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Moving Framesin Applications
Peter J. Olver
University of Minnesota
http://www.math.umn.edu/! olver
Varna, June, 2012
Moving Frames
Classical contributions:M. Bartels (!1800), J. Serret, J. Frenet, G. Darboux,
E. Cotton, Elie Cartan
Modern developments: (1970’s)
S.S. Chern, M. Green, P. Gri!ths, G. Jensen, . . .
The equivariant approach: (1997 – )PJO, M. Fels, G. Marı–Be"a, I. Kogan, J. Cheh,
J. Pohjanpelto, P. Kim, M. Boutin, D. Lewis, E. Mansfield,E. Hubert, O. Morozov, R. McLenaghan, R. Smirnov, J. Yue,A. Nikitin, J. Patera, F. Valiquette, R. Thompson, . . .
“I did not quite understand how he [Cartan] does thisin general, though in the examples he gives theprocedure is clear.”
“Nevertheless, I must admit I found the book, likemost of Cartan’s papers, hard reading.”
— Hermann Weyl
“Cartan on groups and di"erential geometry”Bull. Amer. Math. Soc. 44 (1938) 598–601
Applications of Moving Frames
• Di"erential geometry
• Equivalence
• Symmetry
• Di"erential invariants
• Rigidity
• Joint invariants and semi-di"erential invariants
• Invariant di"erential forms and tensors
• Identities and syzygies
• Classical invariant theory
• Computer vision — object recognition & symmetrydetection
• Invariant numerical methods
• Invariant variational problems
• Invariant submanifold flows
• Poisson geometry & solitons
• Killing tensors in relativity
• Invariants of Lie algebras in quantum mechanics
• Lie pseudo-groups
The Basic Equivalence Problem
M — smooth m-dimensional manifold.
G — transformation group acting on M
• finite-dimensional Lie group
• infinite-dimensional Lie pseudo-group
Equivalence:Determine when two p-dimensional submanifolds
N and N " M
are congruent :
N = g · N for g # G
Symmetry:Find all symmetries,
i.e., self-equivalences or self-congruences :
N = g · N
Classical Geometry — F. Klein
• Euclidean group: G =
!"
#
SE(m) = SO(m) ! Rm
E(m) = O(m) ! Rm
z $%& A · z + b A # SO(m) or O(m), b # Rm, z # R
m
' isometries: rotations, translations , (reflections)
• Equi-a!ne group: G = SA(m) = SL(m) ! Rm
A # SL(m) — volume-preserving
• A!ne group: G = A(m) = GL(m) ! Rm
A # GL(m)
• Projective group: G = PSL(m + 1)acting on Rm " RPm
=' Applications in computer vision
Tennis, Anyone?
Classical Invariant Theory
Binary form:
Q(x) =n$
k=0
%n
k
&
ak xk
Equivalence of polynomials (binary forms):
Q(x) = (!x + ")n Q
%#x + $
!x + "
&
g =
%# $! "
&
# GL(2)
• multiplier representation of GL(2)• modular forms
Q(x) = (!x + ")n Q
%#x + $
!x + "
&
Transformation group:
g : (x, u) $%&%#x + $
!x + ",
u
(!x + ")n
&
Equivalence of functions (' equivalence of graphs
#Q = { (x, u) = (x, Q(x)) } " C2
Moving Frames
Definition.
A moving frame is a G-equivariant map
% : M %& G
Equivariance:
%(g·z) =
'g · %(z) left moving frame
%(z) · g!1 right moving frame
%left(z) = %right(z)!1
The Main Result
Theorem. A moving frame exists ina neighborhood of a point z # M if andonly if G acts freely and regularly near z.
