Infinite Dimensional Symmetry Groups

Transcription

1 Infinite Dimensional Symmetry Groups Peter J. Olver University of Minnesota olver Lausanne, July, 2007

2 Sur la théorie, si importante sans doute, mais pour nous si obscure, des groupes de Lie infinis, nous ne savons rien que ce qui trouve dans les mémoires de Cartan, première exploration à travers une jungle presque impénétrable; mais celle-ci menace de se refermer sur les sentiers déjà tracés, si l on ne procède bientôt à un indispensable travail de défrichement. André Weil, 1947

3 What s the Deal with Infinite Dimensional Groups? Lie invented Lie groups to study symmetry and solution of differential equations. In Lie s time, there were no abstract Lie groups. All groups were realized by their action on a space. Therefore, Lie saw no essential distinction between finitedimensional and infinite-dimensional group actions. However, with the advent of abstract Lie groups, the two subjects have gone in radically different directions. The general theory of finite-dimensional Lie groups has been rigorously formalized and applied. But there is still no generally accepted abstract object that represents an infinite-dimensional Lie pseudo-group!

4 Ehresmann s Trinity 1953: Lie Pseudo-groups Jets Groupoids

5 Ehresmann s Trinity 1953: Lie Pseudo-groups Jets Groupoids

6 Ehresmann s Trinity 1953: Lie Pseudo-groups Jets Groupoids

7 Ehresmann s Trinity 1953: Lie Pseudo-groups Jets Groupoids

8 Lie Pseudo-groups in Action Lie Medolaghi Vessiot Cartan Ehresmann Kuranishi, Spencer, Goldschmidt, Guillemin, Sternberg, Kumpera,... Relativity Noether s (Second) Theorem

9 Gauge theory and field theories: Maxwell, Yang Mills, conformal, string,... Fluid mechanics, metereology: Euler, Navier Stokes, boundary layer, quasi-geostropic,... Solitons (in dimensions): K P, Davey-Stewartson,... Kac Moody Linear and linearizable PDEs Lie groups!

10 Moving Frames In collaboration with Juha Pohjanpelto and Jeongoo Cheh, I have recently established a moving frame theory for infinite-dimensional Lie pseudo-groups mimicking the earlier equivariant approach for finite-dimensional Lie groups developed with Mark Fels and others. The finite-dimensional theory and algorithms have had a very wide range of significant applications, including differential geometry, differential equations, calculus of variations, computer vision, Poisson geometry and solitons, numerical methods, relativity, classical invariant theory,...

11 What s New? In the infinite-dimensional case, the moving frame approach provides new constructive algorithms for: Invariant Maurer Cartan forms Structure equations Moving frames Differential invariants Invariant differential operators Basis Theorem Syzygies and recurrence formulae

12 Further applications: = Symmetry groups of differential equations = Vessiot group splitting; explicit solutions = Gauge theories = Calculus of variations = Numerical methods

13 Symmetry Groups Review System of differential equations: ν (x, u (n) ) = 0, ν = 1, 2,..., k By a symmetry, we mean a transformation that maps solutions to solutions. Lie: To find the symmetry group of the differential equations, work infinitesimally. The vector field v = p i=1 ξ i (x, u) x i + q α=1 ϕ α (x, u) u α is an infinitesimal symmetry if its flow exp(t v) is a oneparameter symmetry group of the differential equation.

14 To find the infinitesimal symmetry conditions, we prolong v to the jet space whose coordinates are the derivatives appearing in the differential equation: where v (n) = ϕ J α = D J p i=1 ( ξ i x i + ϕ α p i=1 q α=1 u α i ξi ) + Φ J α (x, u(n) ; ξ (n), ϕ (n) ) Infinitesimal invariance criterion: n ϕ J α #J=0 u α J p u α J,i ξi i=1 v (n) ( ν ) = 0 whenever = 0. Infinitesimal determining equations: L(x, u; ξ (n), ϕ (n) ) = 0

15 Symmetry generator: Prolongation: where The Korteweg devries equation v = τ(t, x, u) t v (3) = v + ϕ t u t + u xxx + uu x = 0 + ξ(t, x, u) x u t + ϕ x + ϕ(t, x, u) u u x + + ϕ xxx u xxx ϕ t = ϕ t + u t ϕ u u t τ t u 2 t τ u u x ξ t u t u x ξ u ϕ x = ϕ x + u x ϕ u u t τ x u t u x τ u u x ξ x u 2 x ξ u ϕ xxx = ϕ xxx + 3u x ϕ u +

16 Infinitesimal invariance: v (3) (u t + u xxx + uu x ) = ϕ t + ϕ xxx + u ϕ x + u x ϕ = 0 on solutions Infinitesimal determining equations: τ x = τ u = ξ u = ϕ t = ϕ x = 0 ϕ = ξ t 2 3 uτ t ϕ u = 2 3 τ t = 2 ξ x τ tt = τ tx = τ xx = = ϕ uu = 0 General solution: τ = c 1 + 3c 4 t, ξ = c 2 + c 3 t + c 4 x, ϕ = c 3 2c 4 u.

