Moving Frames. Peter J. Olver University of Minnesota olver. IMA, July, i 1

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1 Moving Frames Peter J. Olver University of Minnesota olver IMA, July, 2006 i 1

2 Collaborators Mark Fels Gloria Marí Beffa Irina Kogan Juha Pohjanpelto Jeongoo Cheh i 2

3 Moving Frames Classical contributions: M. Bartels ( 1800), J. Serret, J. Frénet, C. Jordan, G. Darboux, É. Cotton, É. Cartan Modern contributions: P. Griffiths, M. Green, G. Jensen I did not quite understand how he [Cartan] does this in general, though in the examples he gives the procedure is clear. Nevertheless, I must admit I found the book, like most of Cartan s papers, hard reading. Cartan on groups and differential geometry Bull. Amer. Math. Soc. 44 (1938) Hermann Weyl i 1

4 Fels, M., Olver, P.J., References Part I, Acta Appl. Math. 51 (1998) Part II, Acta Appl. Math. 55 (1999) Calabi, Olver, Shakiban, Tannenbaum, Haker, Int. J. Computer Vision 26 (1998) Marí Beffa, G., Olver, P.J., Commun. Anal. Geom. 7 (1999) Olver, P.J., Classical Invariant Theory, Cambridge Univ. Press, 1999 Olver, P.J., Selecta Math. 6 (2000) Berchenko, I.A., Olver, P.J. J. Symb. Comp. 29 (2000) Olver, P.J., Found. Comput. Math. 1 (2001) 3 67 Olver, P.J., Appl. Alg. Engin. Commun. Comput. 11 (2001) Kogan, I.A., Olver, P.J., Acta Appl. Math. 76 (2003) Olver, P.J., Pohjanpelto, J., Selecta Math. 11 (2005) Olver, P.J., Pohjanpelto, J., Moving frames for Lie pseudo groups, Canadian J. Math., to appear Cheh, J., Olver, P.J., Pohjanpelto, J., J. Math. Phys., 46 (2005) Cheh, J., Olver, P.J., Pohjanpelto, J., Found. Comput. Math., to appear i 2

5 Applications of Moving Frames Differential geometry Equivalence Symmetry Differential invariants Rigidity Joint Invariants and Semi-Differential Invariants Invariant differential forms and tensors Identities and syzygies Classical invariant theory Computer vision object recognition symmetry detection Invariant numerical methods Poisson geometry & solitons Lie pseudogroups Invariants of Killing tensors Invariants of Lie algebras i 3

6 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 n-dimensional submanifolds N and N M are congruent: N = g N for g G Symmetry: Self-equivalence or self-congruence: N = g N i 4

7 Classical Geometry Equivalence Problem: Determine whether or not two given submanifolds N and N are congruent under a group transformation: N = g N. Symmetry Problem: Given a submanifold N, find all its symmetries (belonging to the group). Euclidean group G = SE(n) or E(n) isometries of Euclidean space translations, rotations (& reflections) z R z + a R SO(n) or O(n) a R n z R n Equi-affine group: G = SA(n) R SL(n) area-preserving Affine group: R GL(n) Projective group: acting on RP n 1 G = A(n) G = PSL(n) = Applications in computer vision i 5

8 Classical Invariant Theory Binary form: Q(x) = n k=0 ( ) n a k k x k Equivalence of polynomials (binary forms): Q(x) = (γx + δ) n Q ( ) αx + β γx + δ ( α β g = γ δ ) GL(2) multiplier representation of GL(2) modular forms Transformation group: g : (x,u) ( αx + β γx + δ, ) u (γx + δ) n Equivalence of functions equivalence of graphs N Q = { (x, u) = (x,q(x)) } C 2 i 6

9 Moving Frames Definition. A moving frame is a G-equivariant map ρ : M G Equivariance: ρ(g z) = g ρ(z) ρ(z) g 1 left moving frame right moving frame ρ left (z) = ρ right (z) 1 i 7

10 The Main Result Theorem. A moving frame exists in a neighborhood of a point z M if and only if G acts freely and regularly near z. G z = {g g z = z } = Isotropy subgroup free the only group element g G which fixes one point z M is the identity: = G z = {e} for all z M. locally free the orbits all have the same dimension as G: = G z is a discrete subgroup of G. regular all orbits have the same dimension and intersect sufficiently small coordinate charts only once irrational flow on the torus effective the only group element g G which fixes every point z M is the identity: g z = z for all z M iff g = e: G M = \ G z = {e} z M i 8

11 Theorem. A moving frame exists in a neighborhood of a point z M if and only if G acts freely and regularly near z. Necessity: Let z M. Let ρ : M G be a left moving frame. Freeness: If g G z, so g z = z, then by left equivariance: ρ(z) = ρ(g z) = g ρ(z). Therefore g = e, and hence G z = {e} for all z M. Regularity: Suppose z n = g n z z as n By continuity, Hence g n e in G. ρ(z n ) = ρ(g n z) = g n ρ(z) ρ(z) Sufficiency: By construction normalization. Q.E.D. i 9