Isotropy & Freeness
Isotropy subgroup: Gz = { g | g · z = z } for z # M
• free — the only group element g # G which fixes one pointz # M is the identity
=' Gz = {e} for all z # M
• locally free — the orbits all have the same dimension as G=' Gz " G is discrete for all z # M
• regular — the orbits form a regular foliation)* irrational flow on the torus
• e"ective — the only group element which fixes every point inM is the identity: g · z = z for all z # M i" g = e:
G+M =\
z"MGz = {e}
Geometric Construction
z
Oz
Normalization = choice of cross-section to the group orbits
Geometric Construction
z
Oz
K
k
Normalization = choice of cross-section to the group orbits
Geometric Construction
z
Oz
K
k
g = %left(z)
Normalization = choice of cross-section to the group orbits
Geometric Construction
z
Oz
K
k
g = %right(z)
Normalization = choice of cross-section to the group orbits
Algebraic Construction
r = dim G , m = dim M
Coordinate cross-section
K = { z1 = c1, . . . , zr = cr }
left right
w(g, z) = g!1 · z w(g, z) = g · z
g = (g1, . . . , gr) — group parameters
z = (z1, . . . , zm) — coordinates on M
Choose r = dim G components to normalize:
w1(g, z)= c1 . . . wr(g, z)= cr
Solve for the group parameters g = (g1, . . . , gr)
=' Implicit Function Theorem
The solutiong = %(z)
is a (local) moving frame.
The Fundamental Invariants
Substituting the moving frame formulae
g = %(z)
into the unnormalized components of w(g, z) produces thefundamental invariants
I1(z) = wr+1(%(z), z) . . . Im!r(z) = wm(%(z), z)
=' These are the coordinates of the canonical form k # K.
Completeness of Invariants
Theorem. Every invariant I(z) canbe (locally) uniquely written as afunction of the fundamental invariants:
I(z) = H(I1(z), . . . , Im!r(z))
Invariantization
Definition. The invariantization of a functionF : M & R with respect to a right moving frameg = %(z) is the the invariant function I = &(F )defined by
I(z) = F (%(z) · z).
&(z1) = c1, . . . &(zr) = cr, &(zr+1) = I1(z), . . . &(zm) = Im!r(z).
cross-section variables fundamental invariants“phantom invariants”
& [ F (z1, . . . , zm) ] = F (c1, . . . , cr, I1(z), . . . , Im!r(z))
Invariantization amounts to restricting F to the cross-section
I |K = F |Kand then requiring that I = &(F ) be constantalong the orbits.
In particular, if I(z) is an invariant, then &(I) = I.
Invariantization defines a canonical projection
& : functions $%& invariants
Prolongation
Most interesting group actions (Euclidean, a!ne,projective, etc.) are not free!
Freeness typically fails because the dimensionof the underlying manifold is not large enough, i.e.,m < r = dim G.
Thus, to make the action free, we must increasethe dimension of the space via some natural prolonga-tion procedure.
• An e"ective action can usually be made free by:
• Prolonging to derivatives (jet space)
G(n) : Jn(M, p) %& Jn(M, p)
=' di"erential invariants
• Prolonging to Cartesian product actions
G#n : M - · · ·-M %& M - · · ·-M
=' joint invariants
• Prolonging to “multi-space”
G(n) : M (n) %& M (n)
=' joint or semi-di"erential invariants=' invariant numerical approximations
• Prolonging to derivatives (jet space)
G(n) : Jn(M, p) %& Jn(M, p)
=' di"erential invariants
• Prolonging to Cartesian product actions
G#n : M - · · ·-M %& M - · · ·-M
=' joint invariants
• Prolonging to “multi-space”
G(n) : M (n) %& M (n)
=' joint or semi-di"erential invariants=' invariant numerical approximations
Euclidean Plane CurvesSpecial Euclidean group: G = SE(2) = SO(2) ! R2
acts on M = R2 via rigid motions: w = R z + b
To obtain the classical (left) moving frame we invertthe group transformations:
y = cos' (x% a) + sin' (u% b)
v = % sin' (x% a) + cos' (u% b)
()
* w = R!1(z % b)
Assume for simplicity the curve is (locally) a graph:
C = {u = f(x)}
=' extensions to parametrized curves are straightforward
Prolong the action to Jn via implicit di"erentiation:
y = cos' (x% a) + sin' (u% b)
v = % sin' (x% a) + cos' (u% b)
vy =% sin' + ux cos'
cos' + ux sin'
vyy =uxx
(cos' + ux sin' )3
vyyy =(cos' + ux sin' )uxxx % 3u2
xx sin'
(cos' + ux sin' )5
...
Prolong the action to Jn via implicit di"erentiation:
y = cos' (x% a) + sin' (u% b)
v = % sin' (x% a) + cos' (u% b)
vy =% sin' + ux cos'
cos' + ux sin'
vyy =uxx
(cos' + ux sin' )3
vyyy =(cos' + ux sin' )uxxx % 3u2
xx sin'
(cos' + ux sin' )5
...
