Overview

On the space of space curves, only one symplectic structure was known.

We found more.

Symplectic structure is a sandbox

for Hamiltonian dynamics

Space with

a symplectic structure $\Omega$

A function $H$

Dynamics $V_H$

Symplectic structure is a sandbox

for Hamiltonian dynamics

A symplectic structure $\Omega$ is a 2-form on a manifold $X$ s.t.

Symplectic structure is a sandbox

for Hamiltonian dynamics

A symplectic structure $\Omega$ is a 2-form on a manifold $X$ s.t.

• closed i.e. $d\Omega=0$ 

Symplectic structure is a sandbox

for Hamiltonian dynamics

A symplectic structure $\Omega$ is a 2-form on a manifold $X$ s.t.

• non-degenerate i.e. the map $$\flat: TX\to T^*X$$ $$\qquad    v\to \Omega(v, \cdot )$$ is injective

• closed i.e. $d\Omega=0$ 

Symplectic structure is a sandbox

for Hamiltonian dynamics

Symplectic structure is a sandbox

for Hamiltonian dynamics

For $H \colon X\to \mathbb{R}$,

Hamiltonian vector field $V_H\in \Gamma(TM)$ is the one

$$dH = \Omega(V_H, \cdot).$$

Symplectic structure is a sandbox

for Hamiltonian dynamics

For $H \colon X\to \mathbb{R}$,

Hamiltonian vector field $V_H\in \Gamma(TM)$ is the one

$$dH = \Omega(V_H, \cdot).$$

The Hamiltonian dynamics

$$\dot x=V_H(x).$$

Why Hamiltonian dynamics?

Has nice properties.

E.g.

• $H$ is preserved in dynamics.
• If $H$ is invariant under some group action, there is a conserved quantity.

Spaces with symplectic structures

and Hamiltonian dynamics

$T^*M^n$

$T^*\operatorname{SDiff}(M)$

Celestial mechanics

$C^\infty(M,\mathbb{C})$

 Schrödinger equation

 Incompressible fluids

Spaces with symplectic structures

and Hamiltonian dynamics

$T^*M^n$

$T^*\operatorname{SDiff}(M)$

$C^\infty(M,\mathbb{C})$

 Schrödinger equation

 Incompressible fluids

Space curves

Celestial mechanics

The space of space curves

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)=\{ c\colon \mathbb{S}^1\to\mathbb{R}^3, \partial_\theta c\neq 0 \ \forall \theta\in\mathbb{S}^1 \}$$

Parametrized curves

The space of space curves

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)=\{ c\colon \mathbb{S}^1\to\mathbb{R}^3, \partial_\theta c\neq 0 \ \forall \theta\in\mathbb{S}^1 \}$$

Parametrized curves

$$T_c\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)=C^\infty(\mathbb{S}^1,\mathbb{R}^3)$$

Tangent space

The space of space curves

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\operatorname{Diff}^+(\mathbb{S}^1)$$

Unparametrized curves a.k.a. shape space

The space of space curves

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\operatorname{Diff}^+(\mathbb{S}^1)$$

Unparametrized curves a.k.a. shape space

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\Bigg\downarrow \pi$$

$$\operatorname{Diff}^+(\mathbb{S}^1)$$

The space of space curves

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\operatorname{Diff}^+(\mathbb{S}^1)$$

Unparametrized curves a.k.a. shape space

Tangent space

$$T_{[c]}\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=T_c\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\ker d\pi_c$$

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\Bigg\downarrow \pi$$

$$\operatorname{Diff}^+(\mathbb{S}^1)$$

The space of space curves

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\operatorname{Diff}^+(\mathbb{S}^1)$$

Unparametrized curves a.k.a. shape space

Tangent space

$$T_{[c]}\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=T_c\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\ker d\pi_c$$

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\Bigg\downarrow \pi$$

$$\operatorname{Diff}^+(\mathbb{S}^1)$$

$\{a.\partial_\theta c \mid a\in C^\infty(\mathbb{S}^1)\}$

curve tangent

The space of space curves

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\operatorname{Diff}^+(\mathbb{S}^1)$$

