Geodesics on $SL(n)$ with the Hilbert-Schmidt metric, and one application to fluids

Geodesics on $SL(n)$ with the Hilbert-Schmidt metric, and one application to fluids

Willie Wai Yeung Wong
wongwwy@math.msu.edu
Michigan State University

Slides available :   https://slides-n-notes.qnlw.info

Credits page

• Joint work with:
  • Audrey Rosevear, Amherst College
  • Samuel Sottile, Michigan State University
• Support by NSF through the SURIEM (2020) summer REU site at MSU.
• Paper to appear in La Matematica; arXiv:2101.09266.

Audrey will present a poster on this at JMM2022; do stop by and say hi!

Outline

1. Set-up
2. Physics
3. Geometry; prior results
4. Our results
5. Application; what's next?

1. Set-up

Dramatis personae

• $M(n)$ — space of $n\times n$ real matrices
• $S(n)$ — symmetric elements of $M(n)$
• $SL(n)$ — determinant 1 elements of $M(n)$
• $SO(n)$ — elements of $SL(n)$ satisfying $A^T=A^{-1}$
• ${\mathring{S}}_{+}(n)$ — positive definite elements of $S(n)\cap SL(n)$
• $\mathfrak{sl}(n)$ — trace 0 elements of $M(n)$
• $\mathfrak{so}(n)$ — anti-symmetric elements of $M(n)$
• $⟨A,B⟩:=\mathrm{tr}(A{B}^T)$ — Hilbert-Schmidt inner product on $M(n)$

Hilbert-Schmidt metric on $SL(n)$

$M(n)$ with $⟨A,B⟩$ $\cong$ ${\mathbb{R}}^{n\times n}$ with standard metric.

$SL(n)$ is a co-dimension 1;
induced Riemannian metric — the HS metric.

Not bi-invariant!
(Non-compact, but semi-simple, so Killing form is pseudo-Riemannian.)

The Question

What is the geometry of this Riemannian manifold?

1. Geodesic completeness? — ✓
2. Existence / stability of bounded geodesics?
3. Asymptotic behavior and stability of unbounded geodesics?

Why emphasis on geodesics?

2. Physics

Rigid Body

Rigid Body

• Initial/reference body: $\mathrm{\Omega }⋐{\mathbb{R}}^n$
  Center of mass condition: ${\displaystyle {\int }_{\mathrm{\Omega }}y\mathrm{d}y=0}$
• Configuration at time $t$: $x(t)+A(t)\cdot \mathrm{\Omega };x(t)\in {\mathbb{R}}^n,A\in SO(n)$
• Total kinetic energy: $${\int }_{\mathrm{\Omega }}|x^{\prime }(t)+A^{\prime }(t)y{|}^2\mathrm{d}y=\mathrm{vol}(\mathrm{\Omega })|x^{\prime }(t){|}^2+\mathrm{tr}(A^{\prime }(t)I_{\mathrm{\Omega }}{A}^{\prime }(t{)}^T)$$
• Moment of inertia: ${\displaystyle I_{\mathrm{\Omega }}={\int }_{\mathrm{\Omega }}y{y}^T\mathrm{d}y}$

For write-up with more details: click here.

Action Principle

$$S=\int [\mathrm{vol}(\mathrm{\Omega })|x^{\prime }(t){|}^2+\mathrm{tr}(A^{\prime }(t)I_{\mathrm{\Omega }}{A}^{\prime }(t{)}^T)]dt$$

The integrand defines a positive definite quadratic form on the tangent space of ${\mathbb{R}}^n\times SO(n)$

Solutions are geodesics w.r.t. this Riemannian metric.
Note that the metric given by $I_{\mathrm{\Omega }}$ is not the bi-invariant metric on $SO(n)$.

Motion on ${\mathbb{R}}^n$ decouples from motion on $SO(n)$.