Normalization: r = dim G = 3
y = cos' (x% a) + sin' (u% b) = 0
v = % sin' (x% a) + cos' (u% b) = 0
vy =% sin' + ux cos'
cos' + ux sin'= 0
vyy =uxx
(cos' + ux sin' )3
vyyy =(cos' + ux sin' )uxxx % 3u2
xx sin'
(cos' + ux sin' )5
...
Solve for the group parameters:
y = cos' (x% a) + sin' (u% b) = 0
v = % sin' (x% a) + cos' (u% b) = 0
vy =% sin' + ux cos'
cos' + ux sin'= 0
=' Left moving frame % : J1 %& SE(2)
a = x b = u ' = tan!1 ux
a = x b = u ' = tan!1 ux
Di"erential invariants
vyy =uxx
(cos' + ux sin' )3$%& ( =
uxx
(1 + u2x)3/2
vyyy = · · · $%&d(
ds=
(1 + u2x)uxxx % 3uxu2
xx
(1 + u2x)3
vyyyy = · · · $%&d2(
ds2% 3(3 = · · ·
=' recurrence formulae
Contact invariant one-form — arc length
dy = (cos'+ ux sin') dx $%& ds =+
1 + u2x dx
Dual invariant di"erential operator— arc length derivative
d
dy=
1
cos' + ux sin'
d
dx$%&
d
ds=
1+
1 + u2x
d
dx
Theorem. All di"erential invariants are functions ofthe derivatives of curvature with respect toarc length:
(,d(
ds,
d2(
ds2, · · ·
The Classical Picture:
z
t
n
Moving frame % : (x, u, ux) $%& (R, a) # SE(2)
R =1
+1 + u2
x
%1 %ux
ux 1
&
= ( t, n ) a =
%xu
&
Frenet frame
t =dx
ds=
%xs
ys
&
, n = t$ =
%% ys
xs
&
.
Frenet equations = Pulled-back Maurer–Cartan forms:
dx
ds= t,
dt
ds= (n,
dn
ds= %( t.
Equi-a!ne Curves G = SA(2)
z $%& A z + b A # SL(2), b # R2
Invert for left moving frame:
y = " (x% a)% $ (u% b)
v = % ! (x% a) + # (u% b)
()
* w = A!1(z % b)
# " % $ ! = 1
Prolong to J3 via implicit di"erentiation
dy = (" % $ ux) dx Dy =1
" % $ ux
Dx
Prolongation:
y = " (x% a)% $ (u% b)
v = % ! (x% a) + # (u% b)
vy = %! % # ux
" % $ ux
vyy = %uxx
(" % $ ux)3
vyyy = %(" % $ ux) uxxx + 3$ u2
xx
(" % $ ux)5
vyyyy = %uxxxx(" % $ ux)2 + 10$ (" % $ ux)uxx uxxx + 15$2 u3
xx
(" % $ ux)7
vyyyyy = . . .
Normalization: r = dim G = 5
y = " (x% a)% $ (u% b) = 0
v = % ! (x% a) + # (u% b) = 0
vy = %! % # ux
" % $ ux
= 0
vyy = %uxx
(" % $ ux)3= 1
vyyy = %(" % $ ux) uxxx + 3$ u2
xx
(" % $ ux)5= 0
vyyyy = %uxxxx(" % $ ux)2 + 10$ (" % $ ux)uxx uxxx + 15$2 u3
xx
(" % $ ux)7
vyyyyy = . . .
Equi-a!ne Moving Frame
% : (x, u, ux, uxx, uxxx) $%& (A,b) # SA(2)
A =
%# $! "
&
=
%3
+uxx % 1
3 u!5/3xx uxxx
ux3
+uxx u!1/3
xx % 13 u!5/3
xx uxxx
&
b =
%ab
&
=
%xu
&
Nondegeneracy condition: uxx )= 0.
Totally Singular Submanifolds
Definition. A p-dimensional submanifold N " M istotally singular if G(n) does not act freely on jnN for any n . 0.
Theorem. N is totally singular if and only if its symme-try group GN = { g | g · N " N } has dimension > p, and so GN
does not act freely on N itself.
Thus, the totally singular submanifolds are the only onesthat do not admit a moving frame of any order.
