In v a ria n t V a ria tio n a l P ro b le m s In v a ria ...olver/t_/ivfoxt.pdf · In v a ria n t...

Invariant Variational Problems

&

Invariant Curve Flows

Peter J. Olver

University of Minnesota

http://www.math.umn.edu/! olver

Oxford, December, 2008

Basic Notation

x = (x1, . . . , xp) — independent variables

u = (u1, . . . , uq) — dependent variables

u!J = !Ju! — partial derivatives

F (x, u(n)) = F ( . . . xk . . . u!J . . . ) — di!erential function

G — transformation group acting on the space of independent

and dependent variables

Variational Problems

I[u ] =!

L(x, u(n)) dx — variational problem

L(x, u(n)) — Lagrangian

Variational derivative — Euler-Lagrange equations: E(L) = 0

components: E!(L) ="

J

("D)J !L

!u!J

DkF =!F

!xk+

"

!,J

u!J,k

!F

!u!J

— total derivative of F with respect to xk

Invariant Variational Problems

According to Lie, any G–invariant variational problem can be

written in terms of the di!erential invariants:

I[u ] =!

L(x, u(n)) dx =!

P ( . . . DKI! . . . ) !

I1, . . . , I" — fundamental di!erential invariants

D1, . . . ,Dp — invariant di!erential operators

DKI! — di!erentiated invariants

! = "1 # · · · # "p — invariant volume form

If the variational problem is G-invariant, so

I[u ] =!

L(x, u(n)) dx =!

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) $ 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 ] =!

L(x, u(n)) dx =!

P (#,#s,#ss, . . . ) ds

Euler-Lagrange equations

E(L) $ F (#,#s,#ss, . . . ) = 0

Euclidean Curve Examples

Minimal curves (geodesics):

I[u ] =!

ds =! #

1 + u2x dx

E(L) = "# = 0=% straight lines

The Elastica (Euler):

I[u ] =!

12 #

2 ds =! u2

xx dx

(1 + u2x)5/2

E(L) = #ss + 12 #

3 = 0=% elliptic functions

General Euclidean–invariant variational problem

I[u ] =!

L(x, u(n)) dx =!

P (#,#s,#ss, . . . ) 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

I[u ] =!

L(x, u(n)) dx =!

P (#,#s,#ss, . . . ) ds

Euclidean–invariant Euler-Lagrange formula

E(L) = (D2 + #2) E(P ) + #H(P ) = 0

The Elastica: I[u ] =!

12 #

2 ds P = 12 #

2

E(P ) = # H(P ) = "P = " 12 #

2

E(L) = (D2 + #2) #+ # (" 12 #

2 )

= #ss + 12 #

3 = 0

Applications of Moving Frames

• Di!erential geometry

• Equivalence

• Symmetry

• Di!erential invariants

• Rigidity

• Joint invariants and semi-di!erential invariants

• Integral invariants

• Symmetries of di!erential equations

• Factorization of di!erential operators

• Invariant di!erential forms and tensors

• Identities and syzygies

• Classical invariant theory

• Computer vision

& object recognition

& symmetry detection

& structure from motion

• Invariant variational problems

• Invariant numerical methods

• Poisson geometry & solitons

• Killing tensors in relativity

• Invariants of Lie algebras in quantum mechanics

• Lie pseudo-groups

Moving Frames

G — r-dimensional Lie group acting on M

Jn = Jn(M,p) — nth order jet bundle for

p-dimensional submanifolds N = {u = f(x)} ' M

z(n) = (x, u(n)) = ( . . . xi . . . u!J . . . ) — coordinates on Jn

G acts on Jn by prolongation (chain rule)

Definition.

An nth order moving frame is a G-equivariant map

$ = $(n) : V ' Jn "( G

Equivariance:

$(g(n) · z(n)) =

$%&

%'

g · $(z(n)) left moving frame

$(z(n)) · g"1 right moving frame

Note: $left(z(n)) = $right(z

(n))"1

Theorem. A moving frame exists in a neighborhoodof a point z(n) ) Jn if and only if G acts freelyand regularly near z(n).

• free — the only group element g ) G which fixes one point

z ) M is the identity: g · z = z if and only if g = e.