17 Basis for symmetry algebra g KdV : v 1 = t, v 2 = x, v 3 = t x + u, v 4 = 3t t + x x 2u u. The symmetry group G KdV is four-dimensional (x, t, u) (λ 3 t + a, λ x + c t + b, λ 2 u + c)

18 v 1 = t, v 2 = x, v 3 = t x + u, v 4 = 3t t + x x 2u u. Commutator table: v 1 v 2 v 3 v 4 v v 1 v v 1 3 v 2 v 3 0 v v 3 v 3 v 1 3 v 2 2 v 3 0 Entries: [v i, v j ] = k C k ij v k C k ij structure constants of g

19 Navier Stokes Equations u t + u u = p + ν u, u = 0. Symmetry generators: v α = α(t) x + α (t) u α (t) x p v 0 = t s = x x + 2t t u u 2 p p r = x x + u u w h = h(t) p

20 Kadomtsev Petviashvili (KP) Equation Symmetry generators: ( u t u u x u xxx ) x ± 3 4 u yy = 0 v f = f(t) t y f (t) y + ( 1 3 x f (t) 2 9 y2 f (t) ) x + ( 2 3 u f (t) x f (t) 4 27 y2 f (t) ) u, w g = g(t) y 2 3 y g (t) x 4 9 y g (t) u, z h = h(t) x h (t) u. = Kac Moody loop algebra A (1) 4

21 Main Goals Given a system of partial differential equations: Find the structure of its symmetry (pseudo-) group G directly from the determining equations. Find and classify its differential invariants. Use symmetry reduction or group splitting to construct explicit solutions both invariant and non-invariant.

22 Main Goals Given a system of partial differential equations: Find the structure of its symmetry (pseudo-) group G directly from the determining equations. Find and classify its differential invariants. Use symmetry reduction or group splitting to construct explicit solutions.

23 Main Goals Given a system of partial differential equations: Find the structure of its symmetry (pseudo-) group G directly from the determining equations. Find and classify its differential invariants. Use symmetry reduction or group splitting to construct explicit solutions.

24 Main Goals Given a system of partial differential equations: Find the structure of its symmetry (pseudo-) group G directly from the determining equations. Find and classify its differential invariants. Use symmetry reduction or group splitting to construct explicit solutions.

25 Pseudo-groups M smooth (analytic) manifold Definition. A pseudo-group is a collection of local diffeomorphisms ϕ : M M such that Identity: 1 M G, Inverses: ϕ 1 G, Restriction: U dom ϕ = ϕ U G, Continuation: dom ϕ = [ U κ and ϕ U κ G = ϕ G, Composition: im ϕ dom ψ = ψ ϕ G. = small category with inverses

26 Pseudo-groups M smooth (analytic) manifold Definition. A pseudo-group is a collection of local diffeomorphisms ϕ : M M such that Identity: 1 M G, Inverses: ϕ 1 G, Restriction: U dom ϕ = ϕ U G, Continuation: dom ϕ = [ U κ and ϕ U κ G = ϕ G, Composition: im ϕ dom ψ = ψ ϕ G. = small category with inverses

27 Lie Pseudo-groups Definition. A Lie pseudo-group G is a pseudo-group whose transformations are the solutions to an involutive system of partial differential equations: F (z, ϕ (n) ) = 0. called the nonlinear determining equations. = analytic (Cartan Kähler) Key complication: abstract object G

28 A Non-Lie Pseudo-group Acting on M = R 2 : X = ϕ(x) Y = ϕ(y) where ϕ D(R) is any local diffeomorphism. Cannot be characterized by a system of partial differential equations (x, y, X (n), Y (n) ) = 0

29 Theorem. (Johnson, Itskov) Any non-lie pseudo-group can be completed to a Lie pseudo-group with the same differential invariants. Completion of previous example: X = ϕ(x), Y = ψ(y) where ϕ, ψ D(R).

30 Infinitesimal Generators g Lie algebra of infinitesimal generators of the pseudo-group G z = (x, u) local coordinates on M Vector field: v = m a = 1 ζ a (z) z a = p i=1 ξ i x i + q ϕ α α=1 u α Vector field jet: j n v ζ (n) = (... ζ b A... ) ζ b A = #A ζ b z A = k ζ b z a 1 z a k

31 The infinitesimal generators of G are the solutions to the Infinitesimal (Linearized) Determining Equations L(z, ζ (n) ) = 0 ( ) Remark: If G is the symmetry group of a system of differential equations (x, u (n) ) = 0, then ( ) is the (involutive completion of) the usual Lie determining equations for the symmetry group.

32 The Diffeomorphism Pseudo-group M smooth m-dimensional manifold D = D(M) pseudo-group of all local diffeomorphisms Z = ϕ(z) z = (z 1,..., z m ) source coordinates Z = (Z 1,..., Z m ) target coordinates

33 Jets Jets are a fancy name for Taylor polynomials/series. For 0 n : Given a smooth map ϕ : M M, written in local coordinates as Z = ϕ(z), let j n ϕ z denote its n jet at z M, i.e., its n th order Taylor polynomial or series based at z. J n (M, M) is the n th order jet bundle, whose points are the jets. Local coordinates on J n (M, M): (z, Z (n) ) = (... z a... ZA b... ), Zb A = k Z b z a 1 z a k

34 Diffeomorphism Jets The n th order diffeomorphism jet bundle is the subbundle D (n) = D (n) (M) J n (M, M) consisting of n th order jets of local diffeomorphisms ϕ : M M. The Inverse Function Theorem tells us it is defined by the non-vanishing of the Jacobian determinant: det( Z a b ) = det( Za / z b ) 0 A Lie pseudo-group G D defines the subbundle G (n) = {F (z, Z (n) ) = 0} D (n) consisting of the jets of pseudo-group diffeomorphisms, and therefore characterized by the pseudo-group s nonlinear determining equations.