In equi-a!ne geometry, only the straight lines ( uxx / 0 )are totally singular since they admit a three-dimensional equi-a!ne symmetry group.
Equi-a!ne arc length
dy = (" % $ ux) dx $%& ds = 3
+uxx dx
Equi-a!ne curvature
vyyyy $%& ( =5uxxuxxxx % 3u2
xxx
9u8/3xx
vyyyyy $%&d(
ds
vyyyyyy $%&d2(
ds2% 5(2
The Classical Picture:
z
t
n
A =
%3
+uxx % 1
3 u!5/3xx uxxx
ux3
+uxx u!1/3
xx % 13 u!5/3
xx uxxx
&
= ( t, n ) b =
%xu
&
Frenet frame
t =dz
ds, n =
d2z
ds2.
Frenet equations = Pulled-back Maurer–Cartan forms:
dz
ds= t,
dt
ds= n,
dn
ds= ( t.
Equivalence & Invariants
• Equivalent submanifolds N * Nmust have the same invariants: I = I.
Constant invariants provide immediate information:
e.g. ( = 2 (' ( = 2
Non-constant invariants are not useful in isolation,because an equivalence map can drastically alter thedependence on the submanifold parameters:
e.g. ( = x3 versus ( = sinhx
Equivalence & Invariants
• Equivalent submanifolds N * Nmust have the same invariants: I = I.
Constant invariants provide immediate information:
e.g. ( = 2 (' ( = 2
Non-constant invariants are not useful in isolation,because an equivalence map can drastically alter thedependence on the submanifold parameters:
e.g. ( = x3 versus ( = sinhx
Equivalence & Invariants
• Equivalent submanifolds N * Nmust have the same invariants: I = I.
Constant invariants provide immediate information:
e.g. ( = 2 (' ( = 2
Non-constant invariants are not useful in isolation,because an equivalence map can drastically alter thedependence on the submanifold parameters:
e.g. ( = x3 versus ( = sinhx
However, a functional dependency or syzygy amongthe invariants is intrinsic:
e.g. (s = (3 % 1 (' (s = (3 % 1
• Universal syzygies — Gauss–Codazzi
• Distinguishing syzygies.
Theorem. (Cartan) Two submanifolds are (locally)equivalent if and only if they have identicalsyzygies among all their di"erential invariants.
However, a functional dependency or syzygy amongthe invariants is intrinsic:
e.g. (s = (3 % 1 (' (s = (3 % 1
• Universal syzygies — Gauss–Codazzi
• Distinguishing syzygies.
Theorem. (Cartan) Two submanifolds are (locally)equivalent if and only if they have identicalsyzygies among all their di"erential invariants.
However, a functional dependency or syzygy amongthe invariants is intrinsic:
e.g. (s = (3 % 1 (' (s = (3 % 1
• Universal syzygies — Gauss–Codazzi
• Distinguishing syzygies.
Theorem. (Cartan) Two submanifolds are (locally)equivalent if and only if they have identicalsyzygies among all their di"erential invariants.
Finiteness of Generators and Syzygies
0 There are, in general, an infinite number of di"er-ential invariants and hence an infinite numberof syzygies must be compared to establishequivalence.
1 But the higher order syzygies are all consequencesof a finite number of low order syzygies!
Example — Plane Curves
If non-constant, both ( and (s depend on a singleparameter, and so, locally, are subject to a syzygy:
(s = H(() (+)
But then
(ss =d
dsH(() = H %(()(s = H %(()H(()
and similarly for (sss, etc.
Consequently, all the higher order syzygies are generatedby the fundamental first order syzygy (+).
Thus, for Euclidean (or equi-a!ne or projective or . . . )plane curves we need only know a single syzygy between ( and(s in order to establish equivalence!
The Basis Theorem
Theorem. The di"erential invariant algebra I isgenerated by a finite number of di"erential invariants
I1, . . . , I!
and p = dimN invariant di"erential operators
D1, . . . ,Dp
meaning that every di"erential invariant can be locallyexpressed as a function of the generating invariantsand their invariant derivatives:
DJI" = Dj1Dj2
· · · Djn
I".
=' Lie, Tresse, Ovsiannikov, Kumpera
) Moving frames provides a constructive proof.