Unparametrized curves a.k.a. shape space

Tangent space

$$T_{[c]}\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)=T_c\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)/\ker d\pi_c$$

$$\eqsim\{h \in T_c \operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)\mid h\perp \partial_\theta c \}$$

$$\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\operatorname{Imm}(\mathbb{S}^1,\mathbb{R}^3)$$

$$\Bigg\downarrow \pi$$

$$\operatorname{Diff}^+(\mathbb{S}^1)$$

$\{a.\partial_\theta c \mid a\in C^\infty(\mathbb{S}^1)\}$

curve tangent

Canonical symplectic structure on $\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$

The Marsden-Weinstein structure

$$\Omega_{[c]}\colon T_{[ c]}\operatorname{UImm}\times T_{[ c]}\operatorname{UImm}\to \mathbb{R},$$

Canonical symplectic structure on $\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$

$$\Omega_{[c]}\colon T_{[ c]}\operatorname{UImm}\times T_{[ c]}\operatorname{UImm}\to \mathbb{R},$$

The Marsden-Weinstein structure

$$\Omega_{[c]}([h],[k])=\int_{\mathbb{S}^1}\langle D_s c\times h, k\rangle ds$$

where

$$D_s = \frac{\partial_\theta }{|\partial_\theta c|}, \quad ds = |\partial_\theta c| d\theta$$

unit tangent

Canonical symplectic structure on $\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$

$$\Omega_{[c]}\colon T_{[ c]}\operatorname{UImm}\times T_{[ c]}\operatorname{UImm}\to \mathbb{R},$$

Symplectic!

The Marsden-Weinstein structure

$$\Omega_{[c]}([h],[k])=\int_{\mathbb{S}^1}\langle D_s c\times h, k\rangle ds$$

where

$$D_s = \frac{\partial_\theta }{|\partial_\theta c|}, \quad ds = |\partial_\theta c| d\theta$$

unit tangent

Canonical symplectic structure on $\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$

$$\Omega_{[c]}\colon T_{[ c]}\operatorname{UImm}\times T_{[ c]}\operatorname{UImm}\to \mathbb{R},$$

Closed $d\Omega=0$,

Symplectic!

The Marsden-Weinstein structure

$$\Omega_{[c]}([h],[k])=\int_{\mathbb{S}^1}\langle D_s c\times h, k\rangle ds$$

where

$$D_s = \frac{\partial_\theta }{|\partial_\theta c|}, \quad ds = |\partial_\theta c| d\theta$$

unit tangent

Canonical symplectic structure on $\operatorname{UImm}(\mathbb{S}^1,\mathbb{R}^3)$

$$\Omega_{[c]}\colon T_{[ c]}\operatorname{UImm}\times T_{[ c]}\operatorname{UImm}\to \mathbb{R},$$

Closed $d\Omega=0$,

Symplectic!

The Marsden-Weinstein structure

$$\Omega_{[c]}([h],[k])=\int_{\mathbb{S}^1}\langle D_s c\times h, k\rangle ds$$

where

$$D_s = \frac{\partial_\theta }{|\partial_\theta c|}, \quad ds = |\partial_\theta c| d\theta$$

Non-degenerate $\ker \Omega_{[c]}=[0]=\{a.D_s c \mid a \in C^\infty(\mathbb{S}^1)\}$

unit tangent

Hamiltonian flow

E.g. 

Hamiltonian $H = \operatorname{Length}([c])$

Binormal flow

Dynamics $V_H=[D_s c \times D_s^2 c]$

Hamiltonian flow

E.g. 

Hamiltonian $H = \operatorname{Length}([c])$

Binormal flow

Dynamics $V_H=[D_s c \times D_s^2 c]$

A completely integrable system

More symplectic structures?

More symplectic structures?

Remember

A symplectic structure $\Omega$ defines a Hamiltonian dynamics $V_H$ via $$dH=\Omega(V_H,\cdot).$$

More symplectic structures?

A different $\Omega$ defines different $V_H$.

Remember

A symplectic structure $\Omega$ defines a Hamiltonian dynamics $V_H$ via $$dH=\Omega(V_H,\cdot).$$

More symplectic structures?

A different $\Omega$ defines different $V_H$.

But only one $\Omega$ is known on $\operatorname{UImm}$.

Remember

A symplectic structure $\Omega$ defines a Hamiltonian dynamics $V_H$ via $$dH=\Omega(V_H,\cdot).$$

More symplectic structures?

A different $\Omega$ defines different $V_H$.

We found more.