12 Geometrical Construction Normalization = choice of cross-section to the group orbits K z O z k g =ρ(z ) K cross-section to the group orbits O z orbit through z M k K O z unique point in the intersection k is the canonical form of z the (nonconstant) coordinates of k are the fundamental invariants g G unique group element mapping k to z ρ(z) = g left moving frame ρ(h z) = h ρ(z) k = ρ 1 (z) z = ρ right (z) z = freeness i 10

13 Construction of Moving Frames Coordinate cross-section r = dim G m = dim M K = {z 1 = c 1,..., z r = c r } left w(g, z) = g 1 z right w(g, z) = g z Choose r = dim G components to normalize: w 1 (g, z) = c 1... w r (g, z) = c r The solution is a (local) moving frame. g = ρ(z) = Implicit Function Theorem i 11

14 The Fundamental Invariants Substituting the moving frame formulae g = ρ(z) into the unnormalized components of w(g, z) produces the fundamental invariants: I 1 (z) = w r+1 (ρ(z), z)... I m r (z) = w m (ρ(z), z) = These are the coordinates of the canonical form k K. Theorem. Every invariant I(z) can be (locally) uniquely written as a function of the fundamental invariants: I(z) = H(I 1 (z),...,i m r (z)) i 12

15 Invariantization Definition. The invariantization of a function F : M R with respect to a right moving frame ρ is the the invariant function I = ι(f) defined by I(z) = F(ρ(z) z). ι [ F(z 1,...,z m )] = F(c 1,,...c r, I 1 (z),...,i m r (z)) Invariantization amounts to restricting F to the cross-section I K = F K and then requiring that I = ι(f) be constant along the orbits. In particular, if I(z) is an invariant, then ι(i) = I. Invariantization defines a canonical projection ι: functions invariants i 13

16 The Rotation Group G = SO(2) acting on R 2 z = (x,u) g z = ( xcosθ usin θ, xsin θ + u cosθ ) = Free on M = R 2 \ {0} Left moving frame: w(g,z) = g 1 z = (y, v) y = xcosθ + u sin θ v = xsin θ + u cosθ Cross-section K = {u = 0, x > 0 } Normalization equation Left moving frame: v = xsin θ + u cosθ = 0 θ = tan 1 u x = θ = ρ(x, u) SO(2) Fundamental invariant r = ι(x) = x 2 + u 2 Invariantization ι[ F(x,u)] = F(r,0) i 14

17 Prolongation Most interesting group actions (Euclidean, affine, projective, etc.) are not free! An effective action can usually be made free by: Prolonging to derivatives (jet space) G (n) : J n (M,p) J n (M,p) = differential 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-differential invariants = invariant numerical approximations i 15

18 Jet Space Although in use since the time of Lie and Darboux, jet space was first formally defined by Ehresmann in Jet space is the proper setting for the geometry of partial differential equations. M smooth m-dimensional manifold 1 p m 1 J n = J n (M,p) (extended) jet bundle = Defined as the space of equivalence classes of p- dimensional submanifolds under the equivalence relation of n th order contact at a single point. = Can be identified as the space of n th order Taylor polynomials for submanifolds given as graphs u = f(x) i 16

19 Local Coordinates on Jet Space J n = J n (M,p) n th extended jet bundle for p-dimensional submanifolds N M Local coordinates: Assume N = {u = f(x)} is a graph (section). x = (x 1,...,x p ) u = (u 1,...,u q ) independent variables dependent variables p + q = m = dimm z (n) = (x, u (n) ) = (... x i... u α J... ) u α J = J uα 0 #J n induced jet coordinates No bundle structure assumed on M. Projective completion of J n E when E X is a bundle. i 17

20 Prolongation of Group Actions G transformation group acting on M = G maps submanifolds to submanifolds and preserves the order of contact G (n) prolonged action of G on the jet space J n The prolonged group formulae w (n) = (y, v (n) ) = g (n) z (n) are obtained by implicit differentiation: dy i = D y j = p j=1 p i=1 P i j (g, z(1) )dx j Q i j (g,z(1) ) D x i = Q = P T v α J = D y j 1 D y j k (v α ) Differential invariant I : J n R I(g (n) z (n) ) = I(z (n) ) = curvatures i 18

21 Freeness Theorem. If G acts (locally) effectively on M, then G acts (locally) freely on a dense open subset V n J n for n 0. Definition. N M is regular at order n if j n N V n. Corollary. Any regular submanifold admits a (local) moving frame. Theorem. A submanifold is totally singular, j n N J n \ V n for all n, if and only if its symmetry group does not act freely on N. G N = { g g N N } i 19

22 Moving Frames on Jet Space w (n) = (y, v (n) ) = g (n) z (n) right (g (n) ) 1 z (n) left Choose r = dim G jet coordinates z 1,...,z r x i or u α J Coordinate cross-section K J n z 1 = c 1... z r = c r Corresponding lifted differential invariants: w 1,...,w r y i or v α J Normalization Equations w 1 (g, x,u (n) ) = c 1... w r (g, x, u (n) ) = c r Solution: g = ρ (n) (z (n) ) = ρ (n) (x,u (n) ) = moving frame i 20