Signature Curves
Definition. The signature curve S " R2 of a curveC " R2 is parametrized by the two lowest orderdi"erential invariants
S =
' %
( ,d(
ds
& ,
" R2
Theorem. Two regular curves C and C are equiva-lent:
C = g · Cif and only if their signature curves are identical:
S = S
Signature Curves
Definition. The signature curve S " R2 of a curveC " R2 is parametrized by the two lowest orderdi"erential invariants
S =
' %
( ,d(
ds
& ,
" R2
Theorem. Two regular curves C and C are equiva-lent:
C = g · Cif and only if their signature curves are identical:
S = S
Symmetry and Signature
Theorem. The dimension of the symmetry group
GN = { g | g · N " N }
of a nonsingular submanifold N " M equals thecodimension of its signature:
dimGN = dim N % dimS
Corollary. For a nonsingular submanifold N " M ,
0 , dim GN , dim N
=' Only totally singular submanifolds can have largersymmetry groups!
Maximally Symmetric Submanifolds
Theorem. The following are equivalent:
• The submanifold N has a p-dimensional symmetry group
• The signature S degenerates to a point: dimS = 0
• The submanifold has all constant di"erential invariants
• N = H · {z0} is the orbit of a p-dimensional subgroup H " G
=' Euclidean geometry: circles, lines, helices, spheres, cylinders, planes, . . .
=' Equi-a!ne plane geometry: conic sections.
=' Projective plane geometry: W curves (Lie & Klein)
Discrete Symmetries
Definition. The index of a submanifold N equalsthe number of points in N which map to a genericpoint of its signature:
&N = min-
# $!1{w}... w # S
/
=' Self–intersections
Theorem. The cardinality of the symmetry group ofa submanifold N equals its index &N .
=' Approximate symmetries
The Index
$
%&
N S
The Curve x = cos t + 15 cos2 t, y = sin t + 1
10 sin2 t
-0.5 0.5 1
-0.5
0.5
1
The Original Curve
0.25 0.5 0.75 1 1.25 1.5 1.75 2
-2
-1
0
1
2
Euclidean Signature
0.5 1 1.5 2 2.5
-6
-4
-2
2
4
A!ne Signature
The Curve x = cos t + 15 cos2 t, y = 1
2 x + sin t + 110 sin2 t
-0.5 0.5 1
-1
-0.5
0.5
1
The Original Curve
0.5 1 1.5 2 2.5 3 3.5 4
-7.5
-5
-2.5
0
2.5
5
7.5
Euclidean Signature
0.5 1 1.5 2 2.5
-6
-4
-2
2
4
A!ne Signature
Canine Left Ventricle Signature
Original Canine HeartMRI Image
Boundary of Left Ventricle
Smoothed Ventricle Signature
10 20 30 40 50 60
20
30
40
50
60
10 20 30 40 50 60
20
30
40
50
60
10 20 30 40 50 60
20
30
40
50
60
-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
-0.06
-0.04
-0.02
0.02
0.04
0.06
-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
-0.06
-0.04
-0.02
0.02
0.04
0.06
-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
-0.06
-0.04
-0.02
0.02
0.04
0.06
Evolution of Invariants and Signatures
Basic question: If the submanifold evolves according toan invariant evolution equation, how do its di"erentialinvariants & signatures evolve?
Theorem. Under the curve shortening flow Ct = %(n,the signature curve (s = H(t,() evolves according to theparabolic equation
*H
*t= H2 H"" % (3H" + 4(2H
Signature Metrics
• Hausdor"
• Monge–Kantorovich transport
• Electrostatic repulsion
• Latent semantic analysis
• Histograms
• Gromov–Hausdor" & Gromov–Wasserstein
Signatures
s
(
Classical Signature%&
Original curve(
(s
Di"erential invariant signature
Signatures
s
(
Classical Signature%&
Original curve(
(s
Di"erential invariant signature
Occlusions
s
(
Classical Signature%&
Original curve(
(s
Di"erential invariant signature
The Ba"er Jigsaw Puzzle
The Ba"er Solved
=' Dan Ho"
Classical Invariant Theory
M = R2 \ {u = 0}
G = GL(2) =
' %# $! "
& ..... % = # " % $ ! )= 0
,
(x, u) $%&%#x + $
!x + ",
u
(!x + ")n
&
n )= 0, 1
Prolongation:
y =#x + $
! x + "+ = ! x + "
v = +!n u % = # " % $ !
