Invariant Calculus with Vessiot

Transcription

1 Invariant Calculus with Vessiot Irina Kogan (Yale University) August 13, 2002 Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex, I. Kogan, P.J.Olver. Maple package Vessiot for calculus on the jet bundles. Utah State University, I.Anderson et al. 1

2 Group action on jet bundles G M G {p-dimensional submanifolds of M} G J k (M, p) Jet spaces: J k = J k (M, p) are bundles over M. The fiber over z M consists of the equivalence classes of p-dim. submanifolds with k-th order contact at z. Local coordinates on J k : x = (x 1,..., x p ) independent variables, u = (u 1,... u q ) dependent variables, u (k) = (u α J, α = 1... q, J = multi-index, J k) derivatives J = (j 1,..., j p ) 0 j i, J = j j p, (1) J = (j 1,..., j k ) S, 1 j i p, J = k (2) 2

3 Projections: π n k : J n J k for k n: J J k J k 1... J 1 J 0 = M Jets of submanifolds: If N : u α = f α (x) then j k (N) : u α = f α (x), u α J = k f α j 1 x 1... j p x p. Prolongation of the action: g j k (N) = j k (g N). 3

4 Contact structure on J (M, p) Basis of horizontal sub-bundle total derivatives: d dx i = x i + q α=1 uα i + α,j uα Ji u α J horizontal one-forms d x 1,..., d x p u α Tangent Cotangent Basis of vertical sub-bundle vertical derivatives u α J u α,, α = 1... q contact one-forms θ α = du α p i=1 uα i dxi, θ α J = duα J p i=1 uα Ji dxi. 4

5 Bigrading of exterior differential algebra: Grading: Λ = Λ k, where Λ k = one form one form }{{}. k times d: Λ k Λ k+1, d d = 0 = de Rham complex. Bigrading: Λ = Λ s,t, where Λ s,t = hor. 1-form hor. 1-form }{{} cont. } 1-form {{ cont. 1-form } s times t times d: Λ s,t Λ s+1,t Λ s,t+1 d = d H + d V d 2 = (d H + d V ) 2 = 0 d 2 H = 0, d 2 V = 0, d H d V = d V d H Bicomplex 5

6 Variational Bicomplex Tulczyjew, Tsujishita, Vinogradov, Anderson d V. d V. d V. d V. d V.. V d V Λ 0,2 d H Λ 1,2 d V d H... d H Λ p 1,2 d V d H Λ p,2 d V I F 2 V d V Λ 0,1 d H Λ 1,1 d V Λ 0,0 d Λ 1,0 H d H d H d H Λ p 1,1 d V Λ p 1,0 d H d H Λ p,1 Λ p,0 d H I : Λ p,s F s = Λ p,s /Im d H - integration by parts operator v = I d V - variational derivative. I F 1 V 6

7 Integration by parts operator, λ Λ p,s : I(λ) = ( 1 s θα λ; u α + α J ( d dx J ) λ, u α J ). λ = L(x, u (n) )dx d V α,j L u α J θ α J dx I α,j Eα (L)θ α dx E α (L) = 0, α = 1,..., q Euler-Lagrange equations. Λ p,0 Lagrangians d H Λ p 1,0 = (loc) = ker V trivial Lagrangians. F 1 source forms V Λ p,0 = (loc) = ker V Euler-Lagrange forms. 7

8 Invariant structure on J (M, p). Invariant horizontal basis invariant total diff. operators Tangent Invariant vertical basis invariant vertical diff. operators D 1,... { } D p { span Di = span } d dx i invariant horizontal one-forms ω 1,..., ω p H = span { ω i} H = span { d x i} unless the action is projectable Cotangent C α, { } C J α span Cα span { } u α unless the action is projectable invariant contact one-forms θ α, θ J α { θα span, θ } J α = span {θ α, θj α} 8

9 A generating set of differential invariants. Thm. (Tresse) {I 1,..., I n } - invariant functions on J (M, p), s. t. any invariant function ( I = F..., D ) J (I l ),... Problems: non-free differential algebra (with syzygies): ( S..., D ) J (I l ),... = 0 Finding a (minimal) set of generating invariants. Finding a (minimal) set of generating syzygies. Finding invariant basis for differential forms. Solution: invariantization via moving frame method (Olver, Fels). 9