23 The Fundamental Differential Invariants I (n) (z (n) ) = w (n) (ρ (n) (z (n) ), z (n) ) H i (x,u (n) ) = y i (ρ (n) (x,u (n) ),x, u) I α K (x, u(k) ) = v α K (ρ(n) (x,u (n) ),x, u (k) ) Phantom differential invariants w 1 = c 1... w r = c r = normalizations Theorem. Every n th order differential invariant can be locally uniquely written as a function of the non-phantom fundamental differential invariants in I (n). i 21

24 Invariant Differentiation Contact-invariant coframe dy i ω i = p j=1 Invariant differential operators: P i j (ρ(n) (z (n) ),z (n) ) dx i = arc length element Duality: D y j D j = df = p i=1 p i=1 Q i j (ρ(n) (z (n) ),z (n) ) D x i = arc length derivative D i F ω i Theorem. The higher order differential invariants are obtained by invariant differentiation with respect to D 1,...,D p. i 22

25 Euclidean Curves G = SE(2) Assume the curve is (locally) a graph: C = {u = f(x)} Prolong to J 3 via implicit differentiation y = cosθ (x a) + sin θ (u b) v = sin θ (x a) + cosθ (u b) v y = sin θ + u x cosθ cosθ + u x sin θ v yy = u xx (cosθ + u x sin θ ) 3 v yyy = (cosθ + u x sin θ )u xxx 3u2 xx sin θ (cosθ + u x sin θ ) 5. Normalization r = dim G = 3 w = R 1 (z b) y = 0, v = 0, v y = 0 Left moving frame ρ: J 1 SE(2) a = x, b = u, θ = tan 1 u x i 23

26 Differential invariants v yy κ = v yyy v yyyy dκ ds d2 κ ds 2 3κ3 = u xx (1 + u 2 x )3/2 = (1 + u2 x )u xxx 3u x u2 xx (1 + u 2 x )3 Invariant one-form arc length dy = (cosθ + u x sin θ)dx ds = 1 + u 2 x dx Invariant differential operator d dy = 1 cosθ + u x sin θ d dx d ds = u 2 x d dx Theorem. All differential invariants are functions of the derivatives of curvature with respect to arc length: κ, dκ ds, d 2 κ ds 2, i 24

27 Euclidean Curves e 2 e 1 x Moving frame ρ : (x,u, u x ) (R,a) SE(2) ( ) ( 1 1 ux x R = = (e 1 + u 2 u x x 1 1,e 2 ) a = u ) Frenet frame e 1 = dx ds = ( ) xs y s e 2 = e 1 = ( ys x s ) Frenet equations = Maurer Cartan equations: dx ds = e 1 de 1 ds = κe 2 de 2 ds = κe 1 i 25

28 The Replacement Theorem Any differential invariant has the form I = F(x,u (n) ) = F(y, w (n) ) = F(I (n) ) = T.Y. Thomas κ = v yy (1 + v 2 y )2 = u xx (1 + u 2 x )2 ι(x) = ι(u) = (u x ) = 0 ι(u xx ) = κ i 26

29 Equi-affine Curves G = SA(2) z Az + b A SL(2), b R 2 Prolong to J 3 via implicit differentiation dy = (δ u x β) dx D y = 1 δ u x β D x y = δ(x a) β(u b) v = γ(x a) + α(u b) w = A 1 (z b) v y = γ αu x δ βu x v yy = u xx (δ βu x ) 3 v yyy = (δ βu x )u xxx + 3βu2 xx (δ βu x ) 5 v yyyy = u xxxx (δ βu x )2 + 10u xx u xxx β(δ βu x ) + 15u 3 xx β2 (α + βu x ) 7. Nondegeneracy u xx = 0 = Straight lines are totally singular (three-dimensional equi-affine symmetry group) Normalization r = dimg = 5 y = 0, v = 0, v y = 0, v yy = 1, v yyy = 0. i 27

30 Left Moving frame ρ: J 3 SA(2) A = = 3 uxx 1 3 u 5/3 xx u xxx 3 u x uxx uxx 1/3 1 3 u 5/3 xx u xxx ) ( dz ds d 2 z ds 2 b = z = ( ) x u e 2 e 1 x i 28

31 Frenet frame e 1 = dz ds e 2 = d2 z ds 2 Frenet equations = Maurer Cartan equations: dz ds = e 1 de 1 ds = e 2 de 2 ds = κe 1 Equi-affine arc length Invariant differential operator dy ds = 3 uxx dx = 3 z z dt D y d ds = 1 D x = uxx z z D t Equi-affine curvature v 4y κ = 5u xx u xxxx 3u2 xxx 9u 8/3 xx = z s z ss v 5y dκ ds v 6y d2 κ ds 2 5κ2 i 29