35 G (n) = {F (z, Z (n) ) = 0} D (n) Local coordinates on G (n), e.g., the restricted diffeomorphism jet coordinates z c, ZB a, are viewed as the pseudogroup parameters, playing the same role as the local coordinates on a Lie group G. The pseudo-group jet bundle G (n) does not form a group, but rather a groupoid.

36 Double fibration: Groupoid Structure σ (n) M G (n) τ (n) M σ (n) (z, Z (n) ) = z source map τ (n) (z, Z (n) ) = Z target map You are only allowed to multiply h (n) g (n) if σ (n) (h (n) ) = τ (n) (g (n) ) Composition of Taylor polynomials/series is well-defined only when the source of the second matches the target of the first.

37 One-dimensional case: M = R Source coordinate: x Target coordinate: X Local coordinates on D (n) (R) g (n) = (x, X, X x, X xx, X xxx,..., X n ) Jet: X [ h ] = X + X x h X xx h X xxx h3 + = Taylor polynomial/series at a source point x

38 Groupoid multiplication of diffeomorphism jets: (X, X, X X, X XX,... ) (x, X, X x, X xx,... ) = (x, X, X X X x, X X X xx + X XX X 2 x,... ) = Composition of Taylor polynomials/series The higher order terms are expressed in terms of Bell polynomials according to the general Fàa di Bruno formula. The groupoid multiplication (or Taylor composition) is only defined when the source coordinate X of the first multiplicand matches the target coordinate X of the second.

39 Structure of Lie Pseudo-groups The structure of a finite-dimensional Lie group G is specified by its Maurer Cartan forms a basis µ 1,..., µ r for the right-invariant one-forms: dµ k = i<j C k ij µi µ j

40 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a pseudo-group can be identified with the right-invariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

41 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a pseudo-group can be identified with the right-invariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

42 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a pseudo-group can be identified with the right-invariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

43 What should be the Maurer Cartan forms of a Lie pseudo-group? Cartan: Use exterior differential systems and prolongation to determine the structure equations. I propose a direct approach based on the following observation: The Maurer Cartan forms for a pseudo-group can be identified with the right-invariant one-forms on the jet groupoid G ( ). The structure equations can be determined immediately from the infinitesimal determining equations.

44 The Variational Bicomplex The differential one-forms on an infinite jet bundle split into two types: horizontal forms contact forms Definition. A contact form θ is a differential form that vanishes on all jets: θ j n ϕ = 0 for all local diffeomorphisms ϕ D.

45 For the diffeomorphism jet bundle Local coordinates: D ( ) J (M, M) z 1,... z m, Z 1,... Z m,... ZA b,... }{{}}{{}}{{} source target jet Horizontal forms: dz 1,..., dz m Basis contact forms: Θ b A = d G Zb A = dzb A m a = 1 Z a A,a dza

46 One-dimensional case: M = R Local coordinates on D ( ) (R) (x, X, X x, X xx, X xxx,..., X n,... ) Horizontal form: Contact forms: dx Θ = dx X x dx Θ x = dx x X xx dx Θ xx = dx xx X xxx dx. the contact forms vanish when X = ϕ(x)

47 Infinite jet space Local coordinates The Variational Bicomplex J = = Vinogradov, Tsujishita, I. Anderson lim n Jn z ( ) = (x, u ( ) ) = (... x i... u α J... ) Horizontal one-forms dx 1,..., dx p Contact (vertical) one-forms θj α = duα J p i = 1 u α J,i dxi

48 Bigrading of the differential forms on J : Ω = M r = # of dx i Ω r,s r,s s = # of θj α Vertical and Horizontal Differentials d = d H + d V

49 The Variational Bicomplex d H : Ω r,s Ω r+1,s d V : Ω r,s Ω r,s+1 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 π : Ω p,k F k = Ω p,k / d H [ Ω p 1,k ] integration by parts

50 . 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 Ω 0,1 d H Ω 1,1 d H d V d V R Ω 0,0 d H Ω 1,0 d H 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 d H Ω p 1,1 d H Ω p,1 π F 1 d V d V d H Ω p 1,0 d H Ω p,0 E conservation laws Lagrangians PDEs (Euler Lagrange) Helmholtz conditions

51 The Simplest Example. M = R 2 x, u R Horizontal form dx Contact (vertical) forms θ = du u x dx θ x = D x θ = du x u xx dx θ xx = Dx 2 θ = du xx u xxx dx.

52 Differential F = F (x, u, u x, u xx,... ) df = F x dx + F u du + F u x du x + F u xx du xx + Total derivative = (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 +

53 Lagrangian form: λ = L(x, u (n) ) dx Ω 1,0 Vertical derivative variation: dλ = d V λ = d V L dx ( L = u θ + L θ u x + L θ x u xx + xx Integration by parts compute modulo im d H : dλ δλ = ( L u D x = E(L) θ dx L u x + D 2 x ) dx Ω 1,1 ) L θ dx u xx = Euler-Lagrange source form.