10 Euclidean group SE(2) = SO(2) R 2 acts on plane curves u = u(x): x cos(φ)x sin(φ)u + a, u sin(φ)x + cos(φ)u + b. a generating set: κ = u xx, D (1+u = d 2 x )3/2 ds = 1 1+u 2 x d dx, any invariant I = F (κ, κ s, κ ss,... ), no syzygies, ds = 1 + u 2 xd x is not an invariant form: g ds = ds + contact form. invariant horizontal form: ω = ds + u x θ. 1+u 2 x invariant contact forms: θ θ =, θ1 = (1+u2 x )θ x u x u xx θ 1+u 2 (1+u,... 2 x x )2 10

11 SE(2) = SO(2) R 2 acts on surfaces u = u(x, y) in R 3 : x cos(φ)x sin(φ)y + a, y sin(φ)x + cos(φ)y + b, u u a minimal generating set: I = u, ( ) 1 D 1 = d u u 2 y x +u 2 dx u x d dy, D2 = y ( ) d u x dx + u y d dy. 1 u 2 x +u 2 y generating syzygies: D 1 I = 0, I xx = u2 y u xx 2u x u y u xy +u 2 x u yy u 2 x +u2 y 2 Ixx 2 I xx D2 I + 2 D1 D2 I ( D2 I) D2 1 D2 I D 2 I xx = 0, D1 D2 I I xx + D 2 2 I + D2 I D2 D1 D2 D 1 D2 2 I = 0 [ D 1, D 2 ] = I xx D 2 I D 1 + D 1 D2 I D 2 I invariant horizontal forms: D 2, 11

12 1 ω 1 = u 2 x +u 2 y 1 (u y d x u x d y), ω 2 = u 2 x +u 2 y (u x d x + u y d y). 12

13 Thm. (S.Lie) Application: symmetry reduction. (almost) any symmetric system of differential equations can be written in terms of differential invariants. (almost) any symmetric variational problem can be written in terms of differential invariants and invariant differential forms. 13

14 Invariant Variational Problems: for g G: L[u] = L(x, u (n) )dx = Ω Ω [ ] g L(x, u (n) )dx L[u] = L(x, u (n) )dx L[u] = L(..., DJ I l,... ) ω 1 ω p E α = J ( d dx J ) u α J α (x, u (n) ) = 0, α? ( DJ1 I 1,..., D JN I N ) = 0, α = 1,..., q. α = 1,..., N. problem and examples: P. Griffiths, I. Anderson; general formula: P. Olver, I. Kogan; V. Itskov. 14

15 Euclidean invariant Lagrangian: L[u] = u 2 xx 2 (1+u 2 x )5/2 dx 1 2 κ2 ds E = ( ) d 2 dx u xx ( ) d dx u x? = 0 κ ss κ3 = 0 = 1 30 u 3 2 u u u 2 u 1 u 3 20 u 2 u 3 1 u u u 4 u u 4 u 1. 2 (1 + u 12 ) (9/2) 15

16 Invariantization via moving frames. (Fels, Olver) Thm. (Ovsyannikov) The action of G is loc. effective on open subsets n dim G (the order of stabilization), s. t. the prolonged action is loc. free on an open dense V n J n. dim O z (n) = dim G, z (n) V n (local) cross-section K n : codim K n = dim O z (n), K n is transversal to O z (n) for z (n) V n, K n O z (n) consists of at most one point regular action. 16

17 Define ρ : V n G, by the condition ρ(z (n) ) z (n) K n ρ(g z (n) )g z (n) = ρ(z) z (n) ; freeness = ρ(g z (n) ) = ρ(z (n) )g 1 ρ is a G-equivariant map. c.-s. and ρ can be lifted to J k for k > n: 1 ρ(z (k) ) = ρ πn k (z (k) ) G, K k = πn k K n J k, K = (πn ) 1 K n Projection on the c.-s. γ(z) = ρ(z) z: J K for z J. 17