vy =+ ux % n ! u
% +n!1
vyy =+2 uxx % 2(n% 1)! + ux + n(n% 1)!2 u
%2 +n!2
vyyy = · · ·
Normalization:
y =# x + $
! x + "= 0 + = ! x + "
v = +!n u = 1 % = # " % $ !
vy =+ ux % n ! u
% +n!1= 0
vyy =+2 uxx % 2(n% 1) ! + ux + n(n% 1)!2 u
%2 +n!2=
1
n(n% 1)
vyyy = · · ·
Moving frame:
# = u(1!n)/n2
H $ = %x u(1!n)/n2
H
! = 1n u(1!n)/n " = u1/n % 1
n xu(1!n)/n
Hessian:
H = n(n% 1)u uxx % (n% 1)2u2x )= 0
Note: H / 0 if and only if Q(x) = (a x + b)n
=' Totally singular forms
Di"erential invariants:
vyyy $%&J
n2(n% 1)* ( vyyyy $%&
K + 3(n% 2)
n3(n% 1)*
d(
ds
Absolute rational covariants:
J2 =T 2
H3K =
U
H2
H = 12(Q, Q)(2) = n(n% 1)QQ%% % (n% 1)2Q%2 ! QxxQyy %Q2
xy
T = (Q, H)(1) = (2n% 4)Q%H % nQH % ! QxHy %QyHx
U = (Q, T )(1) = (3n% 6)Q%T % nQT % ! QxTy %QyTx
deg Q = n deg H = 2n% 4 deg T = 3n% 6 deg U = 4n% 8
Signatures of Binary Forms
Signature curve of a nonsingular binary form Q(x):
SQ =
'
(J(x)2, K(x)) =
%T (x)2
H(x)3,
U(x)
H(x)2
&,
Nonsingular : H(x) )= 0 and (J %(x), K %(x)) )= 0.
Theorem. Two nonsingular binary forms are equiva-lent if and only if their signature curves are identical.
Maximally Symmetric Binary Forms
Theorem. If u = Q(x) is a polynomial, then thefollowing are equivalent:
• Q(x) admits a one-parameter symmetry group
• T 2 is a constant multiple of H3
• Q(x) 3 xk is complex-equivalent to a monomial
• the signature curve degenerates to a single point
• all the (absolute) di"erential invariants of Q areconstant
• the graph of Q coincides with the orbit of aone-parameter subgroup
Symmetries of Binary Forms
Theorem. The symmetry group of a nonzero binary formQ(x) )/ 0 of degree n is:
• A two-parameter group if and only if H / 0 if and only ifQ is equivalent to a constant. =' totally singular
• A one-parameter group if and only if H )/ 0 and T 2 = cH3
if and only if Q is complex-equivalent to a monomial xk,with k )= 0, n. =' maximally symmetric
• In all other cases, a finite group whose cardinality equalsthe index of the signature curve, and is bounded by
&Q ,'
6n% 12 U = cH2
4n% 8 otherwise
Symmetry–Preserving Numerical Methods
• Invariant numerical approximations to di"erentialinvariants.
• Invariantization of numerical integration methods.
=' Structure-preserving algorithms
Numerical approximation to curvature
ab
cA
B
C
Heron’s formula
0((A, B,C) = 4%
abc= 4
+s(s% a)(s% b)(s% c)
abc
s =a + b + c
2— semi-perimeter
Invariantization of Numerical Schemes
=' Pilwon Kim
Suppose we are given a numerical scheme for integratinga di"erential equation, e.g., a Runge–Kutta Method for ordi-nary di"erential equations, or the Crank–Nicolson method forparabolic partial di"erential equations.
If G is a symmetry group of the di"erential equation, thenone can use an appropriately chosen moving frame to invari-antize the numerical scheme, leading to an invariant numeri-cal scheme that preserves the symmetry group. In challengingregimes, the resulting invariantized numerical scheme can, withan inspired choice of moving frame, perform significantly betterthan its progenitor.
Invariant Runge–Kutta schemes
uxx + xux % (x + 1)u = sinx, u(0) = ux(0) = 1.
Comparison of symmetry reduction and invariantization for
uxx + xux % (x + 1)u = sinx, u(0) = ux(0) = 1.