Affine Motion

• Allow the body to also deform linearly but preserving volume.
• Swap $SO(n)$ with $SL(n)$.
• Everything else the same: $I_{\mathrm{\Omega }}$ induces a Riemannian metric on $SL(n)$ which determines the trajectories.

Why do we only care when $I_{\mathrm{\Omega }}={\mathrm{Id}}_n$?

Incompressible Fluids

• Allow the body to arbitrarily deform, but preserving infinitesimal volume.
• Replace $SO(n)$ by the group(oid) of volume preserving diffeomorphisms.
• For fluid flow inside a fixed domain $\mathrm{\Omega }$: Euler-Arnol'd equations.
• Free boundary problem: requires boundary conditions.
  • At any fixed time: fluid pressure is constant along (moving) boundary.

Relating affine motion to fluid motion

• $SL(n)$ is a subgroup of the group of volume preserving diffeos of ${\mathbb{R}}^n$.
• Boundary conditions satisfied by $SL(n)$ geodesic $⟺$ under the Riemannian metric induced by the geometry of $\mathrm{\Omega }$, $SL(n)$ is totally geodesic.
• Theorem. This holds iff $\mathrm{\Omega }$ is a round ball (and hence $I_{\mathrm{\Omega }}={\mathrm{Id}}_n$).

3. Geometry; prior results

Discrete symmetry

$A\to A^T$ is isometry of $M(n)$, and fixes $SL(n)$, so is isometry on $SL(n)$.

Lemma. $S(n)\cap SL(n)$ is totally geodesic in $SL(n)$.

Corollary. ${\mathring{S}}_{+}(n)$ is totally geodesic in $SL(n)$.

Continuous symmetries

• $SO(n)$ acting on $M(n)$ by matrix multiplication is isometric & fixes $SL(n)$
• Distinct left and right actions!
  • Generate a total of $n(n-1)$ Killing vector fields
  • Physical interpretation: $$ angular momentum and vorticity
• Conjugate action: $A\mapsto OA{O}^{-1}$ (fixes ${\mathring{S}}_{+}(n)$)
• Composite action: $A\mapsto OAO$ (not faithful)

Polar decomposition

Every $A\in SL(n)$ has a unique factorization as $OP$ (or $PO$): $\{\begin{array}{l}O\in SO(n)\\ P\in {\mathring{S}}_{+}(n)\end{array}$

$SL(n)$ foliated by cosets

So the geodesics are easy to describe, right?

Problem: Left actions generate right-invariant vector fields and vice versa

... so even with our conserved quantities, the motions on the two factors do not split.

$n=2$ is special: analysis

• Motion is completely integrable: $\dim (SL(2))=3$ and conservation of
  • Energy
  • Angular momentum
  • Vorticity
• Used in Roberts-Shkoller-Sideris (CMP 2020) where geodesics are fully classified.

Will return to this classification a bit later.

$n=2$ is special: geometry

• $SL(2)$ is a warped product ${\mathring{S}}_{+}(2){\times }_f{\mathbb{S}}^1$
• Use the composite $A\mapsto OAO$ action
  • double cover of $SL(2)$ by ${\mathring{S}}_{+}(2){\times }_{f}SO(2)$.
  • Use $SO(2)=U(1)$ and "take square roots"
• $SO(2)$ is abelian:
  • Composite $OAO$ action is orthogonal to cosets of polar decomposition
  • Conjugate $OA{O}^{-1}$ action is tangent to cosets
• Decompose as integrable motion on ${\mathring{S}}_{+}(2)$ plus fibre motion.
• (Private communication from Ben Schmidt)

Definitions

Throat

Call $SO(n)⊊SL(n)$ the "throat": it describes the points closest to the origin in $M(n)$.

Second fundamental form

Sign convention: $II(X,X)>0$ if it curves away from the origin.

Second f.f. defined via normal pointing away from origin of $M(n)$.