32 Equivalence & Signature Cartan s main idea: The equivalence and symmetry properties of submanifolds will be found by restricting the differential invariants to the submanifold J(x) = I( j n N x ). Equivalent submanifolds should have the same invariants. However, unless an invariant J(x) is constant, it carries little information by itself, since the equivalence map will typically drastically change the dependence of the invariant on the parameter x. = Constant curvature submanifolds However, a functional dependency or syzygy among the invariants is intrinsic: J k (x) = Φ(J 1 (x),...,j k 1 (x)) i 30

33 The Signature Map Equivalence and symmetry properties of submanifolds are governed by the functional dependencies syzygies among the differential invariants. J k (x) = Φ(J 1 (x),...,j k 1 (x)) The syzygies are encoded by the signature map Σ : N S of the submanifold N, which is parametrized by the fundamental differential invariants: Σ(x) = (J 1 (x),...,j m (x)) = ( I 1 N,..., I m N ) The image S = Im Σ is the signature subset (or submanifold) of N. i 31

34 Geometrically, the signature S K is the image of j n N in the cross-section K J n, where n 0 is sufficiently large. Σ : N j n N S K Theorem. Two submanifolds are equivalent: N = g N if and only if their signatures are identical: S = S i 32

35 Signature Curves Definition. The signature curve S R 2 of a curve C R 2 is parametrized by the first two differential invariants κ and κ s S = { ( κ, dκ ds ) } R 2 Theorem. Two curves C and C are equivalent C = g C if and only if their signature curves are identical S = S = object recognition i 33

36 Symmetry Signature map Σ : N S Theorem. Let S denote the signature of the submanifold N. Then the dimension of its symmetry group G N = {g g N N } equals dimg N = dim N dim S Corollary. For a regular submanifold N M, 0 dimg N dim N = Only totally singular submanifolds can have larger symmetry groups! i 34

37 Maximally Symmetric Submanifolds Theorem. The following are equivalent: The submanifold N has a p-dimensional symmetry group The signature S degenerates to a point dim S = 0 The submanifold has all constant differential invariants N = H {z 0 } is the orbit of a p-dimensional subgroup H G = In Euclidean geometry, these are the circles, straight lines, helices, spheres & planes. = In equi-affine plane geometry, these are the conic sections. = In projective geometry, these are the W curves of Lie and Klein. i 35

38 Discrete Symmetries Definition. The index of a submanifold N equals the number of points in N which map to a generic point of its signature S: ι N = min { # Σ 1 {w} w S } = Self intersections Theorem. The cardinality of the symmetry group of N equals its index ι N. = Approximate symmetries i 36

39 Classical Invariant Theory { ( M = R 2 α β \ {u = 0} G = GL(2) = γ δ ) αδ βγ 0 } (x, u) ( αx + β γx + δ, ) u (γx + δ) n n 0,1 σ = γ x + δ = αδ βγ Prolongation: y = αx + β γx + δ v = σ n u v y = σu x nγu σ n 1 v yy = σ2 u xx 2(n 1)γσu x + n(n 1)γ 2 u 2 σ n 2 v yyy = Normalization: y = 0 v = 1 v y = 0 v yy = 1 n(n 1) i 37

40 Moving frame: α = u (1 n)/n H γ = 1 n u(1 n)/n β = xu (1 n)/n H δ = u 1/n 1 n xu(1 n)/n H = n(n 1)uu xx (n 1) 2 u 2 x Hessian Nonsingular form: H 0 Note: H 0 if and only if Q(x) = (ax + b) n = Totally singular forms Differential invariants: v yyy J n 2 (n 1) κ v yyyy K + 3(n 2) n 3 (n 1) dκ ds Absolute rational covariants: J 2 = T 2 H 3 K = U H 2 H = 1 2 (Q, Q)(2) = n(n 1)QQ (n 1) 2 Q 2 Q xx Q yy Q 2 xy T = (Q, H) (1) = (2n 4)Q H nqh Q x H y Q y H x U = (Q, T) (1) = (3n 6)Q T nqt Q x T y Q y T x deg Q = n deg H = 2n 4 deg T = 3n 6 deg U = 4n 8 i 38

41 Signatures of Binary Forms Signature curve of a nonsingular binary form Q(x): S Q = { (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. Signature map Σ: N Q S Q Σ(x) = (J(x) 2, K(x)) Theorem. Two nonsingular binary forms are equivalent if and only if their signature curves are identical. i 39

42 Maximally Symmetric Binary Forms Theorem. If u = Q(x) is a polynomial, then the following are equivalent: Q(x) admits a one-parameter symmetry group T 2 is a constant multiple of H 3 Q(x) x k is complex-equivalent to a monomial the signature curve degenerates to a single point all the (absolute) differential invariants of Q are constant the graph of Q coincides with the orbit of a one-parameter subgroup = diagonalizable i 40

