Difference Moving Frames
Elizabeth Mansfield

Based on joint work with:

Gloria Marı̀ Beffa (Madison, Wisconsin),

Applications with Linyu Peng (Waseda), Peter Hydon (Kent),
Ana Rojo-Echeburúa and Jing Ping Wang (Kent).

This talk:
1. Moving frames - definitions, examples
Why a discrete or difference moving frame?
Joint v. discrete invariants.
Difference moving frames arise naturally.
2. Difference Maurer Cartan invariants and their recurrence
relations.
3. Pointers to the applications: difference variational methods,
discrete integrable systems
Not this talk:1. From local to global: Discrete moving frames on lattice
based “manifolds” and global syzygies
2. Multispace: a coherent framework for discrete and smoothframesSmooth moving frames can be used to describe complete, orgenerating, sets of differential invariants and their relations.
There are excellent algorithms to manipulate quantities derived
from moving frames in symbolic computation environments.
Moving frames are flexible, to allow for ease of computation in
specific applications, and they satisfy equations that allow them
to be obtained numerically (if necessary).
Mansfield, A practical guide to the invariant calculus,Cambridge Univ. Press, 2010.
Fels and Olver, Kogan and Olver, Hubert, Hubert and Kogan, . . .

Moving Frame if G × M → M is a regular, free action
Kh∈Gk•z•diﬀerentorbits

ρ : M → G ρ(z) = h is equivariant: ρ(g · z) = ρ(z)g −1

Calculation of a moving frame
Specify K, the cross-section, as the locus of Φ(z) = 0. Thensolve Φ(g · z) = 0 for g. In practice, solve
φj (g · z) = 0,j = 1, . . . , r = dim(G)for the r independent parameters describing g. Call the solution
ρ(z). Invoke Implicit Function Theorem. Unique solution yields
ρ(g · z) = ρ(z) · g −1.
• local solutions only this way: but see Hubert and Kogan,
FoCM 7 (2007) and J. Symb. Comp., 42 (2007).

Equivariance is the key to success. In particular, we obtain:Invariants: The components of I(z) = ρ(z) · z are invariant.
I(g · z) = ρ(g · z) · (g · z) = ρ(z)g −1g · z = ρ(z) · z.
If I(zi) are the canonical invariants for z = (z1, z2, . . . , zn), andF (z1, z2, . . . , zn) is an invariant, then we have theReplacement rule,
F (z1, z2, . . . , zn) = F (g · z1, g · z2, . . . , g · zn)= F (g · z1, g · z2, . . . , g · zn)|frame= F (I(z1), I(z2), . . . , I(zn))
Also symbolic invariant differentiation formulae and much more.

Main expository example: G = (R \ {0}) n R where the action isscaling and translation of x ∈ R given by(λ, ) · x = x̃ = λx + 
has a standard matrix representationx̃1!

=λ 0 1!

x1!

Joint invariants example: G = (R \ {0}) n R or scaling andtranslation:e k = λuk + 
uk 7→ uOlver: Joint invariant signatures, FoCM 2001

With the normalisation equations
Φ:

e 0 = 1,ue1 = 0u1,
u1 − u0=the frame is
ρ:

λ=−u1
u1 − u0

Easy to see equivariance in standard matrix representation:
1− ũ −ũ
100ũ1ũ1 −ũ01!

=1− u −u100!
u1u1 −u011µ0− µδ1!

.

The canonical invariants given by I(z) = ρ(z) · z are thenuk − u1I(uk ) = (λuk + ) |frame = −u1 − u0

From the computational point of view, for discrete curvature
flows or discrete calculus of variations, such invariants are adisaster as they perform poorly under the shift operator. The
problem is the frame fixes the base point.
Preferably, one would want something more natural, such as
uk+2 − uk+1
Ik =,uk+1 − uk

so thatIk = S k I0 .

This can be achieved with a frame that allows for a shiftingbase point.

Consider the discrete variational problem:
L[u] =X1

22In+2=X1

which is overtly invariant underun+2 − un+1
un+1 − unn 2!2e k = λuk + .
uk 7→ uThere are two conservation laws which can be written in theform






−12
20  In+1 − In un−1 − un−2c = Anvn(I) = In−un−121In
un−1 − un−2

where each An is equivariant, that is, is a frame for each n.

Note: this is a moving frame free calcuation: the point is thatindexed sets of frames arise naturally. In detail:−10  ueen−1 − un−2eAn = 

e−un−1
1 e n−1 − ue n−2u−10  λ(un−1 − un−2)== 
−λun−1 − 
1 λ(un−1 − un−2)
=

1/λ 0
/λ 1!

An



−1!
1/λ 0  un−1 − un−2−un−1
/λ 1 un−1 − un−20 
1 

Picture of a discrete moving frame. If Ki+1 = S(Ki) call this adifference frame.

Since we have a sequence of moving frames, we have aplethora of invariants:
ρs · zr = Irs.
Need some canonical choices! For the differential case, thecomponents of the Maurer Cartan matrices Ki = (Diρ) ρ−1 are(almost) generating. Here, the components ofKr+1= ρr+1ρ−1rr
are denoted the discrete Maurer Cartan invariants.
If Ki+1 = S(Ki) then ρr+1 = S(ρr ) and we have difference
Maurer Cartan invariants.