54 Maurer Cartan Forms The Maurer Cartan forms for the diffeomorphism pseudo-group are the right-invariant one-forms on the diffeomorphism jet groupoid D ( ). Key observation: The target coordinate functions Z a are right-invariant. Thus, when we decompose dz a = σ a + µ a horizontal contact the two constituents are also right-invariant.

55 µ b A = DA Z µb = D Z a 1 D Z anθ b b = 1,..., m, #A 0 Invariant horizontal forms: σ a = d M Z a = m b = 1 Z a b dzb Invariant total differentiation (dual operators): D Z a = m b = 1 ( Z a b ) 1 D z b Invariant contact forms: µ b = d G Z b = Θ b = dz b m a = 1 Z b a dza

56 One-dimensional case: M = R Contact forms: Θ = dx X x dx Θ x = D x Θ = dx x X xx dx Θ xx = D 2 x Θ = dx xx X xxx dx Right-invariant horizontal form: σ = d M X = X x dx Invariant differentiation: D X = 1 X x D x

57 Invariant contact forms: µ = Θ = dx X x dx µ X = D X µ = Θ x X x = dx x X xx dx X x µ XX = D 2 X µ = X x Θ xx X xx Θ x X 3 x µ n = D n X µ = X x dx xx X xx dx x + (X2 xx X x X xxx ) dx X 3 x.

58 The Structure Equations for the Diffeomorphism Pseudo group Maurer Cartan series: µ b [ H ] = A 1 A! µb A HA H = (H 1,..., H m ) formal parameters dµ[ H ] = µ[ H ] ( µ[ H ] dz ) dσ = dµ[ 0 ] = µ[ 0 ] σ

59 Structure equations: One-dimensional case: M = R dσ = µ X σ dµ[ H ] = dµ dh [ H ] (µ[ H ] dz) where σ = X x dx = dx µ µ[ H ] = µ + µ X H µ XX H2 + µ[ H ] dz = σ + µ X H µ XX H2 + dµ[ H ] dh = µ X + µ XX H µ XXX H2 +

60 In components: dσ = µ 1 σ dµ n = µ n+1 σ + = σ µ n+1 n 1 i = 0 [ n+1 2 ] j = 1 ( ) n i n 2j + 1 n + 1 µ i+1 µ n i ( n + 1 j ) µ j µ n+1 j. = Cartan

61 The Maurer Cartan Forms for a Lie Pseudo-group The Maurer Cartan forms for G are obtained by restricting the diffeomorphism Maurer Cartan forms σ a, µ b A to G( ) D ( ). The resulting one-forms are no longer linearly independent.

62 Theorem. The Maurer Cartan forms on G ( ) satisfy the invariant infinitesimal determining equations L(... Z a... µ b A... ) = 0 ( ) obtained from the infinitesimal determining equations by replacing L(... z a... ζ b A... ) = 0 ( ) source variables z a by target variables Z a derivatives of vector field coefficients ζ b A by right-invariant Maurer Cartan forms µ b A

63 The Structure Equations for a Lie Pseudo-group Theorem. The structure equations for the pseudo-group G are obtained by restricting the universal diffeomorphism structure equations dµ[ H ] = µ[ H ] ( µ[ H ] dz ) to the solution space of the linearized involutive system L(... Z a,... µ b A,... ) = 0.

64 The Korteweg devries Equation u t + u xxx + uu x = 0 Diffeomorphism Maurer Cartan forms: µ t, µ x, µ u, µ t T, µt X, µt U, µx T,..., µu U, µt T T, µt T X,...

65 Maurer Cartan determining equations: µ t X = µt U = µx U = µu T = µu X = 0, µ u = µ x T 2 3 Uµt T, µu U = 2 3 µt T = 2 µx X, µ t T T = µt T X = µt XX = = µu UU =... = 0. Basis (dim G KdV = 4): µ 1 = µ t, µ 2 = µ x, µ 3 = µ u, µ 4 = µ t T.

66 Structure equations: dµ 1 = µ 1 µ 4, dµ 2 = µ 1 µ U µ1 µ µ2 µ 4, dµ 3 = 2 3 µ3 µ 4, dµ 4 = 0. dµ i = C i jk µj µ k

67 The structure equations are on the principal bundle G ( ) ; if G is a finite-dimensional Lie group, then G ( ) M G, and the usual Lie group structure equations are found by restriction to the target fibers {Z = c} G.

68 Lie Kumpera Example Linearized determining system X = f(x) U = u f (x) ξ x = ϕ u ξ u = 0 ϕ u = ϕ u

69 Maurer Cartan forms: σ = u U dx = f x dx, τ = U x dx + U u du = u f xx dx + f x du f x 2 µ = dx U u dx = df f x dx, ν = du U x dx U u du = u f x 2 ( df x f xx dx ) µ X = du u du U x dx U = df x f xx dx f x, µ U = 0 ν X = U u (du x U xx dx) U x u (du U x dx) ν U = du u + du U x dx U = u f x 3 ( df xx f xxx dx ) + u f xx f x 4 ( df x f xx dx ) = df x f xx dx f x

70 Right-invariant linearized system: µ X = ν U µ U = 0 ν U = ν U First order structure equations: dµ = dσ = ν σ U, dν = ν X σ ν τ U dν X = ν XX σ ν X (τ + 2 ν) U

71 Action of Pseudo-groups on Submanifolds a.k.a. Solutions of Differential Equations G Lie pseudo-group acting on p-dimensional submanifolds: N = {u = f(x)} M For example, G may be the symmetry group of a system of differential equations (x, u (n) ) = 0 and the submanifolds the graphs of solutions u = f(x).