43 Symmetries of Binary Forms Theorem. The symmetry group of a nonzero binary form Q(x) 0 of degree n is: A two-parameter group if and only if H 0 if and only if Q is equivalent to a constant. = totally singular A one-parameter group if and only if H 0 and T 2 is a constant multiple of H 3 if and only if Q is complexequivalent to a monomial x k, with k 0,n. = maximally symmetric In all other cases, a finite group whose cardinality equals the index ι Q = min { # Σ 1 {w} w S } of the signature curve, and is bounded by ι Q 6n 12 U = ch 2 4n 8 otherwise i 41

44 The Variational Bicomplex = Dedecker, Vinogradov, Tsujishita, I. Anderson Infinite jet space Local coordinates J = lim n Jn z ( ) = (x,u ( ) ) = (... x i... u α J... ) Horizontal one-forms dx 1,...,dx p Contact (vertical) one-forms θ α J = duα J p i=1 u α J,i dxi Intrinsic definition of contact form θ j N = 0 θ = A α J θα J i 42

45 Bigrading of the differential forms on J Ω = M r,s Ω r,s r = # of dx i s = # of θ α J Vertical and Horizontal Differentials d = d H + d V Variational Bicomplex: d H : Ω r,s Ω r+1,s d V : Ω r,s Ω r,s+1 d 2 H = 0 d H d V + d V d H = 0 d2 V = 0 F(x,u (n) ) differential function d H F = p i=1 d V F = α,j (D i F) dx i total differential F u α J θ α J variation i 43

46 The Simplest Example. M = R 2, x,u R Horizontal form dx Contact (vertical) forms θ = du u x dx θ x = du x u xx dx θ xx = du xx u xxx dx Differential F = F(x,u,u x, u xx,... ) df = F x Total derivative dx + F u. du + F u x du x + F u xx du xx + = (D x F) dx + F u θ + F u x θ x + F u xx θ xx + = d H F + d V F D x F = F u u x + F u x u xx + F u xx u xxx + i 44

47 Lagrangian form Vertical derivative λ = L(x,u (n) ) dx Ω 1,1 variation dλ = d V λ = d V L dx ( L = u θ + L θ u x + L θ x u xx + xx ) dx Ω 1,1 Integration by parts d H (A θ) = (D x A)dx θ A θ x dx so = [(D x A) θ + A θ x ] dx A θ x dx (D x A)θ dx mod im d H Variational derivative compute modulo im d H : ( L dλ δλ = u D L x + D 2 L x u x u xx = E(L) θ dx ) θ dx = Euler-Lagrange source form. i 45

48 Variational Derivative p = # independent variables Variation: d V : Ω p,0 Ω p,1 Integration by Parts: π : Ω p,1 F 1 = Ω p,1 /d H Ω p 1,1 = source forms Variational derivative or Euler operator: δ = π d V : Ω p,0 F 1 λ = Ldx q α=1 E α (L) θ α dx Variational Problems Source Forms i 46

49 The Variational Bicomplex..... d V d V d V d V δ Ω 0,3 d H Ω 1,3 d H d V d V Ω 0,2 d H Ω 1,2 d H d V d V d H Ω p 1,3 d H Ω p,3 π F 3 d V d V δ d H Ω p 1,2 d H Ω p,2 π F 2 d V d V δ R Ω 0,1 d H Ω 1,1 d H d H Ω p 1,1 d H Ω p,1 π F 1 d V d V d V d V Ω 0,0 d H Ω 1,0 d H d H Ω p 1,0 d H Ω p,0 p = # independent variables q = # dependent variables δ i 47

50 Invariantization of the Bicomplex ι : Functions Invariants Forms Invariant Forms Fundamental differential invariants H i (x,u (n) ) = ι(x i ) I α K (x, u(l) ) = ι(u α K ) Invariant horizontal forms ϖ i = ι(dx i ) = ω i + σ i Invariant contact forms ϑ α J = ι(θα J ) i 48

51 Invariant Variational Complex Differential forms Ω = M r,s Ω r,s = I. Kogan, PJO Differential d = d H + d V + d W d H : Ω r,s d V : Ω r,s d W : Ω r,s Ω r+1,s Ω r,s+1 Ω r 1,s+2 d 2 H = 0 d H d V + d V d H = 0 d 2 V + d H d W + d W d H = 0 d V d W + d W d V = 0 d 2 W = 0 i 49

52 The Key Formula d ι(ω) = ι(dω) + p k=1 ν κ ι[v κ (Ω)] v 1,...,v r basis for g ν κ = σ µ κ = γ κ + ε κ κ = 1,...,r γ κ Ω 1,0 ε κ Ω 0,1 pull back of the dual basis Maurer Cartan forms via the moving frame section σ : J B All recurrence formulae, syzygies, commutation formulae, etc. are found by applying the key formula for various forms and functions Ω i 50