• locally free — the orbits have the same dimension as G.

• regular — all orbits have the same dimension and intersect

su"ciently small coordinate charts only once

( *+ irrational flow on the torus)

Geometric Construction

z

Oz

Normalization = choice of cross-section to the group orbits


z

Oz

K

k



z

Oz

K

k

g = $left(z)



z

Oz

K

k

g = $right(z)


The Normalization Construction

1. Write out the explicit formulas for theprolonged group action:

w(n)(g, z(n)) = g(n) · z(n)

=% Implicit di"erentiation

2. From the components of w(n), choose r = dim Gnormalization equations :

w1(g, z(n)) = c1 . . . wr(g, z(n)) = cr

3. Solve the normalization equations for the group parametersg = (g1, . . . , gr):

g = $(z(n)) = $(x, u(n))

The solution is the right moving frame.

4. Invariantization: substitute the moving frame formulas

g = $(z(n)) = $(x, u(n))

for the group parameters into the un-normalized components ofw(n) to produce a complete system of functionally independentdi!erential invariants:

I(n)(x, u(n)) = %(z(n)) = w(n)($(z(n)), z(n)))

Euclidean plane curves G = SE(2)

Assume the curve is (locally) a graph:

C = {u = f(x)}

Write out the group transformations

y = x cos&" u sin&+ a

v = x cos& + u sin&+ b

()

* w = R z + c

Prolong to Jn via implicit di!erentiationy = x cos&" u sin& + a v = x cos&+ u sin& + 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

...

Choose a cross-section, or, equivalently a set of r = dimG = 3normalization equations:

y = 0 v = 0 vy = 0

Solve the normalization equations for the group parameters:

& = " tan"1 ux a = "x + uux#

1 + u2x

b =xux " u#

1 + u2x

The result is the right moving frame $ : J1 "( SE(2)

Substitute into the moving frame formulas for the group param-eters into the remaining prolonged transformation formulae toproduce the basic di!erential invariants:

vyy ,"( # =uxx

(1 + u2x)3/2

vyyy ,"(d#

ds=

(1 + u2x)uxxx " 3uxu2

xx

(1 + u2x)3

vyyyy ,"(d2#

ds2+ 3#3 = · · ·

Theorem. All di!erential invariants are functions of thederivatives of curvature with respect to arc length:

#d#

ds

d2#

ds2· · ·

The invariant di!erential operators and invariant di!erentialforms are also substituting the moving frame formulas forthe group parameters:

Invariant one-form — arc length

dy = (cos&" ux sin&) dx ,"( ds =#

1 + u2x dx

Invariant di!erential operator — arc length derivative

d

dy=

1

cos&" ux sin&

d

dx,"(

d

ds=

1#

1 + u2x

d

dx

Euclidean Curves

x

e1

e 2

Left moving frame +$(x, u(1)) = $(x, u(1))"1

a = x b = u +& = tan"1 ux

R =1

#1 + u2

x

,1 "ux

ux 1

-

= ( t n ) +a =

,xu

-

InvariantizationThe process of replacing group parameters in transformation

rules by their moving frame formulae is known asinvariantization.

The invariantization I = %(F ) is the unique invariant functionthat agrees with F on the cross-section: I | K = F | K.

Invariantization respects algebraic operations, and providesa canonical projection that maps objects to their invari-antized counterparts.

% :

Functions "( Invariants

Forms "( Invariant Forms

Di!erential

Operators"(

Invariant Di!erential

Operators

Fundamental di!erential invariants = invariantized jet coordinates

Hi(x, u(n)) = %(xi) I!K(x, u(l)) = %(u!

K)

The constant di!erential invariants, coming from the movingframe normalizations, are known as the phantom invariants .The remaining non-constant di!erential invariants are thebasic invariants and form a complete system of functionallyindependent di!erential invariants.

Invariantization of di!erential functions:

% [ F ( . . . xi . . . u!J . . . ) ] = F ( . . . Hi . . . I!

J . . . )

Replacement Theorem:

If J is a di!erential invariant, then %(J) = J .

J( . . . xi . . . u!J . . . ) = J( . . . Hi . . . I!