72 Prolongation J n = J n (M, p) n th order submanifold jet bundle Local coordinates : z (n) = (x, u (n) ) = (... x i... u α J... ) Prolonged action of G (n) on submanifolds: (x, u (n) ) (X, Û (n) ) Coordinate formulae: Û α J = F J α (x, u(n), g (n) ) = Implicit differentiation.

73 Differential Invariants A differential invariant is an invariant function I : J n R for the prolonged pseudo-group action Invariant differential operators: I(g (n) (x, u (n) )) = I(x, u (n) ) = curvature, torsion,... D 1,..., D p = arc length derivative I(G) the algebra of differential invariants

74 The Basis Theorem Theorem. The differential invariant algebra I(G) is locally generated by a finite number of differential invariants I 1,..., I l and p = dim S invariant differential operators D 1,..., D p meaning that every differential invariant can be locally expressed as a function of the generating invariants and their invariant derivatives: D J I κ = D j1 D j2 D jn I κ. = Lie groups: Lie, Ovsiannikov = Lie pseudo-groups: Tresse, Kumpera, Pohjanpelto O

75 Key Issues Minimal basis of generating invariants: I 1,..., I l Commutation formulae for the invariant differential operators: [ D j, D k ] = p i=1 Syzygies (functional relations) among Y i jk D i = Non-commutative differential algebra the differentiated invariants: Φ(... D J I κ... ) 0 = Codazzi relations

76 Examples of Differential Invariants Euclidean Group on R 3 G = SE(3) = SO(3) R 3 = group of rigid motions z R z + b R SO(3) Induced action on curves and surfaces.

77 Euclidean Curves C R 3 κ curvature: order = 2 τ torsion: order = 3 κ s, τ s, κ ss,... derivatives w.r.t. arc length ds Theorem. Every Euclidean differential invariant of a space curve C R 3 can be written I = H(κ, τ, κ s, τ s, κ ss,... ) Thus, κ and τ generate the differential invariants of space curves under the Euclidean group.

78 Euclidean Curves C R 3 κ curvature: order = 2 τ torsion: order = 3 κ s, τ s, κ ss,... derivatives w.r.t. arc length ds Theorem. Every Euclidean differential invariant of a space curve C R 3 can be written I = H(κ, τ, κ s, τ s, κ ss,... ) Thus, κ and τ generate the differential invariants of space curves under the Euclidean group.

79 Euclidean Curves C R 3 κ curvature: order = 2 τ torsion: order = 3 κ s, τ s, κ ss,... derivatives w.r.t. arc length ds Theorem. Every Euclidean differential invariant of a space curve C R 3 can be written I = H(κ, τ, κ s, τ s, κ ss,... ) Thus, κ and τ generate the differential invariants of space curves under the Euclidean group.

80 Euclidean Curves C R 3 κ curvature: order = 2 τ torsion: order = 3 κ s, τ s, κ ss,... derivatives w.r.t. arc length ds Theorem. Every Euclidean differential invariant of a space curve C R 3 can be written I = F (κ, τ, κ s, τ s, κ ss,... ) Thus, κ and τ generate the differential invariants of space curves under the Euclidean group.

81 Euclidean Surfaces S R 3 H = 1 2 (κ 1 + κ 2 ) mean curvature: order = 2 K = κ 1 κ 2 Gauss curvature: order = 2 D 1 H, D 2 H, D 1 K, D 2 K, D1 2 H,... derivatives with respect to the equivariant Frenet frame on S Theorem. Every Euclidean differential invariant of a non-umbilic surface S R 3 can be written I = F (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Thus, H, K generate the differential invariants of (generic) Euclidean surfaces.

82 Euclidean Surfaces S R 3 H = 1 2 (κ 1 + κ 2 ) mean curvature: order = 2 K = κ 1 κ 2 Gauss curvature: order = 2 D 1 H, D 2 H, D 1 K, D 2 K, D1 2 H,... derivatives with respect to the equivariant Frenet frame on S Theorem. Every Euclidean differential invariant of a non-umbilic surface S R 3 can be written I = F (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Thus, H, K generate the differential invariants of (generic) Euclidean surfaces.

83 Euclidean Surfaces S R 3 H = 1 2 (κ 1 + κ 2 ) mean curvature: order = 2 K = κ 1 κ 2 Gauss curvature: order = 2 D 1 H, D 2 H, D 1 K, D 2 K, D1 2 H,... derivatives with respect to the equivariant Frenet frame on S Theorem. Every Euclidean differential invariant of a non-umbilic surface S R 3 can be written I = F (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Thus, H, K generate the differential invariants of (generic) Euclidean surfaces.