53 Euclidean Curves Lifted invariants y = w (x) = xcos φ u sin φ + a v = w (u) = xcos φ + u sin φ + b v y = w (u x ) = sin φ + u x cosφ cosφ u x sin φ v yy = w (u xx ) = u xx (cosφ u x sin φ) 3 v yyy = w (u xx ) = (cosφ u x sin φ)u xxx 3u2 xx sin φ (cosφ u x sin φ) 5 dy = (cosφ u x sin φ)dx (sin φ) θ + da v dφ d J y = π J (dy) = (cos φ u x sin φ)dx (sin φ) θ D y = 1 cosφ u x sin φ D x θ = du u x dx Normalization y = 0 v = 0 v y = 0 Right moving frame φ = tan 1 u x ρ: J 1 SE(2) a = x + uu x 1 + u 2 x b = xu x u 1 + u 2 x i 51

54 Fundamental normalized differential invariants ι(x) = H = 0 ι(u) = I 0 = 0 ι(u x ) = I 1 = 0 ι(u xx ) = I 2 = κ ι(u xxx ) = I 3 = κ s ι(u xxxx ) = I 4 = κ ss + 3κ 3 phantom diff. invs. Invariant horizontal one-form ι(dx) = σ (d J y) = ϖ = ω + η = 1 + u 2 x dx + u x 1 + u 2 x θ Invariant contact forms ι(θ) = ϑ = θ 1 + u 2 x ι(θ x ) = ϑ 1 = (1 + u2 x )θ x u x u xx θ (1 + u 2 x )2 i 52

55 Prolonged infinitesimal generators v 1 = x v 2 = u v 3 = u x + x u + (1 + u 2 x ) u x + 3u x u xx uxx + d H I = D s I ϖ Horizontal recurrence formula d H ι(f) = ι(d H F) + ι(v 1 (F))γ 1 + ι(v 2 (F))γ 2 + ι(v 3 (F))γ 3 Use phantom invariants 0 = d H H = ι(d H x) + ι(v κ (x))γ κ = ϖ + γ 1, 0 = d H I 0 = ι(d H u) + ι(v κ (u))γ κ = γ 2, 0 = d H I 1 = ι(d H u x ) + ι(v κ (u x ))γ κ = κϖ + γ 3, to solve for γ 1 = ϖ γ 2 = 0 γ 3 = κϖ i 53

56 γ 1 = ϖ γ 2 = 0 γ 3 = κϖ Recurrence formulae κ s ϖ = d H κ = d H (I 2 ) = ι(d H u xx ) + ι(v 3 (u xx ))γ 3 = ι(u xxx dx) ι(3u x u xx ))κϖ = I 3 ϖ κ ss ϖ = d H (I 3 ) = ι(d H u xxx ) + ι(v 3 (u xxx )γ 3 = ι(u xxxx dx) ι(4u x u xxx + 3u 2 xx )κϖ = I 4 3I3 2 ϖ κ = I 2 κ s = I 3 I 2 = κ I 3 = κ s κ ss = I 4 3I 3 2 I 4 = κ ss + 3κ 3 κ sss = I 5 19I 2 2 I 3 I 4 = κ sss + 19κ 2 κ s i 54

57 Vertical recurrence formula d V ι(f) = ι(d V F) + ι(v 1 (F))ε 1 + ι(v 2 (F))ε 2 + ι(v 3 (F))ε 3 Use phantom invariants 0 = d V H = ε 1 0 = d V I 0 = ϑ + ε 2 0 = d V I 1 = ϑ 1 + ε 3 to solve for ε 1 = 0 ε 2 = ϑ = ι(θ) ε 3 = ϑ 1 = ι(θ 1 ) Recurrence formulae d V I 2 = d V κ = ι(θ 2 ) + ι(v 3 (u xx ))ε 3 = ϑ 2 = (D 2 + κ 2 )ϑ, d H ϑ: Dϑ = ϑ 1 Dϑ 1 = ϑ 2 κ 2 ϑ d V ϖ = κϑ ϖ i 55

58 Example (x 1,x 2,u) M = R 3 G = GL(2) (x 1,x 2,u) (αx 1 + βx 2,γx 1 + δx 2,λu) λ = αδ βγ = Classical invariant theory Prolongation (lifted differential invariants): y 1 = λ 1 (δx 1 βx 2 ) y 2 = λ 1 ( γx 1 + αx 2 ) v = λ 1 u v 1 = αu 1 + γu 2 λ v 2 = βu 1 + δu 2 λ Normalization v 11 = α2 u αγu 12 + γ 2 u 22 λ v 12 = αβu 11 + (αδ + βγ)u 12 + γδu 22 λ v 22 = β2 u βδu 12 + δ 2 u 22 λ y 1 = 1 y 2 = 0 v 1 = 1 v 2 = 0 Nondegeneracy x 1 u u + x2 x1 x 2 0 i 56