But only one $\Omega$ is known on $\operatorname{UImm}$.

Remember

A symplectic structure $\Omega$ defines a Hamiltonian dynamics $V_H$ via $$dH=\Omega(V_H,\cdot).$$

Hint 1. Trend in shape analysis

Hint 1. Trend in shape analysis

a Riemannian metric on $\operatorname{Imm}$

$$g(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

Hint 1. Trend in shape analysis

a Riemannian metric on $\operatorname{Imm}$

$$g(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

a suitable operator

$$L\colon T\operatorname{Imm}\to T\operatorname{Imm}.$$

Hint 1. Trend in shape analysis

a Riemannian metric on $\operatorname{Imm}$

$$g(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

New metric $$g^L(h,k)\coloneqq g(h,Lk)$$

a suitable operator

$$L\colon T\operatorname{Imm}\to T\operatorname{Imm}.$$

e.g. $L_c=\operatorname{id}_c$,

$$g^{\operatorname{id}}_c(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

Hint 1. Trend in shape analysis

e.g. $L_c=\operatorname{id}_c$,

$$g^{\operatorname{id}}_c(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

e.g. $L_c=\lambda(c)$ with a conformal factor $\lambda\colon \operatorname{Imm}\to \mathbb{R}_{>0}$

$$g^{\lambda}_c(h,k)=\lambda(c)\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

Hint 1. Trend in shape analysis

e.g. $L_c=\operatorname{id}_c$,

$$g^{\operatorname{id}}_c(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

e.g. Sobolev-type differential operator  $L_c=\operatorname{id}-D_s^2$

$$g^{\operatorname{id}-D_s^2}_c(h,k)=\int_{\mathbb{S}^1} \langle h,k\rangle -\langle h, D_s^2 k \rangle ds = \int_{\mathbb{S}^1} \langle h,k\rangle + \langle D_s h, D_s k \rangle ds$$

Hint 1. Trend in shape analysis

e.g. $L_c=\lambda(c)$ with a conformal factor $\lambda\colon \operatorname{Imm}\to \mathbb{R}_{>0}$

$$g^{\lambda}_c(h,k)=\lambda(c)\int_{\mathbb{S}^1} \langle h,k\rangle ds$$

Hint 2. Liouville form

$\Omega$ is exact

i.e. there is a 1-form $\Theta$ s.t. $\Omega=d\Theta$.

Hint 2. Liouville form

$\Omega$ is exact

$\Theta$ is given by 

$$\Theta_{[c]}([h])=\frac{1}{3}\int_{\mathbb{S}^1}\langle D_s c\times c, h\rangle  ds=g^{\operatorname{id}}\left(\frac{1}{3}D_s c\times c, h\right).$$

i.e. there is a 1-form $\Theta$ s.t. $\Omega=d\Theta$.

Hint 1 + Hint 2

Let's define a 1-form

$$\Theta_{[c]}^{\color{blue}L}([h])\coloneqq g^{\color{blue}L}\left(\frac{1}{3}D_s c\times c, h\right)= \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, {\color{blue}L} h\rangle ds$$

Hint 1 + Hint 2

and a 2-form $$\Omega^L\coloneqq d\Theta^L$$

Let's define a 1-form

$$\Theta_{[c]}^{\color{blue}L}([h])\coloneqq g^{\color{blue}L}\left(\frac{1}{3}D_s c\times c, h\right)= \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, {\color{blue}L} h\rangle ds$$

$$\Omega^L\coloneqq d\Theta^L$$

Does $\Omega^L$ define a symplectic structure on $\operatorname{UImm}$?.

Hint 1 + Hint 2

with  $\Theta_{[c]}^L([h])\coloneqq \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, L h\rangle ds$

Closeness. OK because it is exact  $d\Omega^L=dd\Theta^L=0$

$$\Omega^L\coloneqq d\Theta^L$$

Does $\Omega^L$ define a symplectic structure on $\operatorname{UImm}$?.

Hint 1 + Hint 2

with  $\Theta_{[c]}^L([h])\coloneqq \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, L h\rangle ds$

Closeness. OK because it is exact  $d\Omega^L=dd\Theta^L=0$

$$\Omega^L\coloneqq d\Theta^L$$

Does $\Omega^L$ define a symplectic structure on $\operatorname{UImm}$?.