18 Generalized definition of a moving frame. G M; a moving frame is an equivariant smooth map ρ: M G. G ρ R g 1 G ρ M g M Thm. moving frame the action is regular and free. 18

19 Invariantization ι: functions f : J R : ι(f)(z) = f(ρ(z) z) {X i = ι(x i ), IJ α = ι(uα J )} complete set of functionally indep. inv. differential forms: ι(ω) z = [ρ (z)] ( ) Ω ρ(z) z ω 1-form, v vector field: ι(ω) z ; v z = ω ρ(z) z ; [ρ z ] (v z ) θ J α = ι(θα J ), α = 1,..., q - inv. basis of contact 1-forms. ω i = ι(dx i ), i = 1,..., p - inv. basis of horizontal 1-forms. 1-st order inv. diff. operators = inv. vector fields. Basis dual to { ω i, θ J α}. Basis of total inv. operators ω i ; D j = δj i. 19

20 Invariantization steps: 1. Pull back a form (a function) by the action of g G, 2. Replace g with ρ(z). 20

21 Euclidean action on the plane curves: x x = cos(φ)x sin(φ)u + a, u ū = sin(φ)x + cos(φ)u + b u x ū x = sin(φ) + cos(φ)u x cos(φ) sin(φ)u x, u xx ū xx = u xx (cos(φ) sin(φ)u x ) 3 loc. free and transitive on J 1 c.-s. K 1 = {x = 0, u = 0, u x = 0} moving frame is the solution of x = 0, ū = 0, ū x = 0 : φ = arctan(u x ), a = x + u xu u 2 x + 1, b = u xx u u 2 x + 1 g u xx u u xx = ū xx = (cos(φ) sin(φ)u x ) 3 I 2 = κ = xx (1+u 2 x )3/2 g (dx) = cos(φ)dx sin(φ)du ω = dx+u xdu = 1 + u 2 1+u 2 x dx + u x θ. x 1+u 2 x 21

22 Structure equations. d ι (Ω) = ι (d Ω) + r ρ (µ κ ) ι [v κ (Ω)] κ=1 v 1,... v r basis for infinitesimal generators of G-action v κ (Ω) Lie derivative of Ω with respect to v κ. µ 1,... µ r dual basis of invariant differential forms on G Invariant bigrading Λ = Λs,t. Unless the action is projectable Λ s,t Λ s,t and for s 1: d: Λ s,t Λ s+1,t Λ s,t+1 Λ s 1,t+2 d = d H + dṽ + d W d 2 = (d H + dṽ + d W ) 2 = 0 d 2 H = 0, d 2 Ṽ + d Hd W + d W d H = 0, d H dṽ = dṽ d H, d 2 W = 0 22

23 Algorithm for computing structure equations. input: dependent and independent variables [x 1,..., x p ], [u 1,..., u q ], infinitesimal generators V = [v 1,..., v r ], coordinate cross-section K = [z i 1 = c 1,..., z i r = c r ], order of prolongation k. output: dṽ ω i, d H ω i = c i jk ωj ω k, [ D k, D j ] = c i jk D i, dṽ θα J, d H θα J, Di ( θα J ), dṽ X i, dṽ I α J, d HX i, d HI α J, Dj ( X i ), Dj (I α J ). operations: comments: differentiation and linear algebra. explicit formulae for moving frame, invariants and invariant forms are not needed. 23

24 I. Preliminary computations: V k := k-th prolongations of vectors V ; d := dim J k = p + qc p p+k, L := (r d) matrix of the coefficients of V k, L ξ := (r p)-submatrix of L, corresponding to x i, L φ := (r (d p))-submatrix of L, corresponding to u α J L c := r (d p))-matrix of contact forms such that px {L c } α J;κ = v κ (θj α ) = d V φ α J;κ u α Jid V ξκ i L r := (r r)-submatrix of L of columns i 1,..., i r defining K. i=1 if det L r K = 0 then cross-section is invalid; stop. else define the process of invariantization: ι(expression) := subs {dx i = ω i, θj α = θ J α, z i 1 = c 1,..., z i r = c r, for the rest of variables x i = X i, u α J = IJ α }, expression 24