J . . . )

The Infinite Jet Bundle

Jet bundles

M = J0 -" J1 -" J2 -" · · ·

Inverse limitJ! = lim

n#!Jn

Local coordinates

z(!) = (x, u(!)) = ( . . . xi . . . u!J . . . )

=% Taylor series

Di!erential Forms

Coframe — basis for the cotangent space T.J!:

• Horizontal one-forms

dx1, . . . , dxp

• Contact (vertical) one-forms

'!J = du!

J "p"

i=1

u!J,i dxi

Intrinsic definition of contact form

' | j!N = 0 /% ' ="

A!J '

!J

The Variational Bicomplex=% Dedecker, Vinogradov, Tsujishita, I. Anderson, . . .

Bigrading of the di!erential forms on J!:

#. =M

r,s#r,s

r = # horizontal forms

s = # contact forms

Vertical and Horizontal Di!erentials

d = dH + dV

dH : #r,s "( #r+1,s

dV : #r,s "( #r,s+1

Vertical and Horizontal Di!erentials

F (x, u(n)) — di!erential function

dH F =p"

i=1

(DiF ) dxi — total di!erential

dV F ="

!,J

!F

!u!J

'!J — first variation

dH (dxi) = dV (dxi) = 0,

dH ('!J ) =

p"

i=1

dxi # '!J,i dV ('!

J ) = 0

The Simplest Example

(x, u) ) M = R2

x — independent variable

u — dependent variable

Horizontal formdx

Contact (vertical) forms

' = du " ux dx

'x = dux " uxx dx

'xx = duxx " uxxx dx

...

' = du " ux dx, 'x = dux " uxx dx, 'xx = duxx " uxxx dx

Di!erential:

dF =!F

!xdx +

!F

!udu +

!F

!ux

dux +!F

!uxx

duxx + · · ·

= (DxF ) dx +!F

!u' +

!F

!ux

'x +!F

!uxx

'xx + · · ·

= dH F + dV F

Total derivative:

DxF =!F

!x+!F

!uux +

!F

!ux

uxx +!F

!uxx

uxxx + · · ·

The Variational Bicomplex... ... ... ... ...

dV

!dV

!dV

!dV

!#

!

#0,3 dH" #1,3 dH" · · ·dH" #p"1,3 dH" #p,3 $

" F3

dV

!dV

!dV

!dV

!#

!

#0,2 dH" #1,2 dH" · · ·dH" #p"1,2 dH" #p,2 $

" F2

dV

!dV

!dV

!dV

!#

!

#0,1 dH" #1,1 dH" · · ·dH" #p"1,1 dH" #p,1 $

" F1

dV

!dV

!dV

!dV

!

##

##$E

R ( #0,0 dH" #1,0 dH" · · ·dH" #p"1,0 dH" #p,0

conservation laws Lagrangians PDEs (Euler–Lagrange) Helmholtz conditions

The Variational Derivative

E = ( $ dV

dV — first variation( — integration by parts = mod out by image of dH

#p,0 "(dV

#p,1 "((

F1 = #p,1/ dH #p"1,1

) = Ldx "("

!,J

!L

!u!J

'!J # dx "(

q"

!=1

E!(L) '! # dx

Variational

problem"(

First

variation"(

Euler–Lagrange

source form

The Simplest Example: (x, u) ) M = R2

Lagrangian form: ) = L(x, u(n)) dx ) #1,0

First variation — vertical derivative:

d) = dV ) = dV L # dx

=

,!L

!u' +

!L

!ux

'x +!L

!uxx

'xx + · · ·

-

# dx ) #1,1

Integration by parts — compute modulo im dH :

d) ! *) =

,!L

!u" Dx

!L

!ux

+ D2x

!L

!uxx

" · · ·

-

' # dx ) F1

= E(L) ' # dx

=% Euler-Lagrange source form.

To analyze invariant variational prob-

lems, invariant conservation laws, etc., we

apply the moving frame invariantization

process to the variational bicomplex:

Di!erential Invariants and

Invariant Di!erential Forms

% — invariantization associated with moving frame $.

• Fundamental di!erential invariants

Hi(x, u(n)) = %(xi) I!K(x, u(n)) = %(u!