Invariantization of Crank–Nicolsonfor Burgers’ Equation
ut = ,uxx + u ux
The Calculus of Variations
I[u ] =1
L(x, u(n)) dx — variational problem
L(x, u(n)) — Lagrangian
To construct the Euler-Lagrange equations: E(L) = 0
• Take the first variation:
"(L dx) =$
#,J
*L
*u#J
"u#J dx
• Integrate by parts:
"(Ldx) =$
#,J
*L
*u#J
DJ("u#) dx
/$
#,J
(%D)J *L
*u#J
"u# dx =q$
#=1
E#(L) "u# dx
Invariant Variational Problems
According to Lie, any G–invariant variational problem canbe written in terms of the di"erential invariants:
I[u ] =1
L(x, u(n)) dx =1
P ( . . . DKI# . . . ) !
I1, . . . , I! — fundamental di"erential invariants
D1, . . . ,Dp — invariant di"erential operators
DKI# — di"erentiated invariants
! = -1 4 · · · 4 -p — invariant volume form
If the variational problem is G-invariant, so
I[u ] =1
L(x, u(n)) dx =1
P ( . . . DKI# . . . ) !
then its Euler–Lagrange equations admit G as a symmetrygroup, and hence can also be expressed in terms of the di"er-ential invariants:
E(L) 3 F ( . . . DKI# . . . ) = 0
Main Problem:
Construct F directly from P .
(P. Gri!ths, I. Anderson )
Planar Euclidean group G = SE(2)
( =uxx
(1 + u2x)3/2
— curvature (di"erential invariant)
ds =+
1 + u2x dx — arc length
D =d
ds=
1+
1 + u2x
d
dx— arc length derivative
Euclidean–invariant variational problem
I[u ] =1
L(x, u(n)) dx =1
P ((,(s,(ss, . . . ) ds
Euler-Lagrange equations
E(L) 3 F ((,(s,(ss, . . . ) = 0
Euclidean Curve Examples
Minimal curves (geodesics):
I[u ] =1
ds =1 +
1 + u2x dx
E(L) = %( = 0=' straight lines
The Elastica (Euler):
I[u ] =1
12 (
2 ds =1 u2
xx dx
(1 + u2x)5/2
E(L) = (ss + 12 (
3 = 0=' elliptic functions
General Euclidean–invariant variational problem
I[u ] =1
L(x, u(n)) dx =1
P ((,(s,(ss, . . . ) ds
To construct the invariant Euler-Lagrange equations:
Take the first variation:
"(P ds) =$
j
*P
*(j
"(j ds + P "(ds)
Invariant variation of curvature:
"( = A"("u) A" = D2 + (2
Invariant variation of arc length:
"(ds) = B("u) ds B = %(
=' moving frame recurrence formulae
The Recurrence Formula
For any function or di"erential form &:
d &(&) = &(d&) +r$
k=1
.k 4 & [vk(&)]
v1, . . . ,vr — basis for g — infinitesimal generators
.1, . . . , .r — dual invariantized Maurer–Cartan forms
) ) The .k are uniquely determined by the recurrenceformulae for the phantom di"erential invariants
d &(&) = &(d&) +r$
k=1
.k 4 & [vk(&)]
) ) ) All identities, commutation formulae, syzygies, etc.,among di"erential invariants and, more generally,the invariant variational bicomplex follow from thisuniversal recurrence formula by letting & range overthe basic functions and di"erential forms!
) ) ) Therefore, the entire structure of the di"erential invari-ant algebra and invariant variational bicomplex can becompletely determined using only linear di"erential al-gebra; this does not require explicit formulas for themoving frame, the di"erential invariants, the invariantdi"erential forms, or the group transformations!
d &(&) = &(d&) +r$
k=1
.k 4 & [vk(&)]
) ) ) All identities, commutation formulae, syzygies, etc.,among di"erential invariants and, more generally,the invariant variational bicomplex follow from thisuniversal recurrence formula by letting & range overthe basic functions and di"erential forms!
) ) ) Therefore, the entire structure of the di"erential invari-ant algebra and invariant variational bicomplex can becompletely determined using only linear di"erential al-gebra; this does not require explicit formulas for themoving frame, the di"erential invariants, the invariantdi"erential forms, or the group transformations!