25 matrices of variables and forms: K := [z i 1,..., z i r ](1 r)- is matrix of normalized coordinates. Z := (1 d)- matrix of all coordinates, I := ι(z) is (1 d)-matrix of invariants. ω := [ ω 1,..., ω p ] invariant horizontal forms θ := [θ α J, α = 1,..., q, J k] (1 d)-matrix of contact forms. θ := [ θ α J, α = 1,..., q, J k] = ι (θ)-invariant contact forms. II. Structure equations: d H ω = ι [ (d H K) L 1 r (d H L ξ ) ] dṽ ω = ι [ (d H K) L 1 r d H θ = ι [ dh θ (d H K) L 1 r L c ] dṽ θ = ι [ (dv K) L 1 r L c ] (d V L ξ ) + (d V K) L 1 r (d H L ξ ) ] = D ) i i ( θ ω i 25

26 III. Differentials of fundamental invariants. d H I = ι [ d H Z (d H K) L 1 r L ] = D i i (I) ω i dṽ I = ι [ d V Z (d V K) L 1 r L ] 26

27 SE(2) acting on curves in R 2. > restart;with(vessiot):read(ivbproc);with(linalg): > coord_frame([x],[u],fr1); frame name : fr1 > vecte:=[[1,0],[0,1],[-u[0],x]]; sectione:=[x=0,u[0]=0,u[1]=0]; vecte := [[1, 0], [0, 1], [ u 0, x]] sectione := [x = 0, u 0 = 0, u 1 = 0] > ivb(vecte,sectione,3); 27

28 INVARIANT HORIZONTAL FORMS : invariant vertical differentails [ dv Dx = u2 Dx Cu [0] ] invariant horizontal differentails [0] 28

29 INVARIANT CONTACT FORMS invariant vertical differentials : dv Cu [0] = u 2 Cu [0] Cu [1], dv Cu [1] = u 3 Cu [0] Cu [1], dv Cu [2] = u 4 Cu [0] Cu [1], dv Cu [3] = u 5 Cu [0] Cu [1] 6 u 2 Cu [1] Cu [2] invariant horizontal differentials : dh Cu [0] = u 2 2 Dx Cu [0] + Dx Cu [1], dh Cu [1] = u 3 u 2 Dx Cu [0] + Dx Cu [2], dh Cu [2] = u 4 u 2 Dx Cu [0] 3 u 2 2 Dx Cu [1] + Dx Cu [3] Lie derivatives w.r.t inv. diff. operators : 29

30 D 1 Cu [0] = u 2 2 Cu [0] + Cu [1] D 1 Cu [1] = u 3 u 2 Cu [0] + Cu [2] D 1 Cu [2] = u 4 u 2 Cu [0] 3 u 2 2 Cu [1] + Cu [3] D 1 Cu [3] = u 5 u 2 Cu [0] 4 u 3 u 2 Cu [1] 6 u 2 2 Cu [2] + Cu [4] DIFFERENTIALS OF THE FUNDAMENTAL INVARIANTS vertical dv u 2 = Cu [2] dv u 3 = 3 u 2 2 Cu [1] + Cu [3] horizontal 30

31 dh u 2 = u 3 Dx dh u 3 = ( 3 u u 4 ) Dx Lie derivatives w.r.t inv. diff. operators : D 1 u 2 = u 3 D 1 u 3 = 3 u u 4 31

32 SE(2) acting on independent variables for surfaces in Rˆ3. > coord_frame([x,y],[u],fr1): > vectors:=[[1,0,0],[0,1,0],[-y,x,0]]; section:=[x=0,y=0,u[1,0]=0]; > ivb(vectors,section,2); INVARIANT HORIZONTAL FORMS : invariant vertical differentails [ dv Dx = Dy Cu [1, 0] u 0, 1, dv Dy = Dx Cu [1, 0] u 0, 1 ] invariant horizontal differentails [ dh Dx = u 2, 0 Dx Dy u 0, 1, dh Dy = u 1, 1 Dx Dy u 0, 1 ] 32