84 Euclidean Surfaces S R 3 H = 1 2 (κ 1 + κ 2 ) mean curvature: order = 2 K = κ 1 κ 2 Gauss curvature: order = 2 D 1 H, D 2 H, D 1 K, D 2 K, D1 2 H,... derivatives with respect to the equivariant Frenet frame on S Theorem. Every Euclidean differential invariant of a non-umbilic surface S R 3 can be written I = F (H, K, D 1 H, D 2 H, D 1 K, D 2 K, D 2 1 H,... ) Thus, H, K generate the differential invariants of (generic) Euclidean surfaces.

85 Euclidean Surfaces Theorem. The algebra of Euclidean differential invariants for a non-degenerate surface is generated by the mean curvature through invariant differentiation. K = Φ(H, D 1 H, D 2 H,... )

86 Euclidean Surfaces Theorem. The algebra of Euclidean differential invariants for a non-degenerate surface is generated by the mean curvature through invariant differentiation. K = Φ(H, D 1 H, D 2 H,... )

87 Applications of Differential Invariants Every (regular) G-invariant system of differential equations can be expressed in terms of the differential invariants: F (... D J I κ... ) = 0 Every G-invariant variational problem can be expressed in terms of the differential invariants and an invariant volume form: I[u] = L(... D J I κ... ) Ω Question: How to go directly from the differential invariant form of the variational problem to the differential invariant form of the Euler Lagrange equations? (See Kogan O.)

88 Characterization of moduli spaces Integration of invariant ordinary differential equations. Symmetry reduction and group splitting (Vessiot s method) for finding explicit solutions to partial differential equations. Equivalence and symmetry of solutions/submanifolds differential invariant signatures. Image processing. Design of symmetry-preserving numerical algorithms.

89 Computing Differential Invariants The infinitesimal method: v(i) = 0 for every infinitesimal generator v g = Requires solving differential equations. Moving frames. (Cartan; PJO Fels Pohjanpelto ) Completely algebraic. Can be adapted to arbitrary group and pseudo-group actions. Describes the complete structure of the differential invariant algebra I(G) using only linear algebra & differentiation! Prescribes differential invariant signatures for equivalence and symmetry detection.

90 Moving Frames for Pseudo Groups In the finite-dimensional Lie group case, a moving frame is defined as an equivariant map ρ (n) : J n G = All classical moving frames can be thus interpreted.

91 However, we do not have an appropriate abstract object to represent our pseudo-group G. Consequently, the moving frame will be an equivariant section ρ (n) : J n H (n) of the pulled-back pseudo-group jet groupoid: G (n) H (n) M J n.

92 Moving Frames for Pseudo Groups Definition. A (right) moving frame of order n is a rightequivariant section ρ (n) : V n H (n) defined on an open subset V n J n. = Groupoid action. Proposition. A moving frame of order n exists if and only if G (n) acts freely and regularly.

93 Freeness For Lie group actions, freeness means no isotropy. For infinite-dimensional pseudo-groups, this definition cannot work, and one must restrict to the transformation jets of order n, using the n th order isotropy subgroup: Definition. G (n) = { g (n) G (n) z (n) z g (n) z (n) = z (n) } At a jet z (n) J n, the pseudo-group G acts freely if G (n) z (n) = { 1 (n) z } locally freely if G (n) z (n) is a discrete subgroup of G (n) z the orbits have dim = r n = dim G (n) z

94 Freeness Theorem Theorem. If n 1 and G (n) acts locally freely at z (n) J n, then it acts locally freely at any z (k) J k with π n k(z(k) ) = z (n) for all k > n.

95 The Normalization Algorithm To construct a moving frame : I. Compute the prolonged pseudo-group action u α K U α K = F α K (x, u(n), g (n) ) by implicit differentiation. II. Choose a cross-section to the pseudo-group orbits: u α κ J κ = c κ, κ = 1,..., r n = fiber dim G (n)

96 III. Solve the normalization equations U α κ J κ = F α κ J κ (x, u (n), g (n) ) = c κ for the n th order pseudo-group parameters g (n) = ρ (n) (x, u (n) ) IV. Substitute the moving frame formulas into the unnormalized jet coordinates u α K = F K α(x, u(n), g (n) ). The resulting functions form a complete system of n th order differential invariants IK α (x, u(n) ) = FK α (x, u(n), ρ (n) (x, u (n) ))

97 Invariantization A moving frame induces an invariantization process, denoted ι, that projects functions to invariants, differential operators to invariant differential operators; differential forms to invariant differential forms, etc. Geometrically, the invariantization of an object is the unique invariant version that has the same cross-section values. Algebraically, invariantization amounts to replacing the group parameters in the transformed object by their moving frame formulas.

98 Invariantization In particular, invariantization of the jet coordinates leads to a complete system of functionally independent differential invariants: ι(x i ) = H i Phantom differential invariants: ι(u α J ) = Iα J I α κ J κ = c κ The non-constant invariants form a functionally independent generating set for the differential invariant algebra I(G)

99 Replacement Theorem I(... x i... u α J... ) = ι( I(... xi... u α J... ) ) = I(... H i... I α J... ) Differential forms = invariant differential forms ι( dx i ) = ω i i = 1,..., p Differential operators = invariant differential operators ι ( D x i ) = D i i = 1,..., p

100 Recurrence Formulae Invariantization and differentiation do not commute The recurrence formulae connect the differentiated invariants with their invariantized counterparts: D i I α J = Iα J,i + M α J,i = M α J,i correction terms

101 Once established, the recurrence formulae completely prescribe the structure of the differential invariant algebra I(G) thanks to the functional independence of the non-phantom normalized differential invariants. The recurrence formulae can be explicitly determined using only the infinitesimal generators and linear differential algebra!