59 First order moving frame ( ) ( ) α β x 1 u = 2 γ δ x 2 u 1 Normalized differential invariants J 1 = 1 J 2 = 0 u I = x 1 u 1 + x 2 u 2 I 1 = 1 I 2 = 0 I 11 = (x1 ) 2 u x 1 x 2 u 12 + (x 2 ) 2 u 22 x 1 u 1 + x 2 u 2 I 12 = x1 u 2 u 11 + (x 1 u 1 x 2 u 2 )u 12 + x 2 u 1 u 22 x 1 u 1 + x 2 u 2 I 22 = (u 2 )2 u 11 2u 1 u 2 u 12 + (u 1 ) 2 u 22 x 1 u 1 + x 2 u 2 Phantom differential invariants Generating differential invariants I 1 I 2 I I 11 I 12 I 22 Invariant differential operators D 1 = x 1 D 1 + x 2 D 2 D 2 = u 2 D 1 + u 1 D 2 scaling process Jacobian process i 57

60 Recurrence formulae D 1 J 1 = δ = 0 D 2 J1 = δ = 0 D 1 J 2 = δ = 0 D 2 J2 = δ = 0 D 1 I = I 1 I(1 + I 11 ) = I(1 + I 11 ) D 2 I = I 2 I I 12 = I I 12 D 1 I 1 = I 11 I 11 = 0 D 2 I 1 = I 12 I 12 = 0 D 1 I 2 = I 12 I 12 = 0 D 2 I 2 = I 22 I 22 = 0 D 1 I 11 = I (1 I 11 )I 11 D 2 I 11 = I (2 I 11 )I 12 D 1 I 12 = I 112 I 11 I 12 D 2 I 12 = I (1 I 11 )I 22 D 1 I 22 = I (I 11 1)I 22 2I 2 12 D 2 I 22 = I 222 I 12 I 22 Syzygies = Use I to generate I 11 and I 12 D 1 I 12 D 2 I 11 = 2I 12 D 1 I 22 D 2 I 12 = 2(I 11 1)I 22 2I 2 12 (D 1 ) 2 I 22 (D 2 ) 2 I 11 = = 2I 22 D 1 I 11 + (5I 12 2)D 1 I 12 + (3I 11 5)D 1 I 22 (2I 11 5)(I 11 1)I (I 11 1)I 2 12 Commutation formulae [D 1, D 2 ] = I 12 D 1 + (I 11 1)D 2 i 58

61 Invariant Variational Problems I[u] = L(x,u (n) ) dx = P(... D K I α... ) ω I 1,...,I l D K I α fundamental differential invariants differentiated invariants ω = ω 1 ω p contact-invariant volume form Invariant Euler-Lagrange equations E(L) = F(... D K I α... ) = 0 Problem. Construct F directly from P. = P. Griffiths, I. Anderson i 59

62 Example. Planar Euclidean group G = SE(2) Invariant variational problem P(κ,κ s, κ ss,... ) ds Euler-Lagrange equations E(L) = F(κ, κ s, κ ss,... ) = 0 The Elastica (Euler): I[u] = 1 2 κ2 ds = u 2 xx dx (1 + u 2 x )5/2 Euler-Lagrange equation E(L) = κ ss κ3 = 0 = elliptic functions i 60

63 P(κ,κ s, κ ss,... ) ds Invariantized Euler operator E = n=0 ( D) n κ n D = d ds Invariantized Hamiltonian operator H(P) = i>j κ i j ( D) j P κ i P Invariant Euler-Lagrange formula E(L) = (D 2 + κ 2 ) E(P) + κ H(P). Elastica P = 1 2 κ2 E(P) = κ H(P) = P = 1 2 κ2 E(L) = κ ss κ3 = 0 i 61

64 Euler-Lagrange Equations Integration by Parts: π : Ω p,1 F 1 = Ω p,1 /d H Ω p 1,1 Variational derivative or Euler operator: = Source forms δ = π d V : Ω p,0 F 1 Variational Problems Source Forms Hamiltonian δ : λ = L dx H(L) = m α=1 i>j 0 q α=1 u α i j ( D L x )j u α i E α (L) θ α dx L i 62

65 The Simplest Example. M = R 2 x, u R Lagrangian form λ = L(x, u (n) )dx Vertical derivative dλ = d V λ ( L = u θ + L θ u x + L θ x u xx + xx Integration by parts d H (A θ) = (D x A)dx θ Aθ x dx ) = [(D x A)θ + A θ x ] dx dx Ω 1,1 Variational derivative δλ = ( L u D x L u x + D 2 x = E(L)θ dx F 1 ) L θ dx u xx i 63

66 Invariant Lagrangian Plane Curves P(κ,κ s,...) ϖ κ fundamental differential invariant (curvature) ϖ = ω + η fully invariant horizontal form ω = ds contact-invariant arc length Invariant integration by parts d V (P ϖ) = E(P) d V κ ϖ H(P) d V ϖ Vertical differentiation formulae d V κ = A(ϑ) d V ϖ = B(ϑ) ϖ A Eulerian operator B Hamiltonian operator = The explicit formulae follow from our fundamental recurrence formula, based on the infinitesimal generators of the action. Invariant Euler-Lagrange equation A E(P) B H(P) = 0 i 64

67 General Framework Fundamental differential invariants I 1,...,I l Invariant horizontal coframe ϖ 1,...,ϖ p Dual invariant differential operators D 1,...,D p Invariant volume form ϖ = ϖ 1 ϖ p Differentiated invariants I,K α = DK J α = D k1 D kn J α = order is important! i 65