K)

• Invariant horizontal forms

+i = %(dxi)

• Invariant contact forms

,!J = %('!

J )

The Invariant “Quasi–Tricomplex”

Di!erential forms#. =

M

r,s

.#r,s

Di!erentiald = dH + dV + dW

dH : .#r,s "( .#r+1,s

dV : .#r,s "( .#r,s+1

dW : .#r,s "( .#r"1,s+2

Key fact: invariantization and di!erentiation do not commute:

d %(#) *= %(d#)

The Universal Recurrence Formula

d %(#) = %(d#) +r"

%=1

-% # % [v%(#)]

v1, . . . ,vr — basis for g — infinitesimal generators

-1, . . . , -r — invariantized dual Maurer–Cartan forms

=% uniquely determined by the recurrence formulae for thephantom di!erential invariants

d %(#) = %(d#) +r"

%=1

-% # % [v%(#)]

. . . All identities, commutation formulae, syzygies, etc.,among di!erential invariants and, more generally,the invariant variational bicomplex follow from thisuniversal formula by letting # range over the basicfunctions and di!erential forms!

. . . Moreover, determining the structure of the di!erentialinvariant algebra and invariant variational bicomplexrequires only linear di!erential algebra, and not anyexplicit formulas for the moving frame, the di!erentialinvariants, the invariant di!erential forms, or the grouptransformations!

Euclidean plane curves

Fundamental normalized di!erential invariants

%(x) = H = 0

%(u) = I0 = 0

%(ux) = I1 = 0

(%%%%)

%%%%*

phantom di!. invs.

%(uxx) = I2 = # %(uxxx) = I3 = #s %(uxxxx) = I4 = #ss + 3#3

In general:

%( F (x, u, ux, uxx, uxxx, uxxxx, . . . )) = F (0, 0, 0,#,#s,#ss + 3#3, . . . )

Invariant arc length form

dy = (cos&" ux sin&) dx " (sin&) '

+ = %(dx) = " + /

=#

1 + u2x dx +

ux#1 + u2

x

'

=% ' = du " ux dx

Invariant contact forms

, = %(') ='

#1 + u2

x

,1 = %('x) =(1 + u2

x) 'x " uxuxx'

(1 + u2x)2

Prolonged infinitesimal generators

v1 = !x, v2 = !u, v3 = "u !x + x !u + (1 + u2x) !ux

+ 3uxuxx !uxx+ · · ·

Basic recurrence formula

d%(F ) = %(dF ) + %(v1(F )) -1 + %(v2(F )) -2 + %(v3(F )) -3

Use phantom invariants

0 = dH = %(dx) + %(v1(x)) -1 + %(v2(x)) -2 + %(v3(x)) -3 = + + -1,

0 = dI0 = %(du) + %(v1(u)) -1 + %(v2(u)) -2 + %(v3(u)) -3 = ,+ -2,

0 = dI1 = %(dux) + %(v1(ux)) -1 + %(v2(ux)) -2 + %(v3(ux)) -3 = #+ + ,1 + -3,

to solve for the Maurer–Cartan forms:

-1 = "+, -2 = ",, -3 = "#+ " ,1.

-1 = "+, -2 = ",, -3 = "#+ " ,1.

Recurrence formulae:

d# = d%(uxx) = %(duxx) + %(v1(uxx)) -1 + %(v2(uxx)) -2 + %(v3(uxx)) -3

= %(uxxx dx + 'xx) " %(3uxuxx) (#+ + ,1) = I3+ + ,2.

Therefore,D# = #s = I3, dV # = ,2 = (D2 + #2),

where the final formula follows from the contact form recurrence formulae

d, = d%('x) = + # ,1, d,1 = d%(') = + # (,2 " #2,) " #,1 # ,

which imply,1 = D,, ,2 = D,1 + #2 , = (D2 + #2),

Similarly,

d+ = %(d2x) + -1 # %(v1(dx)) + -2 # %(v2(dx)) + -3 # %(v3(dx))

= (#+ + ,1) # %(ux dx + ') = #+ # ,+ ,1 # ,.