Integrate by parts:
"(P ds) / [ E(P )A("u)%H(P )B("u) ] ds
/ [A+E(P )% B+H(P ) ] "u ds = E(L) "u ds
Invariantized Euler–Lagrange expression
E(P ) =&$
n=0
(%D)n *P
*(n
D =d
ds
Invariantized Hamiltonian
H(P ) =$
i>j
(i!j (%D)j *P
*(i
% P
Euclidean–invariant Euler-Lagrange formula
E(L) = A+E(P )% B+H(P ) = (D2 + (2) E(P ) + (H(P ) = 0.
The Elastica:
I[u ] =1
12 (
2 ds P = 12 (
2
E(P ) = ( H(P ) = %P = % 12 (
2
E(L) = (D2 + (2) ( + ( (% 12 (
2 ) = (ss + 12 (
3 = 0
Evolution of Invariants and Signatures
G — Lie group acting on R2
C(t) — parametrized family of plane curves
G–invariant curve flow:
dC
dt= V = I t + J n
• I, J — di"erential invariants
• t — “unit tangent”
• n — “unit normal”
• The tangential component I t only a"ects the underlyingparametrization of the curve. Thus, we can set I to beanything we like without a"ecting the curve evolution.
Normal Curve Flows
Ct = J n
Examples — Euclidean–invariant curve flows
• Ct = n — geometric optics or grassfire flow;
• Ct = (n — curve shortening flow;
• Ct = (1/3 n — equi-a!ne invariant curve shortening flow:Ct = nequi!a!ne ;
• Ct = (s n — modified Korteweg–deVries flow;
• Ct = (ss n — thermal grooving of metals.
Intrinsic Curve Flows
Theorem. The curve flow generated by
v = I t + J n
preserves arc length if and only if
B(J) + D I = 0.
D — invariant arc length derivative
B — invariant arc length variation
"(ds) = B("u) ds
Normal Evolution of Di#erential Invariants
Theorem. Under a normal flow Ct = J n,
*(
*t= A"(J),
*(s
*t= A"s
(J).
Invariant variations:
"( = A"("u), "(s = A"s
("u).
A" = A — invariant variation of curvature;
A"s
= DA + ((s — invariant variation of (s.
Euclidean–invariant Curve Evolution
Normal flow: Ct = J n
*(
*t= A"(J) = (D2 + (2) J,
*(s
*t= A"s
(J) = (D3 + (2D + 3((s)J.
Warning : For non-intrinsic flows, *t and *s do not commute!
Theorem. Under the curve shortening flow Ct = %(n,the signature curve (s = H(t,() evolves according to theparabolic equation
*H
*t= H2 H"" % (3H" + 4(2H
Smoothed Ventricle Signature
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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2
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Intrinsic Evolution of Di#erential Invariants
Theorem.
Under an arc-length preserving flow,
(t = R(J) where R = A% (sD!1B (+)
In surprisingly many situations, (*) is a well-known integrableevolution equation, and R is its recursion operator!
=' Hasimoto
=' Langer, Singer, Perline
=' Marı–Be"a, Sanders, Wang
=' Qu, Chou, Anco, and many more ...
Euclidean plane curves
G = SE(2) = SO(2) ! R2
A = D2 + (2 B = %(
R = A% (sD!1B = D2 + (2 + (sD
!1 · (
(t = R((s) = (sss + 32 (
2(s
=' modified Korteweg-deVries equation
Equi-a!ne plane curves
G = SA(2) = SL(2) ! R2
A = D4 + 53 (D
2 + 53 (sD + 1
3 (ss + 49 (
2
B = 13 D
2 % 29 (
R = A% (sD!1B
= D4 + 53 (D
2 + 43 (sD + 1
3 (ss + 49 (
2 + 29 (sD
!1 · (
(t = R((s) = (5s + 53 ((sss + 5
3 (s(ss + 59 (
2(s
=' Sawada–Kotera equation
Recursion operator: 2R = R · (D2 + 13 ( + 1
3 (sD!1)
Euclidean space curves
G = SE(3) = SO(3) ! R3
A =
3
44445
D2s + ((2 % /2)
2/
(D2
s +3(/s % 2(s/
(2Ds +
(/ss % (s/s + 2(3/
(2
%2/Ds % /s
1
(D3
s %(s
(2D2
s +(2 % /2
(Ds +
(s/2 % 2(//s
(2
6
77778
B = (( 0 )
R = A%%(s
/s
&
D!1B%(t
/t
&
= R%(s
/s
&
=' vortex filament flow (Hasimoto)