Commuting Hamiltonian Flows of Curves in Real Space Forms

Albert Chern
TU Berlin
Franz Pedit
UMass Amherst
Ulrich Pinkall
TU Berlin

Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infinitesimal invariant, we describe the flows directly as vector fields on the manifold of space curves. This manifold carries a canonical symplectic form introduced by Marsden and Weinstein. Our flows are precisely the symplectic gradients of a natural hierarchy of invariants, beginning with length, total torsion, and elastic energy. There are a number of advantages to our geometric approach. For instance, the real part of the spectral curve is geometrically realized as the motion of the monodromy axis when varying total torsion. This insight provides a new explicit formula for the hierarchy of Hamiltonians. We also interpret the complex spectral curve in terms of curves in hyperbolic space and Darboux transforms. Furthermore, we complete the hierarchy of Hamiltonians by adding area and volume. These allow for the characterization of elastic curves as solutions to an isoperimetric problem: elastica are the critical points of length while fixing area and volume.

The paper is based on a lecture course the third author gave during his stay at the Yau Institute at Tsinghua University in Beijing during Spring 2018. The development of the material was also supported by SFB Transregio 109 "Discretization in Geometry and Dynamics" at Technical University Berlin. Software support for the images was provided by SideFX.

Background

What can we measure from a space curve \(\gamma\colon I\rightarrow \Bbb R^3\)? The first thing that comes up in one's mind is the total length of the curve \(L=\int_I dx\), where \(dx\) is the arclength measure. For reasons that come later, we label this geometric invariant \[E_1 = L.\] One can also measure the area vector (Kepler) of a space curve \[E_{-1} = {1\over 2}\int_I\gamma\times d\gamma,\] whose component \(E_{-1}^\upsilon = (\upsilon,E_{-1})\) along a unit vector \(\upsilon\) is the enclosed area of the 2D image of the curve projected on \(\upsilon^\bot\). One can even talk about the volume of a space curve. Given an origin and an axis \(\upsilon\), \[E_{-2}^\upsilon = {1\over 2}\int_I \left(|\gamma-(\gamma,\upsilon)|^2 d\gamma,\upsilon\right)\] is proportional to the (signed) volume of the solid torus generated by revolving the space curve around the given axis. In answering the question of determining the shape of bent beams posed by his uncle Jakob Bernoulli, Daniel Bernoulli in 1742 realized that these bent beams are the minimizers of the so called bending energy \[E_3 = \int_I|\gamma^{\prime\prime}|^2dx,\] where \((\cdot)^\prime = {d\over dx}\) is the derivative with respect to arclength. Euler then classified in 1744 all planar elastic curves. To generalize 2D Bernoulli–Euler elastica to a realistic model for 3D elastic rods, Binét in 1844 argued that one must include the total torsion \(E_2\) in its energy.

Here, the total torsion is defined for quasi-periodic curves. A quasi-periodic space curve is given by \(\gamma\colon\Bbb R\rightarrow\Bbb R^3\) where there exists a translation \(\mathfrak{t}\colon\Bbb R\rightarrow\Bbb R\) in the domain and a \(\Bbb R^3\) rigid motion \(h(p) = Ap + a\) so that \(\gamma\circ\mathfrak{t} = h\circ\gamma\). Let \(\nu\) be an arbitrary unit normal vector field of \(\gamma\) sharing the same quasi-periodicity; that is, \(\nu\circ\mathfrak{t} = A\circ\nu\). Then the total torsion is given by the total turning angle \[E_2 = \int_I (\nu',T\times\nu)dx\] where \(T = \gamma'\) is the tangent vector, and \(I\) is the fundamental domain for \(\Bbb R/\tau\). This quantity is independent (up to an integer multiple of \(2\pi\)) of the choice of \(\nu\). For those who learned the notion of torsion from Frenet theory: the Frenet torsion is the turning speed \((\nu',T\times\nu)\) of the special frame \(\nu = \gamma^{\prime\prime}/|\gamma^{\prime\prime}|\), in which case the total torsion can be computed by the total Frenet torsion. Note that Frenet theory works only when the curve has no inflection point (\(\gamma^{\prime\prime}=0\)), which is not a natural framework for discussing many important space curves. Hence it is important to think of the total torsion as a geometric invariant which does not depend on the Frenet theory. Every quasi-periodic framed curve (a curve endowed with a \(\nu\)) has its own frame-dependent torsion, whose integral depends only on the curve geometry independent of \(\nu\).

In addition to the geometric invariants \(E_k\) mentioned above, an important dynamical equation for space curves were discovered in the early 20th century. Da Rios and Levi-Civita in 1906 discovered that a thin vortex filament in a nearly inviscid fluid evolves according to the so-called vortex filament equation (a.k.a. binormal equation, localized induction, Da Rios equation, smoke ring flow) \[\dot{\gamma} = Y_1 \equiv \gamma^{\prime}\times\gamma^{\prime\prime}\mod C^{\infty}_{\frak t}(\Bbb R)T\] which is modulo any smooth periodic function multiple of the tangent vector as they do not change the shape of the curve. As described by Marsden and Weinstein in 1983, \(Y_1\) is in fact the symplectic gradient of total length \(E_1\) with respect to the (pre-)symplectic form \(\sigma(\dot\gamma,\mathring\gamma) = \int_I {\rm det}(d\gamma,\dot\gamma,\mathring\gamma)\) on the space of curves. The Marsden–Weinstein form is degenerate exactly in the direction of \(C^{\infty}_{\frak t}(\Bbb R)T\), which is "good enough" for inducing well-defined symplectic gradients up to tangential flows.