102 Korteweg devries Equation Prolonged Symmetry Group Action: T = e 3λ 4 (t + λ 1 ) X = e λ 4 (λ 3 t + x + λ 1 λ 3 + λ 2 ) U = e 2λ 4 (u + λ 3 ) U T = e 5λ 4 (u t λ 3 u x ) U X = e 3λ 4 u x U T T = e 8λ 4 (u tt 2λ 3 u tx + λ 3 2 u xx ) U T X = D X D T U = e 6λ 4 (u tx λ 3 u xx ) U XX = e 4λ 4 u xx.

103 Cross Section: T = e 3λ 4 (t + λ 1 ) = 0 X = e λ 4 (λ 3 t + x + λ 1 λ 3 + λ 2 ) = 0 U = e 2λ 4 (u + λ 3 ) = 0 U T = e 5λ 4 (u t λ 3 u x ) = 1 Moving Frame: λ 1 = t, λ 2 = x, λ 3 = u, λ 4 = 1 5 log(u t + uu x )

104 Moving Frame: λ 1 = t, λ 2 = x, λ 3 = u, λ 4 = 1 5 log(u t + uu x ) Invariantization: ι(u K ) = U K λ1 = t,λ 2 = x,λ 3 = u,λ 4 =log(u t +uu x )/5 Phantom Invariants: H 1 = ι(t) = 0 H 2 = ι(x) = 0 I 00 = ι(u) = 0 I 10 = ι(u t ) = 1

105 Normalized differential invariants: u I 01 = ι(u x ) = x (u t + uu x ) 3/5 I 20 = ι(u tt ) = u tt + 2uu tx + u2 u xx (u t + uu x ) 8/5 I 11 = ι(u tx ) = I 02 = ι(u xx ) = u tx + uu xx (u t + uu x ) 6/5 u xx (u t + uu x ) 4/5 I 03 = ι(u xxx ) =. u xxx u t + uu x

106 Invariantization: ι( F (t, x, u, u t, u x, u tt, u tx, u xx,... ) ) = F (ι(t), ι(x), ι(u), ι(u t ), ι(u x ), ι(u tt ), ι(u tx ), ι(u xx ),... ) = F (H 1, H 2, I 00, I 10, I 01, I 20, I 11, I 02,... ) = F (0, 0, 0, 1, I 01, I 20, I 11, I 02,... ) Replacement Theorem: 0 = ι(u t + u u x + u xxx ) = 1 + I 03 = u t + uu x + u xxx u t + uu x. Invariant horizontal one-forms: ω 1 = ι(dt) = (u t + uu x ) 3/5 dt, ω 2 = ι(dx) = u(u t + uu x ) 1/5 dt + (u t + uu x ) 1/5 dx.

107 Invariant differential operators: D 1 = ι(d t ) = (u t + uu x ) 3/5 D t + u(u t + uu x ) 3/5 D x, D 2 = ι(d x ) = (u t + uu x ) 1/5 D x. Commutation formula: [ D 1, D 2 ] = I 01 D 1 Recurrence formulae: D 1 I 01 = I I I 01 I 20, D 2 I 01 = I I I 01 I 11, D 1 I 20 = I I I 01 I I2 20, D 2 I 20 = I I 01 I I2 01 I I 11 I 20, D 1 I 11 = I 21 + I I 01 I I 11 I 20, D 2 I 11 = I 12 + I 01 I I2 01 I I2 11, D 1 I 02 = I I 01 I I 02 I 20, D 2 I 02 = I I2 01 I I 02 I 11,..

108 Generating differential invariants: I 01 = ι(u x ) = Fundamental syzygy: u x (u t + uu x ) 3/5, I 20 = ι(u tt ) = u tt + 2uu tx + u2 u xx (u t + uu x ) 8/5. D 2 1 I I 01 D 1 I 20 D 2 I 20 + ( 1 5 I I 01) D1 I 01 D 2 I I 01 I I2 01 I I3 01 = 0.

109 Horizontal coframe Lie Tresse Kumpera Example X = f(x), Y = y, U = u f (x) d H X = f x dx, d H Y = dy, Implicit differentiations D X = 1 f x D x, D Y = D y.

110 Prolonged pseudo-group transformations on surfaces S R 3 X = f Y = y U = u f x U X = u x f 2 x U XX = u xx f 3 x u f xx f 3 x 3u x f xx f 4 x U Y = u y f x u f xxx f 4 x + 3u f 2 xx f 5 x U XY = u xy f 2 x Coordinate cross-section u y f xx f 3 x U Y Y = u yy f x = action is free at every order. X = f = 0, U = u f x = 1, U X = u x f 2 x u f xx f 3 x = 0, U XX = = 0.