In particular,dV + = "#, #+

Key recurrence formulae:

dV # = (D2 + #2), dV + = "# , #+

Plane Curves

Invariant Lagrangian:

+) = L(x, u(n)) dx = P (#,#s, . . .)+

Euler–Lagrange form:dV

+) ! E(L), #+

Invariant Integration by Parts Formula

F dV (DH) #+ ! " (DF ) dV H #+ " (F · DH) dV +

dV+) = dV P #+ + P dV +

="

n

!P

!#n

dV #n #+ + P dV +

! E(P ) dV # #+ + H(P ) dV +

Vertical di!erentiation formulae

dV # = A(,) A — “Eulerian operator”

dV + = B(,) #+ B — “Hamiltonian operator”

dV+) ! E(P ) A(,) #+ + H(P ) B(,) #+

!/A.E(P ) " B.H(P )

0, #+

Invariant Euler-Lagrange equation

A.E(P ) " B.H(P ) = 0

Euclidean Plane Curves

dV # = (D2 + #2),

Eulerian operator

A = D2 + #2 A. = D2 + #2

dV + = "# , #+

Hamiltonian operator

B = "# B. = "#

Euclidean–invariant Euler-Lagrange formula

E(L) = A.E(P ) " B.H(P ) = (D2 + #2) E(P ) + #H(P ).

Invariant Plane Curve Flows

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”

t, n — basis of the invariant vector fields dual to the invariantone-forms:

0 t ;+ 1 = 1, 0n ;+ 1 = 0,

0 t ;, 1 = 0, 0n ;, 1 = 1.

Ct = V = I t + J n

• 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.

• There are two principal choices of tangential component:

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

dV + = B(,) #+

B — invariant Hamiltonian operator

Normal Evolution of Di!erential Invariants

Theorem. Under a normal flow Ct = J n,

!#

!t= A%(J),

!#s

!t= A%s

(J).

Invariant variations:

dV # = A%(,), dV #s = A%s(,).

A% = A — invariant linearization operator of curvature;

A%s= DA% + ##s — invariant linearization operator 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!

Grassfire flow: J = 1

!#

!t= #2,

!#s

!t= 3##s, . . .

=% caustics

Signature Curves

Definition. The signature curve S ' R2 of a curve C ' R2 is

parametrized by the two lowest order di!erential invariants

S =

1 ,

# ,d#

ds

- 2

' R2

Equivalence and Signature Curves

Theorem. Two curves C and C are equivalent:

C = g · C

if and only if their signature curves are identical:

S = S

=% object recognition

Euclidean Signature Evolution

Evolution of the Euclidean signature curve

#s = $(t,#).

Grassfire flow:!$

!t= 3#$" #2 !$

!#.

Curve shortening flow:

!$

!t= $2$%% " #3$% + 4#2$.

Modified Korteweg-deVries flow:

!$

!t= $3$%%% + 3$2$%$%% + 3#$2.

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

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 integrable

evolution equation, and R is its recursion operator!

=% Hasimoto

=% Langer, Singer, Perline

=% Marı–Be!a, Sanders, Wang

=% Qu, Chou, and many more ...

Euclidean plane curves

G = SE(2) = SO(2) ! R2

dV # = (D2 + #2),, dV + = "# , #+

=% 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

dV # = A(,), dV + = B(,) #+

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 + 2##ss + 43 #

2s + +5

9 #2 #s

=% Sawada–Kotera equation

Euclidean space curves

G = SE(3) = SO(3) ! R3

,dV #dV 0

-

= A

,,1

,2

-

dV + = B

,,1

,2

-

#+

A =

3

44445

D2s + (#2 " 02)

20

#D2

s +3#0s " 2#s0

#2Ds +

#0ss " #s0s + 2#30

#2

"20Ds " 0s

1

#D3

s "#s

#2D2

s +#2 " 02

#Ds +

#s02 " 2#00s

#2

6

77778

B = (# 0 )

Recursion operator:

R = A"

,#s

0s

-

D"1B

,#t

0t

-

= R

,#s

0s

-

=% vortex filament flow

=% nonlinear Schrodinger equation (Hasimoto)

In v a ria n t V a ria tio n a l P ro b le m s In v a ria ...olver/t_/ivfoxt.pdf · In v a ria n t...

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