Now, using the Marsden–Weinstein form, we may take each \(E_k\) as the Hamiltonian and take its symplectic gradient \(Y_k\). That is, \(\sigma(Y_k,\mathring\gamma) = \mathring E\), where \(\mathring E\) is the variation with respect to the variation \(\mathring\gamma\). In order to drop the modulo signs, we choose a "canonical" representative for the symplectic gradient. The tangential flow component will be chosen in such a way so that the curve is arclength preserving over time. That is \((Y_k',T)=0\). This can be achieved for all examples below as they all happen to be total length preserving. In the case of \(Y_1\), \[Y_1 = \gamma'\times\gamma^{\prime\prime}.\]

As a simple exercise one shows that the symplectic gradient of \(E_{-1}^\upsilon\) (area vector in the \(\upsilon\) component) is a pure translation \[Y_{-1} = \upsilon.\] The symplectic gradient of \(E_{-2}^\upsilon\) is a pure rotation \[Y_{-2} = \upsilon\times\gamma.\] The symplectic gradient of the total torsion \(E_2\), whose physical interpretation is given by Holm and Stechmann in 2004, is the helicity filament flow \[Y_2 = -\gamma^{\prime\prime\prime}-{3\over 2}|\gamma^{\prime\prime}|^2\gamma^\prime\] which can be useful for modeling magnetic filaments in magnetohydrodynamics.

In the process of deriving the symplectic gradient \(Y_k\), one takes the variation of \(E_k\). If instead of using the symplectic form one uses the standard \(L^2\) metric \(\langle\!\langle\dot\gamma,\mathring\gamma\rangle\!\rangle :=\int_I (\dot\gamma,\mathring\gamma)dx\), then these variations are turned into \(L^2\) gradients \(G_k\) which satisfies \(\langle\!\langle G_k,\mathring\gamma\rangle\!\rangle = \mathring E_k\). One then finds \begin{align*} & G_{-2} = \gamma^\prime\times(\upsilon\times\gamma) && Y_{-2} = \upsilon\times\gamma\\ & G_{-1} = \gamma'\times\upsilon && Y_{-1} = \upsilon\\ & G_0 = 0 && Y_0 = \gamma'\\ & G_1 = -\gamma^{\prime\prime} && Y_1 = \gamma'\times\gamma^{\prime\prime}\\ & G_2 = -\gamma'\times\gamma^{\prime\prime\prime} && Y_2 = -\gamma^{\prime\prime\prime} - {3\over 2}|\gamma^{\prime\prime}|^2\gamma^\prime\\ & G_3 = \left(\gamma^{\prime\prime\prime}+{3\over 2}|\gamma^{\prime\prime}|^2\gamma^\prime\right)^\prime && Y_3 = -\gamma^\prime\times\gamma^{\prime\prime\prime\prime} - {3\over 2}|\gamma^{\prime\prime}|^2\gamma^\prime\times\gamma^{\prime\prime} + {\rm det}(\gamma',\gamma^{\prime\prime},\gamma^{\prime\prime\prime})\gamma^\prime. \end{align*} Here we add \(G_0\), \(Y_0\), the gradient and symplectic gradient of some tautological constant energy. The curious recursion pattern then emerges: \[G_{k+1} = -Y_k^\prime,\quad Y_k^\prime + T\times Y_{k+1},\quad Y_0 = T.\]

This recursion continues for \(k>3\), giving a hierarchy of flows \(Y_k\). In fact, for all \(k,\ell>0\),

That is, each \(Y_k, Y_\ell\) commute up to tangential flow. The dynamics of curves evolved by each (linear combinations) of \(Y_k\)'s is a complete integrable system.

This has not been a surprise since 1972, which is when Hasimoto discovered the following. As a curve evolves according to vortex filament equation, its complex curvature evolves according to the 1D non-linear Schrödinger equation — an integrable system. In the context of the non-linear Schrödinger equation and soliton theory, Faddeev and Takhtajan (1987) gave a detailed calculation of the recursions and the invariants \(E_k\). The generated hierarchy also coincides with the modified Korteweg—de Vries (mKdV) hierarchy. In 1991, Langer and Perline brought the results from the non-linear Schrödinger soliton theory to space curves. Relating to the non-linear Schrödinger theory through the Hasimoto transformation has been a standard approach in the study of the integrable system of space curves.

Contribution

We give a new exposition to the space curve hierarchy. In contrast to traditional approach, where one works with the complex curvature (Hasimoto transforms), we describe the flows \(Y_k\) directly as vector fields on the manifold of space curves. At the level of space curves, with the loop group method, where one studies the generating function \(\hat Y_\lambda = \sum_{k}Y_k\lambda^{-k}\) a.k.a. spectral curve, we show that \(Y_k\) commute not only modulo tangential flows, but genuinely commute as vector fields.

We add \(E_{-1}\), \(E_{-2}\) to the list of Hamiltonians. This enables new isoperimetric characterizations of finite gap curves (curves with finite spectral genus). For example, as it surprisingly turns out, the shortest space curves with fixed area and volume (critical \(E_1\) constraining \(E_{-1}\) and \(E_{-2}\)) are exactly the 3D Euler elastica (critical \(E_3\) constraining \(E_1\) and \(E_2\)), which is a significant reduction of order.

In contrast to previous work, where \(E_k\) are calculated from the non-linear Schrödinger hierarchy, we derive a new explicit formula for these invariants. The basic ingredient comes from realizing the spectral curve \(\hat Y_\lambda\) as the motion of the monodromy axis when varying the total torsion. Aided by Gauß–Bonnet Theorem \(\hat E_\lambda\) is expressed in terms of the area of a spherical region.

While a real parameter \(\lambda\) controls the total torsion of a space curve, a complex \(\lambda\) "bends" the underlying space into a hyperbolic 3-space. This reveals a relation between \(\hat Y_\lambda\) and Darboux transforms of the space curve.