111 Moving frame f = 0, f x = u, f xx = u x, f xxx = u xx. Differential invariants U Y J = u y u U XY J 1 = uu xy u x u y u 3 U Y Y J 2 = u yy u Invariant horizontal forms d H X = f x dx u dx, d H Y = dy dy, Invariant differentiations D 1 = 1 u D x D 2 = D y

112 Higher order differential invariants: D m 1 Dn 2 J Recurrence formulae: J,1 = D 1 J = uu xy u x u y u 3 = J 1, J,2 = D 2 J = uu yy u2 y u 2 = J 2 J 2. D 1 J = J 1, D 2 J = J 2 J 2, D 1 J 1 = J 3, D 2 J 1 = J 4 3 J J 1, D 1 J 2 = J 4, D 2 J 2 = J 5 J J 2,

113 The Master Recurrence Formula p d H IJ α = ( D i IJ α ) ωi = i=1 p i=1 I α J,i ωi + ψ α J where ψ α J = ι( ϕ α J ) = Φα J (... Hi... IJ α... ;... γb A... ) are the invariantized prolonged vector field coefficients, which are particular linear combinations of γa b = ι(ζb A ) invariantized Maurer Cartan forms prescribed by the invariantized prolongation map. The invariantized Maurer Cartan forms are subject to the invariantized determining equations: L(H 1,..., H p, I 1,..., I q,..., γ b A,... ) = 0

114 d H I α J = p i=1 I α J,i ωi + ψ α J (... γb A... ) Step 1: Solve the phantom recurrence formulas 0 = d H I α J = p i=1 I α J,i ωi + ψ α J (... γb A... ) for the invariantized Maurer Cartan forms: γ b A = p i=1 J b A,i ωi ( ) Step 2: Substitute ( ) into the non-phantom recurrence formulae to obtain the explicit correction terms.

115 Only uses linear differential algebra based on the specification of cross-section. Does not require explicit formulas for the moving frame, the differential invariants, the invariant differential operators, or even the Maurer Cartan forms!

116 The Korteweg devries Equation (continued) Recurrence formula: di jk = I j+1,k ω 1 + I j,k+1 ω 2 + ι(ϕ jk ) Invariantized Maurer Cartan forms: ι(τ) = λ, ι(ξ) = µ, ι(ϕ) = ψ = ν, ι(τ t ) = ψ t = λ t,... Invariantized determining equations: λ x = λ u = µ u = ν t = ν x = 0 ν = µ t ν u = 2 µ x = 2 3 λ t λ tt = λ tx = λ xx = = ν uu = = 0 Invariantizations of prolonged vector field coefficients: ι(τ) = λ, ι(ξ) = µ, ι(ϕ) = ν, ι(ϕ t ) = I 01 ν 5 3 λ t, ι(ϕ x ) = I 01 λ t, ι(ϕ tt ) = 2I 11 ν 8 3 I 20 λ t,...

117 Phantom recurrence formulae: 0 = d H H 1 = ω 1 + λ, 0 = d H H 2 = ω 2 + µ, 0 = d H I 00 = I 10 ω 1 + I 01 ω 2 + ψ = ω 1 + I 01 ω 2 + ν, 0 = d H I 10 = I 20 ω 1 + I 11 ω 2 + ψ t = I 20 ω 1 + I 11 ω 2 I 01 ν 5 3 λ t, = Solve for λ = ω 1, µ = ω 2, ν = ω 1 I 01 ω 2, λ t = 3 5 (I 20 + I 01 )ω (I 11 + I2 01 )ω2. Non-phantom recurrence formulae: d H I 01 = I 11 ω 1 + I 02 ω 2 I 01 λ t, d H I 20 = I 30 ω 1 + I 21 ω 2 2I 11 ν 8 3 I 20 λ t, d H I 11 = I 21 ω 1 + I 12 ω 2 I 02 ν 2I 11 λ t, d H I 02 = I 12 ω 1 + I 03 ω I 02 λ t,.

118 D 1 I 01 = I I I 01 I 20, D 2 I 01 = I I I 01 I 11, D 1 I 20 = I I I 01 I I2 20, D 2 I 20 = I I 01 I I2 01 I I 11 I 20, D 1 I 11 = I 21 + I I 01 I I 11 I 20, D 2 I 11 = I 12 + I 01 I I2 01 I I2 11, D 1 I 02 = I I 01 I I 02 I 20, D 2 I 02 = I I2 01 I I 02 I 11,..

119 Lie Tresse Kumpera Example (continued) X = f(x), Y = y, U = u f (x) Phantom recurrence formulae: 0 = dh = ϖ 1 + γ, 0 = di 10 = J 1 ϖ 2 + ϑ 1 γ 2, 0 = di 00 = J ϖ 2 + ϑ γ 1, 0 = di 20 = J 3 ϖ 2 + ϑ 3 γ 3, Solve for pulled-back Maurer Cartan forms: γ = ϖ 1, γ 2 = J 1 ϖ 2 + ϑ 1, Recurrence formulae: dy = ϖ 2 γ 1 = J ϖ 2 + ϑ, γ 3 = J 3 ϖ 2 + ϑ 3, dj = J 1 ϖ 1 + (J 2 J 2 ) ϖ 2 + ϑ 2 J ϑ, dj 1 = J 3 ϖ 1 + (J 4 3 J J 1 ) ϖ 2 + ϑ 4 J ϑ 1 J 1 ϑ, dj 2 = J 4 ϖ 1 + (J 5 J J 2 ) ϖ 2 + ϑ 5 J 2 ϑ,