Geodesic equation

Let $X,Y\in T_{A}SL(n)$, then $$II(X,Y)=\frac{\mathrm{tr}(A^{-1}X{A}^{-1}Y)}{|A^{-1}|}$$ and the geodesic equation reads $$\ddot{A}=\frac{\mathrm{tr}(A^{-1}\dot{A}{A}^{-1}\dot{A})}{\mathrm{tr}(A^{-1}{A}^{-T})}{A}^{-T}.$$

$n=2$ geodesic classification

• The only bounded geodesic orbits the throat
• There exists semi-bounded geodesics converging to the throat
• All unbounded geodesics are "asymptotically linear" ($II(\dot{\gamma },\dot{\gamma })\to 0$)

How much of these survive in higher dimensions?

$n=3$ prior results

Analyses of some special explicit solutions and their asymptotics

Sideris (ARMA 2017)

$n$-independent prior results

• Virial Identity: let $A:t\to M(n)$ a $SL(n)$-geodesic $$\frac{d^2}{d{t}^2}|A{|}^2=\frac{|\dot{A}{|}^2}{|A|}+\frac{n}{|A{|}^2}II(\dot{A},\dot{A}).$$
• Proposition: if additionally $A$ is such that $II(\dot{A},\dot{A})>0$ for all $t>t_0$, then $A$ is unbounded and asymptotically linear.

Sideris (ARMA 2017)

4. Our results

Higher dimensions are more curvy

• Theorem. $SL(2)$ has bounded sectional curvature.
• Theorem. When $n\ge 3$, given $A\in SL(n)$ and $W\subset T_{A}SL(n)$ a plane, denote by $k(W)$ its sectional curvature. Then:
  1. $k(W)\le |A^{-1}{|}^2$ for all $(A,W)$;
  2. There exists a sequence $(A_i,W_i)$ with
     $|A_i^{-1}|\to \mathrm{\infty }$ and $lim infk(W_i)|A_i^{-1}{|}^{-2}\ge \frac{1}{4}$.
  (Remark: the function $A\mapsto |A^{-1}|$ is proper.)

Linear solutions are linearly stable

• Linear solutions: those whose images are lines in $M(n)$.
  All in the form on $A(t)=A_0(\mathrm{Id}+tM)$ where $M$ nilpotent.
• Linear stability: Jacobi fields along such curves grow at most linearly.
• Key step: showing for such $A(t)$ the quantity $$|\mathrm{Riem}(\dot{A},J,\dot{A},K)|\lesssim \frac{|J||K|}{(1+t^4)}.$$

Bounded geodesics abound in $n\ge 4$

• Theorem. When $n=2m\ge 4$, for every $\eta \ge |\mathrm{Id}|$, there exists a bounded geodesic $A(t)$ with $|A(t)|\equiv \eta$.
• In fact, we can write $A(t)=A_0\exp (Ct)$ with $A_0\in {\mathring{S}}_{+}(n)$ and $C\in \mathfrak{so}(n)$.
• Remark: these are the only "non-linear" "exponential" solutions.
• Physical picture: Stationary fluid solution; an ellipsoid with "paired" axes, spinning separately in each plane.
• When the rotational speeds are not "tuned", the solutions remain bounded, but pulsates.

Refinement of Virial argument

$$\frac{d^2}{d{t}^2}|A{|}^2=\frac{|\dot{A}{|}^2}{|A|}+\frac{n}{|A{|}^2}II(\dot{A},\dot{A}).$$

Lemma. When $n=2$, we have $\frac{d^2}{d{t}^2}|A(t){|}^2\le 0⟹|A(t)|=|\mathrm{Id}|$.

Corollary. All geodesics except for the throat rotation are unbounded.

Emphatically false when $n\ge 3$.