Knn+1 (z) =====

ρn+1 (z)ρn (z)−1 = ρn+1 (z)g −1 gρn (z)−1
ρn+1 (g · z)ρn (g · z)−1Knn+1 (g · z) so Knn+1 is invariant
Knn+1 (ρn (z) · z) setting g = ρn (z)ρn+1 (ρn (z) · z) as ρn (ρn (z) · z) = e

In this way, action by the group element Knn+1 (z) provides a change ofcoordinates from one set of generating invariants to another.

For the scaling and translation example, we calculate a
difference moving frame using
Φn :e n = 1,ue n+1 = 0u

giving in the standard matrix representation,matrix rep(ρn) =− u 1−unn+10!
un+1
un+1 −un1n
We have by definition ρn · un+k = In+kand thenun+2 −un+1=  un+1−unnKn+10
un+2 −un+1− u=n+1 −un

1nn−In+2
In+201!

Without solving for the frame: the normalisation equations andnin matrix form are:definition of In+k!

ρn · · ·un−1 un un+1 un+2 un+3··· =11111nnn
In−1
1 0 In+2In+3······

11 111for all n, so that
ρn+1un+1 un+211!

=

1 0
1 1!

yieldingn0 In+211!

= ρnρ−1n+1 ρn+1un+1 un+211!

= ρnρ−1n+1|{znKn+1}

1 0
1 1!

!

This shows we have, for all n,nρnρ−1=Kn+1 =n+1nn−In+2In+20!

1

as before, only this time we didn’t solve for the frame and have
nsymbolically: we don’t need to know what it is in
used In+2n .
order to obtain Kn in terms of the symbolic invariants In+kFinally,n+1 I nS(Ikn) = S(ρn · zk ) = ρn+1 · zk+1 = ρn+1ρ−1n ρn · zk+1 = Kn
k+1

gives the recurrence relations between the Maurer Cartan
invariants and their shifts.

The theory of discrete moving frames and the symbolic
calculus of difference invariants is developed in
ELM, Marı́ Beffa and Wang, FoCM, 2013; Marı́ Beffa andELM, FoCM, 2018.

Application 1: Calculus of Variations
Given a difference Lagrangian which is invariant under anr-dimensional Lie symmetry group, use the symbolic calculus ofdifference invariants to find the difference Euler Lagrange
equations directly in terms of the invariants.
Further, it is proved Noether’s difference laws can be written as
0=Xi(Si − id) (vi(I)Ad(ρ0))

1. where Ad(ρ) is the (left) Adjoint representation of anequivariant map from M n → G
2. and vi(I) is a (row) vector of invariants,
proving the equivariance of the laws.

In 1-d problems, the laws give additional, algebraic equations
for the frame, easing the integration problem for the extremals.

1. Solve the Euler Lagrange equations for the generating
invariants, then use the recurrence relations to obtain all
the needed additional Irs = ρs · zr .
2. Solve the algebraic equation c = v(I)Ad(ρ) and asnecessary, the recurrence equation ρn+1 = Knn+1ρn, for ρn.
Note v(I), Knn+1 are known, since the invariants have beensolved for.
s
3. Obtain zr from the algebraic equation, zr = ρ−1s · ( Ir ) .
ELM, Rojo-Echeburúa, Hydon and Peng, arxiv.org/abs/1804.00317; ELM,
Rojo-Echeburúa, arxiv.org/abs/1808.03606

The key to getting the finite difference approximation to match
all the smooth laws, bypassing the Ge and Marsden theorem,
seems to be, to match the smooth with the discrete moving
frames.
Example: Euler’s Elastica:uxx
L[u] = κ2 ds,
κ=ds =
3/2 ,2(1+ux )3 = 0.
Euler Lagrange equation: κss + 1
κ2
invariant under SO(2) n R2.

Rq

1 + u2x dx

Method used: match famous frame for SE(2) with an obviously
matching discrete frame, in order to read off the discrete
invariants which matches the Euclidean curvature, as well as
the infinitesimal arclength, to build the discrete Lagrangian.

Discrete integrable systems
Just as smooth integrable systems can occur in pairs, with one
being the Lie symmetry reduction of another, e.g. vortex
filament equations and the nonlinear Schrödinger equation, so
can discrete integrable systems.
We have for an evolving difference frame, the identity∂ ∂t )K − K N t ,K0 =ρ1ρ−1=S(N00 000∂t∂t

N0t =!

∂ρ0 ρ−10∂tyielding a linear difference syzygy operator H satisfying∂
κ0 = H σ0t∂twhere κ0 is a (vector) of Maurer Cartan invariants and σ0t are
the generating t-differential invariants, σ0t = ρ0(z) · z0,t.

Given two evolutions, and the (easy to compute) syzygies
between the σ0t and the σ0s being written asC(σ0t , σ0s ) =∂∂ s
σ0 − σ0t + Mst∂t∂sthen we have both∂κ0 = H σ0t ,∂tand remarkably,∂κ0 = H σ0s∂s[∂t, ∂s] κ0 = H C(σ0t , σ0s ),showing commuting flows factor through the syzygy operator,mirroring the smooth result.
ELM and P. van der Kamp, J. Geom. Phys., (2006)
Rojo-Echeburúa, ELM and JP Wang, in preparation

Conjecture (JP Wang): the syzygy operator H ispre-Hamiltonian.
Much more about discrete moving frames and discrete
integrability can be said, for example:
Gloria Marı̀ Beffa and Jing Ping Wang, (2013) Nonlinearity.
B. Wang, X-K Chang, X-B Hu and S-H Li, (2018) J. Phys., A.
SIDE ’13 Poster: Bao Wang.

Thank you!!