68 Eulerian operator d V I α = q β=1 A α β (ϑβ ) A = ( A α β ) = m q matrix of invariant differential operators Hamiltonian operator complex d V ϖ j = q β=1 B j i,β (ϑβ ) ϖ i B j i = ( B j i,β ) = p 2 row vectors of invariant differential operators ϖ (i) = ( 1) i 1 ϖ 1 ϖ i 1 ϖ i+1 ϖ p Twist invariants Twisted adjoint d H ϖ (i) = Z i ϖ D i = (D i + Z i ) i 66

69 Invariant variational problem P(I (n) )ϖ Invariant Eulerian E α (P) = K D K P I α,k Invariant Hamiltonian tensor H i j (P) = P δi j + q α=1 J,K I α,j,j D K P I α,j,i,k, Invariant Euler-Lagrange equations A E(P) p i,j=1 (Bi) j Hj i (P) = 0. i 67

70 Euclidean Surfaces S M = R 3 coordinates z = (x,y, u) Group: G = E(3) z R z + a, R O(3) Normalization coordinate cross-section x = y = u = u x = u y = u xy = 0. Left moving frame a = z R = ( t 1 t 2 n ) t 1,t 2 TS n Frenet frame unit normal i 68

71 Fundamental differential invariants κ 1 = ι(u xx ) κ 2 = ι(u yy ) = principal curvatures Frenet coframe ϖ 1 = ι(dx 1 ) = ω 1 + η 1 ϖ 2 = ι(dx 2 ) = ω 2 + η 2 Invariant differential operators D 1 D 2 = Frenet differentiation Fundamental Syzygy: Use the recurrence formula to compare ι(u xxyy ) with κ 1,22 = D2 2 ι(u xx ) κ 2,11 = D2 1 ι(u yy ) κ 1,22 κ2,11 + κ1,1 κ2,1 + κ1,2 κ2,2 2(κ2,1 )2 2(κ 1,2 )2 κ 1 κ 2 κ 1 κ 2 (κ 1 κ 2 ) = 0 = Codazzi equations i 69

72 Twisted adjoints D 1 = (D 1 + Z 1 ) Z 1 = κ2,1 κ 1 κ 2 D 2 = (D 2 + Z 2 ) Z 2 = κ1,2 κ 2 κ 1 Gauss curvature Codazzi equations: K = κ 1 κ 2 = D 1(Z 1 ) + D 2(Z 2 ) = (D 1 + Z 1 )Z 1 (D 2 + Z 2 )Z 2 K is an invariant divergence = Gauss Bonnet Theorem! i 70

73 Invariant contact form ϑ = ι(θ) = ι(du u x dx u y dy) Invariant vertical derivatives d V κ 1 = ι(θ xx ) = ( D1 2 + Z 2 D 2 + (κ1 ) 2 )ϑ d V κ 2 = ι(θ yy ) = ( D2 2 + Z 1 D 1 + (κ2 ) 2 )ϑ Eulerian operator ( D 2 A = 1 + Z 2 D 2 + (κ 1 ) 2 D2 2 + Z 1 D 1 + (κ2 ) 2 ) d V ϖ 1 = κ 1 ϑ ϖ 1 + d V ϖ 2 = Hamiltonian operator complex 1 κ 1 κ 2( D 1 D 2 Z 2 D 1 )ϑ ϖ2, 1 κ 2 κ 1( D 2 D 1 Z 1 D 2 )ϑ ϖ1 κ 2 ϑ ϖ 2, B 1 1 = κ1, B 2 2 = κ 2, B 1 2 = 1 κ 1 κ 2( D 1 D 2 Z 2 D 1 ) = B2 1 i 71

74 Euclidean-invariant variational problem P(κ (n) )ω 1 ω 2 = P(κ (n) ) da Euler-Lagrange equations Special case: P(κ 1,κ 2 ) E(L) = A E(P) B H(P) = 0, E(L) = [(D 1 ) 2 + D 2 Z 2 + (κ 1 ) 2 ] L κ 1 + Minimal surfaces: P = 1 + [ (D 2 ) 2 + D 1 Z 1 + (κ 2 ) 2 ] L κ 2 (κ1 + κ 2 ) L. (κ 1 + κ 2 ) = 2H = 0 Minimizing mean curvature: P = H = 1 2 (κ1 + κ 2 ) [ 1 2 (κ 1 ) 2 + (κ 2 ) 2 (κ 1 + κ 2 ) 2 ] = κ 1 κ 2 = K = 0. Willmore surfaces: P = 1 2 (κ1 ) (κ2 ) 2 (κ 1 + κ 2 ) (κ1 + κ 2 )(κ 1 κ 2 ) 2 = 2 H + 4(H 2 K)H = 0 Laplace Beltrami operator = (D 1 + Z 1 )D 1 + (D 2 + Z 2 )D 2 = D 1 D 1 D 2 D 2 i 72