous argument of $\Phi _t{\bar{\zeta }}$ with $t\in [0,1]$. Let

$$\begin{aligned} \Delta \Theta ({\bar{\zeta }}) = \frac{\Theta _{{\bar{\zeta }}}(1) - \Theta _{{\bar{\zeta }}}(0)}{2\pi }, \end{aligned}$$

and

$$\begin{aligned} I^\tau (P):= \{ \Delta \Theta ({\bar{\zeta }}):0\ne {\bar{\zeta }} \in {\mathbb {R}}^2\}, \end{aligned}$$

be the interval containing the argument variations of all initial conditions.

The length of $I^\tau (P)$ is less than $\frac{1}{2}$ and thus, for each $\epsilon >0$ small, either $I^\tau (P)-\epsilon$ contains an integer k or is contained in between two consecutive integers k and $k+1$. In the first case we define ${{\widetilde{\mu }}}^\tau (P) := 2k$ and in the second case, ${{\widetilde{\mu }}}^\tau (P)= 2k+1$.

It is a simple exercise to check that

$$\begin{aligned} \mathrm{CZ}^\tau (P) = {{\widetilde{\mu }}}^\tau (P). \end{aligned}$$

Moreover, P is nondegenerate if and only if the boundary of $I^\tau (P)$ does not contain an integer. For a proof to these facts, see [22].

Appendix D: The quaternionic trivialization

Let $S \subset {\mathbb {R}}^4$ be a regular energy level of a Hamiltonian function H, that is $S=H^{-1}(c), c \in {\mathbb {R}},$ and $\nabla H|_S$ never vanishes. The quaternion group induces an orthonormal frame

$$\begin{aligned} TS = \mathrm{span}\{X_1,X_2,X_3\}|_S, \end{aligned}$$

(13)

spanned by the vector fields

$$\begin{aligned} X_i=A_i \dfrac{\nabla H}{|\nabla H|} \subset TS, \,\,\, i=1,2,3. \end{aligned}$$

(14)

Here, the $4\times 4$ matrices $A_i,i=1,2,3,$ are

$$\begin{aligned} A_1 = \left( \begin{array}{cc} 0 &{} J \\ J &{} 0 \end{array} \right) \quad A_2 =\left( \begin{array}{cc} J &{} 0 \\ 0 &{} -J \end{array} \right) \quad A_3 = \left( \begin{array}{cc} 0 &{} I \\ -I &{} 0 \end{array} \right) , \end{aligned}$$

where 0, I and J are the $2 \times 2$ matrices

$$\begin{aligned} 0= \left( \begin{array}{cc} 0 &{} 0 \\ 0 &{} 0 \end{array} \right) \quad I = \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array} \right) \quad J = \left( \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array} \right) . \end{aligned}$$

(15)

Observe that $X_3$ is parallel to the Hamiltonian vector field $X_H=A_3 \nabla H$.

Denote by $\phi _t$ the Hamiltonian flow of $X_H$ restricted to S. Since $d\phi _t: TS \rightarrow TS$ preserves the line bundle ${\mathbb {R}}X_3$, one may restrict the study of the linearized flow to

$$\begin{aligned} TS/{\mathbb {R}}X_3 \simeq \mathrm{span} \{X_1,X_2\}. \end{aligned}$$

As discussed in Sect. 1.1, if S has contact-type then $X_H|_S$ is parallel to a Reeb vector field R of a contact form $\lambda$ on S. In this case, the contact structure $\xi =\ker \lambda$ is transverse to R and as a result we have $\xi \simeq \mathrm{span} \{X_1,X_2\}$. This makes the quaternionic trivialization a useful tool for estimating Conley–Zehnder indices.

The orthonormal frame (13) induces a trivialization $\Psi : TS \rightarrow S \times {\mathbb {R}}^3$

$$\begin{aligned} \Psi : TS \ni a_1X_1 + a_2X_2 +a_3X_3 \mapsto (a_1,a_2,a_3)\in {\mathbb {R}}^3 \end{aligned}$$

(16)

which provides a simple form to analyze the transverse linearized flow along Hamiltonian trajectories.

Proposition D.1

In coordinates $(a_1,a_2)\in {\mathbb {R}}^2$ induced by the trivialization (16), a solution to the linearized flow along a non-constant trajectory of $X_H$, projected to $TS/{\mathbb {R}}X_3$, satisfies the equation

$$\begin{aligned} \left( \begin{array}{c}\dot{a}_1 \\ \dot{a}_2 \\ \end{array} \right) =-J M\left( \begin{array}{c} a_1 \\ a_2 \end{array} \right) , \end{aligned}$$

(17)

where J is given in (15), M is the symmetric matrix

$$\begin{aligned} M = \left( \begin{array}{cc} \kappa _{11} + \kappa _{33} &{} \kappa _{12} \\ \kappa _{12} &{} \kappa _{22}+\kappa _{33} \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned} \kappa _{ij} = \langle \mathbf{H} X_i,X_j \rangle . \end{aligned}$$

The matrix $\mathbf{H}=\mathbf{H}(x)$ is the Hessian of H at $x\in S$, $X_i, i=1,2,3$, is given by (14), and $\langle \cdot ,\cdot \rangle$ is the standard inner product on ${\mathbb {R}}^4$.

Proof

Let $x(t)\in S$ be a Hamiltonian trajectory, that is a solution to

$$\begin{aligned} \dot{x} = A_3 \nabla H(x). \end{aligned}$$

(18)

A solution $y(t)\in T_{x(t)}S$ to the linearized flow $d\phi _t:TS\rightarrow TS$ along x(t) satisfies the linear differential equation

$$\begin{aligned} \dot{y} = A_3 \mathbf{H}(x) y. \end{aligned}$$

(19)

Substituting $y=a_1X_1 + a_2X_2+a_3X_3$ in (19), we obtain

$$\begin{aligned} \sum _{i=1}^3 (\dot{a}_iX_i+a_i\dot{X}_i) = \sum _{i=1}^3 a_i A_3 \mathbf{H} X_i. \end{aligned}$$

(20)

We may assume for simplicity that $|\nabla H|=1$. In particular, it follows from (14) and (18) that

$$\begin{aligned} \dot{X}_i = A_i \mathbf{H} {\dot{x}}=A_i \mathbf{H}A_3 \nabla H=A_i\mathbf{H}X_3. \end{aligned}$$

(21)

Taking the inner product of the expression (20) with $X_1$, using (21) and the relations

$$\begin{aligned} \begin{aligned} A_i^T&=-A_i , \quad A_1A_2 = A_3, \quad A_2A_3 = A_1, \quad A_3A_1=A_2,\\ \langle X_i,X_j\rangle&= \delta _{ij} \Rightarrow \langle \dot{X}_i, X_i\rangle =0, \quad \langle \dot{X}_i, X_j\rangle = - \langle \dot{X}_j,X_i\rangle \quad \forall i,j, \end{aligned} \end{aligned}$$

we obtain

$$\begin{aligned} \dot{a}_1 =- a_2\langle \mathbf{H}X_3,X_3\rangle -a_1\langle \mathbf{H}X_1,X_2\rangle - a_2\langle \mathbf{H}X_2,X_2 \rangle . \end{aligned}$$

This is precisely the expression for $\dot{a}_1$ in (17). Analogously one obtains

$$\begin{aligned} \dot{a}_2 =a_1\langle \mathbf{H}X_1,X_1\rangle +a_1\langle \mathbf{H}X_3,X_3\rangle + a_2\langle \mathbf{H}X_1,X_2 \rangle . \end{aligned}$$