Hint 1 + Hint 2

with  $\Theta_{[c]}^L([h])\coloneqq \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, L h\rangle ds$

Non-degeneracy on $\operatorname{UImm}$ ?

i.e. $\ker \Omega_c^L = [0] = \{a. D_s c \mid a \in C^\infty(\mathbb{S}^1)\}$

Closeness. OK because it is exact  $d\Omega^L=dd\Theta^L=0$

$$\Omega^L\coloneqq d\Theta^L$$

Does $\Omega^L$ define a symplectic structure on $\operatorname{UImm}$?.

Non-degeneracy on $\operatorname{UImm}$ ?

i.e. $\ker \Omega_c^L = [0] = \{a. D_s c \mid a \in C^\infty(\mathbb{S}^1)\}$

Need to check case by case.

Hint 1 + Hint 2

with  $\Theta_{[c]}^L([h])\coloneqq \frac{1}{3} \int_{\mathbb{S}^1} \langle D_s c \times c, L h\rangle ds$

Symplectic structure

induced by a conformal factor

$$\Omega^\lambda\coloneqq d\Theta^\lambda$$

Theorem

With a conformal factor $\lambda\colon \operatorname{UImm}\to \mathbb{R}_{>0}$,

is a symplectic structure on $\operatorname{UImm}$ 

Symplectic structure

induced by a conformal factor

$$\Omega^\lambda\coloneqq d\Theta^\lambda$$

Theorem

With a conformal factor $\lambda\colon \operatorname{UImm}\to \mathbb{R}_{>0}$,

is a symplectic structure on $\operatorname{UImm}$ 

if $\Theta^\lambda$ is NOT invariant under scaling ($c\to c+t c$ with $t \in \mathbb{R}^\times$).

Symplectic structure

induced by a conformal factor

$$\Omega^\lambda\coloneqq d\Theta^\lambda$$

Theorem

With a conformal factor $\lambda\colon \operatorname{UImm}\to \mathbb{R}_{>0}$,

is a symplectic structure on $\operatorname{UImm}$ 

If $\Theta^\lambda$ is invariant under scaling, $\Omega^\lambda$ is NOT symplectic on $\operatorname{UImm}$,

but becomes symplectic in another space! (Check out our paper)

if $\Theta^\lambda$ is NOT invariant under scaling ($c\to c+t c$ with $t \in \mathbb{R}^\times$).

Open problem

Is there a non-conformal $L$ making $\Omega^L\coloneqq d\Theta^L$ symplectic?

Open problem

Is there a non-conformal $L$ making $\Omega^L\coloneqq d\Theta^L$ symplectic?

Conjecture

Squared curvature $L|_\theta=1+\kappa^2_c(\theta)$ defines a symplectic structure.

Open problem

Is there a non-conformal $L$ making $\Omega^L\coloneqq d\Theta^L$ symplectic?

Conjecture

Squared curvature $L|_\theta=1+\kappa^2_c(\theta)$ defines a symplectic structure.

Result in preparation

Squared scale $L|_\theta=|c(\theta)|^2$ defines a symplectic structure.

Simulation of Hamiltonian dynamics

Remember, different symplectic structures induce different dynamics from the same Hamiltonian.

(Marsden-Weinstein)

$\Omega_c^{\operatorname{id}}$

$\Omega_c^{\operatorname{Length}(c)^{-1/10}}$

$\Omega_c^{\operatorname{Length}(c)^{2}}$

Simulation of Hamiltonian dynamics

Hamiltonian dynamics $V_H$ of total squared-scale  $$H(c)=\int_{\mathbb{S}^1}|c|^2 ds$$

$H$ is preserved in all of them.

Future work

Future work

Our paper has a disclaimer

We do not guarantee any correctness of (even short-time behavior) dynamics in our simulation

Future work

Symplectic structure on the space of discrete space curves?

Toward a symplectic integrator for space curves

Our paper has a disclaimer

We do not guarantee any correctness of (even short-time behavior) dynamics in our simulation

Future work

Symplectic structure on the space of discrete space curves?

Toward a symplectic integrator for space curves

Looking for a collaborator!

Our paper has a disclaimer

We do not guarantee any correctness of (even short-time behavior) dynamics in our simulation

Thanks!

Preprint: arXiv 2407.19908

Sadashige.Ishida@ist.ac.at