Virial revisited

Theorem. [sufficient conditions for unboundedness]

1. Solution is tangent to ${\mathring{S}}_{+}(n)$ (or another coset)
2. Solution has vanishing total angular momentum
3. Solution has vanishing total vorticity

New formulation of equations

Given solution $A(t)$, define $$\beta =A^{-1}{A}^{-T},\omega =A^T\dot{A}+{\dot{A}}^{T}A,\zeta =A^T\dot{A}-{\dot{A}}^{T}A$$

• $\beta$: "square" of the "radial component" in polar decomposition
• $\omega$: "radial" velocity
• $\zeta$: "angular" velocity

Above is the vorticity version; angular momentum version moves the transpose to the other factor.

Cannot do both simultaneously.

New formulation of equations

Solves problem with Killing vector field not pointing in the fibre direction.

$$\begin{array}{rl}\frac{d}{dt}\beta & =-\beta \omega \beta \\ \frac{d}{dt}\omega & =\frac{1}{2}(\omega +\zeta {)}^T\beta (\omega +\zeta )+\underset{\ge 0}{\underbrace{\frac{\mathrm{tr}\omega \beta \omega \beta }{2\mathrm{tr}\beta }}}I+\underset{\le 0}{\underbrace{\frac{\mathrm{tr}\zeta \beta \zeta \beta }{2\mathrm{tr}\beta }}}I\\ \frac{d}{dt}\zeta & =0\end{array}$$

Compatibility condition: $\mathrm{tr}\beta \omega =0$.

Preserves "block diagonal structure"

Block diagonal solutions

Assume $\beta ,\omega ,\zeta$ decompose into $2\times 2$ blocks (with one $1\times 1$ block when $n$ is odd) along the diagonal.

A boundedness criterion

Theorem. Let $n$ be even. If the initial data of $\beta ,\omega$ are built from $2\times 2$ blocks that are pure-trace, and if $\zeta$ has no vanishing bocks, then the solution is bounded.

When $n=2$, hypothesis only possible with rotation at throat.

"Swirling and shear flows"

Theorem. If the initial data has the form $${\beta }_0=\left(\begin{array}{c}{b}_1{\mathrm{Id}}_{2m}\\ & b_2{\mathrm{Id}}_{n-2m}\end{array}\right),{\omega }_0=\left(\begin{array}{c}{w}_1{\mathrm{Id}}_{2m}\\ & w_2{\mathrm{Id}}_{n-2m}\end{array}\right)$$ and writing $\epsilon$ for the antisymmetric $2\times 2$ matrix $${\zeta }_0=\left(\begin{array}{c}z(\underset{m\text{ copies}}{\underbrace{\epsilon \oplus \cdots \oplus \epsilon }})\\ & 0\end{array}\right);$$ then the corresponding solution is unbounded, and asymptotically linear.

Sideris [ARMA 2017] described $n=3$ and $m=1$ with explicit integration

Further generalization?

5. Application; what's next?

Asymptotically linear solutions can be unstable

• Let $A(t)$ be a swirling and shear flow: we can show that $II(\dot{A},\dot{A})$ is eventually positive.
• So $A(t)$ is "asymptotically linear"
• "Small perturbations" $${\zeta }_0=\left(\begin{array}{c}z(\underset{m\text{ copies}}{\underbrace{\epsilon \oplus \cdots \oplus \epsilon }})\\ & 0\end{array}\right)⟹\left(\begin{array}{c}z(\underset{m\text{ copies}}{\underbrace{\epsilon \oplus \cdots \oplus \epsilon }})\\ & \delta (\epsilon \oplus \cdots \oplus \epsilon )\end{array}\right)$$
• Non-vanishing rotation $⟹$ motion is bounded

Connection to fluids

• Bounded motions must enter regime with negative second fundamental form.
• Physical instability: Rayleigh-Taylor. (Require pressure just inside the boundary be positive.)

Further questions

• Conjecture. When $n$ is odd, $SL(n)$ has no bounded geodesics.
• Conjecture. The only solutions with $|A(t)|$ constant are those we found.
• Question. Do there exist unbounded geodesics which are not asymptotically linear when $n>2$?
• Question. Do there exist "heteroclinic orbits"?

Thank you!