33 Lie algebra of invariant differential operators : > invd := [[Lie_alg, InvH, [p]], [seq(invdstr[t],t=1..p*(p^2-p)/2)]]; > Lie_alg_init(invD,[Delta],[omega]):frameInformation(): [ [[ invd := [Lie alg, InvH, [2]], [1, 2, 1], u ] [ 2, 0, [1, 2, 2], u ]]] 1, 1 u 0, 1 u 0, 1 frame name : InvH library name : Koszul frame Frame Jet Variables : [z1, z2 ] Frame Labels [ 1, 2 ] 33

34 CoFrame Labels [ω1, ω2 ] Horizontal Coframe Labels [] Vertical Coframe Labels [] ext d :, ω1, u 2, 0 ω1 ω2 u 0, 1 ext d :, ω2, u 1, 1 ω1 ω2 u 0, 1 34

35 Lie derivatives of invariants w.r.t inv. diff. operators : D 1 u 0, 0 = 0 D 2 u 0, 0 = u 0, 1 D 1 u 0, 1 = u 1, 1 D 2 u 0, 1 = u 0, 2 D 1 u 2, 0 = 2 u 1, 1 u 2, 0 + u 3, 0 D 2 u 2, 0 = 2 u 1, 1 + u 2, 1 u 0, 1 u 0, 1 D 1 u 1, 1 = u 2, 0 2 u 2, 0 u 0, 2 u 0, 1 + u 2, 1 D 2 u 1, 1 = u 1, 1 (u 2, 0 u 0, 2 ) u 0, 1 + u 1, 2 D 1 u 0, 2 = 2 u 1, 1 u 2, 0 + u 1, 2 D 2 u 0, 2 = 2 u 2 1, 1 + u 0, 3 u 0, 1 u 0, 1 syzygies : D 2 u 2, 0 D 1 u 1, 1 = 2 u 1, u 2, 0 2 u 2, 0 u 0, 2 u 0, 1 D 2 u 1, 1 D 1 u 0, 2 = u 1, 1 (u 2, 0 + u 0, 2 ) u 0,

36 Invariant integration by parts for plane curves. G J(R 2, 1), ω invariant differential form, D dual invariant ( i differential operator, κ generating invariant, κ i = D) κ. λ = L (κ, κ 1,..., κ n ) ω dṽ λ = dṽ L ω + L dṽ ω = n i=0 L κ i (dṽ κ i ) ω + L dṽ ω (mod d H) (dṽ κ i ) ω = dṽ Dκi 1 ω = (dṽ d Hκ i 1 ) κ i dṽ ω = (d HdṼ κ i 1 ) κ i dṽ ω = n i=0 d H ( L n i=0 ( D ) κ (dṽ i κ i 1 ) ) ( L ( n i=0 L κ i κ i L ) dṽ ω κ i ) (dṽ κ i 1 ) ω (... ) dṽ ω. repeat! 36

37 ) ) dṽ λ E ( L dṽ κ ω H ( L dṽ ω. ) E ( L = n i=0 ( D ) i L, κ i ( L) H = n i>j 0 κ i j ( D ) j L κ L. i λ = L(x, u, u 1,..., u m ) dx λ = L (κ, κ 1,..., κ n ) ω d V (dx) = 0 dṽ ( ω) = B( θ 0 ) ω d V (u) = θ 0 dṽ (κ) = A( θ 0 ) ) dṽ λ [A E ( L B H ( L)] θ0 ω. 37

38 Euclidean group SE(2) = SO(2) R 2 acts on plane curves u = u(x): x cos(φ)x sin(φ)u + a, u sin(φ)x + cos(φ)u + b. dṽ κ = θ 2 = d ds θ 1 + κ 2 θ0 = [ ( d ) 2 ] ds + κ 2 θ0, dṽ ω = κ θ 0 ω. {[ ( dṽ λ = d ) 2 ) ds + κ 2] E ( L + κh ( L)} θ0 ω 38